Properties

Label 384.6.f.e.191.10
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 191.10
Root \(4.85801 + 4.85801i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.6.f.e.191.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.14396 + 15.5464i) q^{3} +71.6821 q^{5} -100.489i q^{7} +(-240.383 - 35.5689i) q^{9} +O(q^{10})\) \(q+(-1.14396 + 15.5464i) q^{3} +71.6821 q^{5} -100.489i q^{7} +(-240.383 - 35.5689i) q^{9} -148.424i q^{11} -381.250i q^{13} +(-82.0014 + 1114.40i) q^{15} +2205.04i q^{17} -1736.63 q^{19} +(1562.25 + 114.955i) q^{21} +3556.80 q^{23} +2013.33 q^{25} +(827.958 - 3696.40i) q^{27} +5772.58 q^{29} -5121.78i q^{31} +(2307.47 + 169.791i) q^{33} -7203.28i q^{35} +11029.0i q^{37} +(5927.07 + 436.134i) q^{39} +1457.33i q^{41} +18616.8 q^{43} +(-17231.1 - 2549.66i) q^{45} +12455.1 q^{47} +6708.93 q^{49} +(-34280.5 - 2522.47i) q^{51} -4842.48 q^{53} -10639.4i q^{55} +(1986.63 - 26998.4i) q^{57} -18265.7i q^{59} -5459.70i q^{61} +(-3574.29 + 24155.9i) q^{63} -27328.8i q^{65} +22202.1 q^{67} +(-4068.83 + 55295.5i) q^{69} -10310.5 q^{71} +61295.6 q^{73} +(-2303.16 + 31300.0i) q^{75} -14915.0 q^{77} -31323.4i q^{79} +(56518.7 + 17100.3i) q^{81} +46611.6i q^{83} +158062. i q^{85} +(-6603.59 + 89742.9i) q^{87} +99794.6i q^{89} -38311.5 q^{91} +(79625.4 + 5859.11i) q^{93} -124485. q^{95} +115614. q^{97} +(-5279.30 + 35678.6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) 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
\(3\) −1.14396 + 15.5464i −0.0733850 + 0.997304i
\(4\) 0 0
\(5\) 71.6821 1.28229 0.641144 0.767420i \(-0.278460\pi\)
0.641144 + 0.767420i \(0.278460\pi\)
\(6\) 0 0
\(7\) 100.489i 0.775129i −0.921843 0.387565i \(-0.873316\pi\)
0.921843 0.387565i \(-0.126684\pi\)
\(8\) 0 0
\(9\) −240.383 35.5689i −0.989229 0.146374i
\(10\) 0 0
\(11\) 148.424i 0.369848i −0.982753 0.184924i \(-0.940796\pi\)
0.982753 0.184924i \(-0.0592039\pi\)
\(12\) 0 0
\(13\) 381.250i 0.625678i −0.949806 0.312839i \(-0.898720\pi\)
0.949806 0.312839i \(-0.101280\pi\)
\(14\) 0 0
\(15\) −82.0014 + 1114.40i −0.0941008 + 1.27883i
\(16\) 0 0
\(17\) 2205.04i 1.85052i 0.379334 + 0.925260i \(0.376153\pi\)
−0.379334 + 0.925260i \(0.623847\pi\)
\(18\) 0 0
\(19\) −1736.63 −1.10363 −0.551815 0.833967i \(-0.686064\pi\)
−0.551815 + 0.833967i \(0.686064\pi\)
\(20\) 0 0
\(21\) 1562.25 + 114.955i 0.773039 + 0.0568829i
\(22\) 0 0
\(23\) 3556.80 1.40197 0.700987 0.713174i \(-0.252743\pi\)
0.700987 + 0.713174i \(0.252743\pi\)
\(24\) 0 0
\(25\) 2013.33 0.644264
\(26\) 0 0
\(27\) 827.958 3696.40i 0.218574 0.975820i
\(28\) 0 0
\(29\) 5772.58 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(30\) 0 0
\(31\) 5121.78i 0.957231i −0.878025 0.478616i \(-0.841139\pi\)
0.878025 0.478616i \(-0.158861\pi\)
\(32\) 0 0
\(33\) 2307.47 + 169.791i 0.368851 + 0.0271413i
\(34\) 0 0
\(35\) 7203.28i 0.993939i
\(36\) 0 0
\(37\) 11029.0i 1.32444i 0.749312 + 0.662218i \(0.230385\pi\)
−0.749312 + 0.662218i \(0.769615\pi\)
\(38\) 0 0
\(39\) 5927.07 + 436.134i 0.623991 + 0.0459154i
\(40\) 0 0
\(41\) 1457.33i 0.135393i 0.997706 + 0.0676967i \(0.0215650\pi\)
−0.997706 + 0.0676967i \(0.978435\pi\)
\(42\) 0 0
\(43\) 18616.8 1.53545 0.767723 0.640782i \(-0.221390\pi\)
0.767723 + 0.640782i \(0.221390\pi\)
\(44\) 0 0
\(45\) −17231.1 2549.66i −1.26848 0.187694i
\(46\) 0 0
\(47\) 12455.1 0.822434 0.411217 0.911537i \(-0.365104\pi\)
0.411217 + 0.911537i \(0.365104\pi\)
\(48\) 0 0
\(49\) 6708.93 0.399175
\(50\) 0 0
\(51\) −34280.5 2522.47i −1.84553 0.135800i
\(52\) 0 0
\(53\) −4842.48 −0.236798 −0.118399 0.992966i \(-0.537776\pi\)
−0.118399 + 0.992966i \(0.537776\pi\)
\(54\) 0 0
\(55\) 10639.4i 0.474252i
\(56\) 0 0
\(57\) 1986.63 26998.4i 0.0809898 1.10065i
\(58\) 0 0
\(59\) 18265.7i 0.683134i −0.939857 0.341567i \(-0.889042\pi\)
0.939857 0.341567i \(-0.110958\pi\)
\(60\) 0 0
\(61\) 5459.70i 0.187864i −0.995579 0.0939322i \(-0.970056\pi\)
0.995579 0.0939322i \(-0.0299437\pi\)
\(62\) 0 0
\(63\) −3574.29 + 24155.9i −0.113459 + 0.766781i
\(64\) 0 0
\(65\) 27328.8i 0.802300i
\(66\) 0 0
\(67\) 22202.1 0.604237 0.302118 0.953270i \(-0.402306\pi\)
0.302118 + 0.953270i \(0.402306\pi\)
\(68\) 0 0
\(69\) −4068.83 + 55295.5i −0.102884 + 1.39819i
\(70\) 0 0
\(71\) −10310.5 −0.242736 −0.121368 0.992608i \(-0.538728\pi\)
−0.121368 + 0.992608i \(0.538728\pi\)
\(72\) 0 0
\(73\) 61295.6 1.34624 0.673120 0.739533i \(-0.264954\pi\)
0.673120 + 0.739533i \(0.264954\pi\)
\(74\) 0 0
\(75\) −2303.16 + 31300.0i −0.0472793 + 0.642527i
\(76\) 0 0
\(77\) −14915.0 −0.286680
\(78\) 0 0
\(79\) 31323.4i 0.564678i −0.959315 0.282339i \(-0.908890\pi\)
0.959315 0.282339i \(-0.0911104\pi\)
\(80\) 0 0
\(81\) 56518.7 + 17100.3i 0.957149 + 0.289595i
\(82\) 0 0
\(83\) 46611.6i 0.742675i 0.928498 + 0.371338i \(0.121101\pi\)
−0.928498 + 0.371338i \(0.878899\pi\)
\(84\) 0 0
\(85\) 158062.i 2.37290i
\(86\) 0 0
\(87\) −6603.59 + 89742.9i −0.0935367 + 1.27117i
\(88\) 0 0
\(89\) 99794.6i 1.33546i 0.744402 + 0.667731i \(0.232735\pi\)
−0.744402 + 0.667731i \(0.767265\pi\)
\(90\) 0 0
\(91\) −38311.5 −0.484982
\(92\) 0 0
\(93\) 79625.4 + 5859.11i 0.954650 + 0.0702464i
\(94\) 0 0
\(95\) −124485. −1.41517
\(96\) 0 0
\(97\) 115614. 1.24762 0.623810 0.781576i \(-0.285584\pi\)
0.623810 + 0.781576i \(0.285584\pi\)
\(98\) 0 0
\(99\) −5279.30 + 35678.6i −0.0541362 + 0.365864i
\(100\) 0 0
\(101\) 127872. 1.24731 0.623653 0.781701i \(-0.285648\pi\)
0.623653 + 0.781701i \(0.285648\pi\)
\(102\) 0 0
\(103\) 109422.i 1.01628i −0.861275 0.508138i \(-0.830334\pi\)
0.861275 0.508138i \(-0.169666\pi\)
\(104\) 0 0
\(105\) 111985. + 8240.25i 0.991259 + 0.0729403i
\(106\) 0 0
\(107\) 102575.i 0.866131i 0.901362 + 0.433066i \(0.142568\pi\)
−0.901362 + 0.433066i \(0.857432\pi\)
\(108\) 0 0
\(109\) 77992.7i 0.628764i 0.949297 + 0.314382i \(0.101797\pi\)
−0.949297 + 0.314382i \(0.898203\pi\)
\(110\) 0 0
\(111\) −171461. 12616.7i −1.32086 0.0971937i
\(112\) 0 0
\(113\) 116524.i 0.858462i −0.903195 0.429231i \(-0.858785\pi\)
0.903195 0.429231i \(-0.141215\pi\)
\(114\) 0 0
\(115\) 254959. 1.79774
\(116\) 0 0
\(117\) −13560.7 + 91645.8i −0.0915832 + 0.618939i
\(118\) 0 0
\(119\) 221582. 1.43439
\(120\) 0 0
\(121\) 139021. 0.863213
\(122\) 0 0
\(123\) −22656.2 1667.12i −0.135028 0.00993585i
\(124\) 0 0
\(125\) −79687.2 −0.456156
\(126\) 0 0
\(127\) 141481.i 0.778376i 0.921158 + 0.389188i \(0.127244\pi\)
−0.921158 + 0.389188i \(0.872756\pi\)
\(128\) 0 0
\(129\) −21296.9 + 289425.i −0.112679 + 1.53131i
\(130\) 0 0
\(131\) 171204.i 0.871637i −0.900035 0.435818i \(-0.856459\pi\)
0.900035 0.435818i \(-0.143541\pi\)
\(132\) 0 0
\(133\) 174512.i 0.855455i
\(134\) 0 0
\(135\) 59349.8 264966.i 0.280275 1.25128i
\(136\) 0 0
\(137\) 138147.i 0.628842i −0.949284 0.314421i \(-0.898190\pi\)
0.949284 0.314421i \(-0.101810\pi\)
\(138\) 0 0
\(139\) −300034. −1.31714 −0.658572 0.752518i \(-0.728839\pi\)
−0.658572 + 0.752518i \(0.728839\pi\)
\(140\) 0 0
\(141\) −14248.1 + 193632.i −0.0603543 + 0.820216i
\(142\) 0 0
\(143\) −56586.7 −0.231406
\(144\) 0 0
\(145\) 413790. 1.63441
\(146\) 0 0
\(147\) −7674.74 + 104300.i −0.0292934 + 0.398098i
\(148\) 0 0
\(149\) −383874. −1.41652 −0.708260 0.705952i \(-0.750519\pi\)
−0.708260 + 0.705952i \(0.750519\pi\)
\(150\) 0 0
\(151\) 481507.i 1.71854i 0.511522 + 0.859270i \(0.329082\pi\)
−0.511522 + 0.859270i \(0.670918\pi\)
\(152\) 0 0
\(153\) 78430.9 530053.i 0.270868 1.83059i
\(154\) 0 0
\(155\) 367140.i 1.22745i
\(156\) 0 0
\(157\) 592981.i 1.91996i 0.280072 + 0.959979i \(0.409642\pi\)
−0.280072 + 0.959979i \(0.590358\pi\)
\(158\) 0 0
\(159\) 5539.60 75283.2i 0.0173774 0.236160i
\(160\) 0 0
\(161\) 357420.i 1.08671i
\(162\) 0 0
\(163\) −482827. −1.42339 −0.711693 0.702490i \(-0.752071\pi\)
−0.711693 + 0.702490i \(0.752071\pi\)
\(164\) 0 0
\(165\) 165404. + 12171.0i 0.472973 + 0.0348030i
\(166\) 0 0
\(167\) 710600. 1.97167 0.985835 0.167719i \(-0.0536400\pi\)
0.985835 + 0.167719i \(0.0536400\pi\)
\(168\) 0 0
\(169\) 225942. 0.608527
\(170\) 0 0
\(171\) 417456. + 61770.1i 1.09174 + 0.161543i
\(172\) 0 0
\(173\) 208298. 0.529140 0.264570 0.964366i \(-0.414770\pi\)
0.264570 + 0.964366i \(0.414770\pi\)
\(174\) 0 0
\(175\) 202317.i 0.499388i
\(176\) 0 0
\(177\) 283966. + 20895.2i 0.681292 + 0.0501318i
\(178\) 0 0
\(179\) 294330.i 0.686597i −0.939226 0.343298i \(-0.888456\pi\)
0.939226 0.343298i \(-0.111544\pi\)
\(180\) 0 0
\(181\) 442331.i 1.00358i −0.864990 0.501789i \(-0.832675\pi\)
0.864990 0.501789i \(-0.167325\pi\)
\(182\) 0 0
\(183\) 84878.9 + 6245.68i 0.187358 + 0.0137864i
\(184\) 0 0
\(185\) 790580.i 1.69831i
\(186\) 0 0
\(187\) 327281. 0.684411
\(188\) 0 0
\(189\) −371448. 83200.8i −0.756387 0.169423i
\(190\) 0 0
\(191\) −311241. −0.617325 −0.308662 0.951172i \(-0.599881\pi\)
−0.308662 + 0.951172i \(0.599881\pi\)
\(192\) 0 0
\(193\) −642830. −1.24223 −0.621116 0.783719i \(-0.713320\pi\)
−0.621116 + 0.783719i \(0.713320\pi\)
\(194\) 0 0
\(195\) 424865. + 31263.0i 0.800137 + 0.0588768i
\(196\) 0 0
\(197\) 269484. 0.494729 0.247364 0.968923i \(-0.420435\pi\)
0.247364 + 0.968923i \(0.420435\pi\)
\(198\) 0 0
\(199\) 675602.i 1.20937i −0.796466 0.604683i \(-0.793300\pi\)
0.796466 0.604683i \(-0.206700\pi\)
\(200\) 0 0
\(201\) −25398.3 + 345163.i −0.0443419 + 0.602608i
\(202\) 0 0
\(203\) 580081.i 0.987982i
\(204\) 0 0
\(205\) 104464.i 0.173614i
\(206\) 0 0
\(207\) −854993. 126512.i −1.38687 0.205213i
\(208\) 0 0
\(209\) 257758.i 0.408175i
\(210\) 0 0
\(211\) −897849. −1.38834 −0.694172 0.719809i \(-0.744229\pi\)
−0.694172 + 0.719809i \(0.744229\pi\)
\(212\) 0 0
\(213\) 11794.8 160291.i 0.0178132 0.242081i
\(214\) 0 0
\(215\) 1.33449e6 1.96888
\(216\) 0 0
\(217\) −514684. −0.741978
\(218\) 0 0
\(219\) −70119.7 + 952928.i −0.0987938 + 1.34261i
\(220\) 0 0
\(221\) 840670. 1.15783
\(222\) 0 0
\(223\) 1.05027e6i 1.41429i 0.707070 + 0.707144i \(0.250017\pi\)
−0.707070 + 0.707144i \(0.749983\pi\)
\(224\) 0 0
\(225\) −483969. 71611.9i −0.637325 0.0943037i
\(226\) 0 0
\(227\) 23541.7i 0.0303230i −0.999885 0.0151615i \(-0.995174\pi\)
0.999885 0.0151615i \(-0.00482624\pi\)
\(228\) 0 0
\(229\) 1.18323e6i 1.49101i −0.666500 0.745505i \(-0.732208\pi\)
0.666500 0.745505i \(-0.267792\pi\)
\(230\) 0 0
\(231\) 17062.2 231875.i 0.0210380 0.285907i
\(232\) 0 0
\(233\) 97276.9i 0.117387i −0.998276 0.0586935i \(-0.981307\pi\)
0.998276 0.0586935i \(-0.0186935\pi\)
\(234\) 0 0
\(235\) 892805. 1.05460
\(236\) 0 0
\(237\) 486967. + 35832.7i 0.563156 + 0.0414389i
\(238\) 0 0
\(239\) −1.31214e6 −1.48589 −0.742943 0.669355i \(-0.766571\pi\)
−0.742943 + 0.669355i \(0.766571\pi\)
\(240\) 0 0
\(241\) 294315. 0.326414 0.163207 0.986592i \(-0.447816\pi\)
0.163207 + 0.986592i \(0.447816\pi\)
\(242\) 0 0
\(243\) −330504. + 859102.i −0.359055 + 0.933316i
\(244\) 0 0
\(245\) 480910. 0.511857
\(246\) 0 0
\(247\) 662090.i 0.690517i
\(248\) 0 0
\(249\) −724644. 53321.8i −0.740673 0.0545012i
\(250\) 0 0
\(251\) 683226.i 0.684510i 0.939607 + 0.342255i \(0.111191\pi\)
−0.939607 + 0.342255i \(0.888809\pi\)
\(252\) 0 0
\(253\) 527916.i 0.518517i
\(254\) 0 0
\(255\) −2.45730e6 180816.i −2.36650 0.174135i
\(256\) 0 0
\(257\) 400744.i 0.378472i 0.981932 + 0.189236i \(0.0606011\pi\)
−0.981932 + 0.189236i \(0.939399\pi\)
\(258\) 0 0
\(259\) 1.10829e6 1.02661
\(260\) 0 0
\(261\) −1.38763e6 205324.i −1.26087 0.186569i
\(262\) 0 0
\(263\) 1.33787e6 1.19269 0.596343 0.802730i \(-0.296620\pi\)
0.596343 + 0.802730i \(0.296620\pi\)
\(264\) 0 0
\(265\) −347119. −0.303643
\(266\) 0 0
\(267\) −1.55145e6 114161.i −1.33186 0.0980029i
\(268\) 0 0
\(269\) −2.23657e6 −1.88452 −0.942262 0.334878i \(-0.891305\pi\)
−0.942262 + 0.334878i \(0.891305\pi\)
\(270\) 0 0
\(271\) 1.06106e6i 0.877642i −0.898574 0.438821i \(-0.855396\pi\)
0.898574 0.438821i \(-0.144604\pi\)
\(272\) 0 0
\(273\) 43826.7 595606.i 0.0355904 0.483674i
\(274\) 0 0
\(275\) 298826.i 0.238280i
\(276\) 0 0
\(277\) 423903.i 0.331946i −0.986130 0.165973i \(-0.946924\pi\)
0.986130 0.165973i \(-0.0530764\pi\)
\(278\) 0 0
\(279\) −182176. + 1.23119e6i −0.140114 + 0.946921i
\(280\) 0 0
\(281\) 1.17156e6i 0.885115i −0.896740 0.442558i \(-0.854071\pi\)
0.896740 0.442558i \(-0.145929\pi\)
\(282\) 0 0
\(283\) −607065. −0.450577 −0.225289 0.974292i \(-0.572332\pi\)
−0.225289 + 0.974292i \(0.572332\pi\)
\(284\) 0 0
\(285\) 142406. 1.93530e6i 0.103852 1.41136i
\(286\) 0 0
\(287\) 146446. 0.104947
\(288\) 0 0
\(289\) −3.44233e6 −2.42442
\(290\) 0 0
\(291\) −132258. + 1.79739e6i −0.0915566 + 1.24426i
\(292\) 0 0
\(293\) −657552. −0.447467 −0.223734 0.974650i \(-0.571825\pi\)
−0.223734 + 0.974650i \(0.571825\pi\)
\(294\) 0 0
\(295\) 1.30932e6i 0.875975i
\(296\) 0 0
\(297\) −548636. 122889.i −0.360905 0.0808392i
\(298\) 0 0
\(299\) 1.35603e6i 0.877185i
\(300\) 0 0
\(301\) 1.87079e6i 1.19017i
\(302\) 0 0
\(303\) −146281. + 1.98796e6i −0.0915336 + 1.24394i
\(304\) 0 0
\(305\) 391363.i 0.240896i
\(306\) 0 0
\(307\) 1.78991e6 1.08389 0.541944 0.840415i \(-0.317688\pi\)
0.541944 + 0.840415i \(0.317688\pi\)
\(308\) 0 0
\(309\) 1.70112e6 + 125174.i 1.01354 + 0.0745795i
\(310\) 0 0
\(311\) 1.83869e6 1.07797 0.538986 0.842314i \(-0.318807\pi\)
0.538986 + 0.842314i \(0.318807\pi\)
\(312\) 0 0
\(313\) −2.47967e6 −1.43065 −0.715324 0.698793i \(-0.753721\pi\)
−0.715324 + 0.698793i \(0.753721\pi\)
\(314\) 0 0
\(315\) −256213. + 1.73154e6i −0.145487 + 0.983234i
\(316\) 0 0
\(317\) −613487. −0.342892 −0.171446 0.985194i \(-0.554844\pi\)
−0.171446 + 0.985194i \(0.554844\pi\)
\(318\) 0 0
\(319\) 856790.i 0.471409i
\(320\) 0 0
\(321\) −1.59468e6 117342.i −0.863796 0.0635610i
\(322\) 0 0
\(323\) 3.82933e6i 2.04229i
\(324\) 0 0
\(325\) 767580.i 0.403102i
\(326\) 0 0
\(327\) −1.21251e6 89220.5i −0.627069 0.0461419i
\(328\) 0 0
\(329\) 1.25160e6i 0.637493i
\(330\) 0 0
\(331\) −685408. −0.343858 −0.171929 0.985109i \(-0.555000\pi\)
−0.171929 + 0.985109i \(0.555000\pi\)
\(332\) 0 0
\(333\) 392289. 2.65117e6i 0.193863 1.31017i
\(334\) 0 0
\(335\) 1.59149e6 0.774806
\(336\) 0 0
\(337\) 2.57486e6 1.23504 0.617518 0.786557i \(-0.288138\pi\)
0.617518 + 0.786557i \(0.288138\pi\)
\(338\) 0 0
\(339\) 1.81154e6 + 133299.i 0.856147 + 0.0629982i
\(340\) 0 0
\(341\) −760197. −0.354030
\(342\) 0 0
\(343\) 2.36310e6i 1.08454i
\(344\) 0 0
\(345\) −291663. + 3.96370e6i −0.131927 + 1.79289i
\(346\) 0 0
\(347\) 3.51690e6i 1.56797i −0.620782 0.783984i \(-0.713185\pi\)
0.620782 0.783984i \(-0.286815\pi\)
\(348\) 0 0
\(349\) 1.15782e6i 0.508833i −0.967095 0.254417i \(-0.918117\pi\)
0.967095 0.254417i \(-0.0818835\pi\)
\(350\) 0 0
\(351\) −1.40925e6 315659.i −0.610550 0.136757i
\(352\) 0 0
\(353\) 3.87000e6i 1.65301i 0.562932 + 0.826503i \(0.309673\pi\)
−0.562932 + 0.826503i \(0.690327\pi\)
\(354\) 0 0
\(355\) −739078. −0.311257
\(356\) 0 0
\(357\) −253481. + 3.44481e6i −0.105263 + 1.43052i
\(358\) 0 0
\(359\) 4.01806e6 1.64543 0.822716 0.568453i \(-0.192458\pi\)
0.822716 + 0.568453i \(0.192458\pi\)
\(360\) 0 0
\(361\) 539784. 0.217998
\(362\) 0 0
\(363\) −159035. + 2.16128e6i −0.0633469 + 0.860885i
\(364\) 0 0
\(365\) 4.39380e6 1.72627
\(366\) 0 0
\(367\) 1.02337e6i 0.396612i −0.980140 0.198306i \(-0.936456\pi\)
0.980140 0.198306i \(-0.0635440\pi\)
\(368\) 0 0
\(369\) 51835.6 350317.i 0.0198181 0.133935i
\(370\) 0 0
\(371\) 486617.i 0.183549i
\(372\) 0 0
\(373\) 826829.i 0.307711i −0.988093 0.153856i \(-0.950831\pi\)
0.988093 0.153856i \(-0.0491691\pi\)
\(374\) 0 0
\(375\) 91158.9 1.23885e6i 0.0334750 0.454926i
\(376\) 0 0
\(377\) 2.20079e6i 0.797491i
\(378\) 0 0
\(379\) 1.29909e6 0.464559 0.232279 0.972649i \(-0.425382\pi\)
0.232279 + 0.972649i \(0.425382\pi\)
\(380\) 0 0
\(381\) −2.19953e6 161849.i −0.776278 0.0571212i
\(382\) 0 0
\(383\) 2.14371e6 0.746740 0.373370 0.927683i \(-0.378202\pi\)
0.373370 + 0.927683i \(0.378202\pi\)
\(384\) 0 0
\(385\) −1.06914e6 −0.367606
\(386\) 0 0
\(387\) −4.47516e6 662181.i −1.51891 0.224750i
\(388\) 0 0
\(389\) 2.45427e6 0.822335 0.411167 0.911560i \(-0.365121\pi\)
0.411167 + 0.911560i \(0.365121\pi\)
\(390\) 0 0
\(391\) 7.84288e6i 2.59438i
\(392\) 0 0
\(393\) 2.66161e6 + 195850.i 0.869287 + 0.0639651i
\(394\) 0 0
\(395\) 2.24533e6i 0.724081i
\(396\) 0 0
\(397\) 2.55038e6i 0.812136i 0.913843 + 0.406068i \(0.133100\pi\)
−0.913843 + 0.406068i \(0.866900\pi\)
\(398\) 0 0
\(399\) −2.71305e6 199635.i −0.853149 0.0627776i
\(400\) 0 0
\(401\) 57308.6i 0.0177975i 0.999960 + 0.00889875i \(0.00283260\pi\)
−0.999960 + 0.00889875i \(0.997167\pi\)
\(402\) 0 0
\(403\) −1.95268e6 −0.598919
\(404\) 0 0
\(405\) 4.05138e6 + 1.22579e6i 1.22734 + 0.371345i
\(406\) 0 0
\(407\) 1.63697e6 0.489840
\(408\) 0 0
\(409\) −147846. −0.0437020 −0.0218510 0.999761i \(-0.506956\pi\)
−0.0218510 + 0.999761i \(0.506956\pi\)
\(410\) 0 0
\(411\) 2.14770e6 + 158035.i 0.627146 + 0.0461475i
\(412\) 0 0
\(413\) −1.83550e6 −0.529517
\(414\) 0 0
\(415\) 3.34122e6i 0.952324i
\(416\) 0 0
\(417\) 343226. 4.66446e6i 0.0966586 1.31359i
\(418\) 0 0
\(419\) 2.93840e6i 0.817666i −0.912609 0.408833i \(-0.865936\pi\)
0.912609 0.408833i \(-0.134064\pi\)
\(420\) 0 0
\(421\) 242806.i 0.0667657i −0.999443 0.0333829i \(-0.989372\pi\)
0.999443 0.0333829i \(-0.0106281\pi\)
\(422\) 0 0
\(423\) −2.99398e6 443013.i −0.813576 0.120383i
\(424\) 0 0
\(425\) 4.43946e6i 1.19222i
\(426\) 0 0
\(427\) −548641. −0.145619
\(428\) 0 0
\(429\) 64732.9 879721.i 0.0169817 0.230782i
\(430\) 0 0
\(431\) 1.23003e6 0.318949 0.159474 0.987202i \(-0.449020\pi\)
0.159474 + 0.987202i \(0.449020\pi\)
\(432\) 0 0
\(433\) 965893. 0.247577 0.123788 0.992309i \(-0.460496\pi\)
0.123788 + 0.992309i \(0.460496\pi\)
\(434\) 0 0
\(435\) −473359. + 6.43296e6i −0.119941 + 1.63000i
\(436\) 0 0
\(437\) −6.17685e6 −1.54726
\(438\) 0 0
\(439\) 7.17684e6i 1.77735i −0.458542 0.888673i \(-0.651628\pi\)
0.458542 0.888673i \(-0.348372\pi\)
\(440\) 0 0
\(441\) −1.61271e6 238629.i −0.394875 0.0584289i
\(442\) 0 0
\(443\) 1.14564e6i 0.277356i 0.990338 + 0.138678i \(0.0442853\pi\)
−0.990338 + 0.138678i \(0.955715\pi\)
\(444\) 0 0
\(445\) 7.15348e6i 1.71245i
\(446\) 0 0
\(447\) 439136. 5.96786e6i 0.103951 1.41270i
\(448\) 0 0
\(449\) 4.35308e6i 1.01902i −0.860466 0.509508i \(-0.829827\pi\)
0.860466 0.509508i \(-0.170173\pi\)
\(450\) 0 0
\(451\) 216303. 0.0500750
\(452\) 0 0
\(453\) −7.48571e6 550824.i −1.71391 0.126115i
\(454\) 0 0
\(455\) −2.74625e6 −0.621886
\(456\) 0 0
\(457\) 4.73821e6 1.06126 0.530632 0.847602i \(-0.321955\pi\)
0.530632 + 0.847602i \(0.321955\pi\)
\(458\) 0 0
\(459\) 8.15071e6 + 1.82568e6i 1.80577 + 0.404476i
\(460\) 0 0
\(461\) −3.48712e6 −0.764214 −0.382107 0.924118i \(-0.624801\pi\)
−0.382107 + 0.924118i \(0.624801\pi\)
\(462\) 0 0
\(463\) 5.31702e6i 1.15270i 0.817203 + 0.576350i \(0.195523\pi\)
−0.817203 + 0.576350i \(0.804477\pi\)
\(464\) 0 0
\(465\) 5.70772e6 + 419993.i 1.22414 + 0.0900762i
\(466\) 0 0
\(467\) 3.34272e6i 0.709265i 0.935006 + 0.354632i \(0.115394\pi\)
−0.935006 + 0.354632i \(0.884606\pi\)
\(468\) 0 0
\(469\) 2.23107e6i 0.468362i
\(470\) 0 0
\(471\) −9.21874e6 678346.i −1.91478 0.140896i
\(472\) 0 0
\(473\) 2.76319e6i 0.567882i
\(474\) 0 0
\(475\) −3.49640e6 −0.711029
\(476\) 0 0
\(477\) 1.16405e6 + 172242.i 0.234248 + 0.0346611i
\(478\) 0 0
\(479\) 3.01866e6 0.601140 0.300570 0.953760i \(-0.402823\pi\)
0.300570 + 0.953760i \(0.402823\pi\)
\(480\) 0 0
\(481\) 4.20479e6 0.828670
\(482\) 0 0
\(483\) 5.55660e6 + 408874.i 1.08378 + 0.0797483i
\(484\) 0 0
\(485\) 8.28748e6 1.59981
\(486\) 0 0
\(487\) 2.35174e6i 0.449332i 0.974436 + 0.224666i \(0.0721290\pi\)
−0.974436 + 0.224666i \(0.927871\pi\)
\(488\) 0 0
\(489\) 552335. 7.50624e6i 0.104455 1.41955i
\(490\) 0 0
\(491\) 2.24809e6i 0.420832i 0.977612 + 0.210416i \(0.0674819\pi\)
−0.977612 + 0.210416i \(0.932518\pi\)
\(492\) 0 0
\(493\) 1.27287e7i 2.35868i
\(494\) 0 0
\(495\) −378431. + 2.55752e6i −0.0694183 + 0.469144i
\(496\) 0 0
\(497\) 1.03609e6i 0.188152i
\(498\) 0 0
\(499\) −4.93445e6 −0.887131 −0.443565 0.896242i \(-0.646287\pi\)
−0.443565 + 0.896242i \(0.646287\pi\)
\(500\) 0 0
\(501\) −812898. + 1.10473e7i −0.144691 + 1.96635i
\(502\) 0 0
\(503\) −8.42902e6 −1.48545 −0.742724 0.669598i \(-0.766466\pi\)
−0.742724 + 0.669598i \(0.766466\pi\)
\(504\) 0 0
\(505\) 9.16616e6 1.59941
\(506\) 0 0
\(507\) −258468. + 3.51259e6i −0.0446567 + 0.606886i
\(508\) 0 0
\(509\) 6.54347e6 1.11947 0.559737 0.828670i \(-0.310902\pi\)
0.559737 + 0.828670i \(0.310902\pi\)
\(510\) 0 0
\(511\) 6.15955e6i 1.04351i
\(512\) 0 0
\(513\) −1.43786e6 + 6.41928e6i −0.241225 + 1.07694i
\(514\) 0 0
\(515\) 7.84361e6i 1.30316i
\(516\) 0 0
\(517\) 1.84863e6i 0.304176i
\(518\) 0 0
\(519\) −238285. + 3.23830e6i −0.0388310 + 0.527713i
\(520\) 0 0
\(521\) 388674.i 0.0627323i −0.999508 0.0313661i \(-0.990014\pi\)
0.999508 0.0313661i \(-0.00998579\pi\)
\(522\) 0 0
\(523\) −252292. −0.0403319 −0.0201659 0.999797i \(-0.506419\pi\)
−0.0201659 + 0.999797i \(0.506419\pi\)
\(524\) 0 0
\(525\) 3.14531e6 + 231443.i 0.498041 + 0.0366476i
\(526\) 0 0
\(527\) 1.12937e7 1.77138
\(528\) 0 0
\(529\) 6.21449e6 0.965531
\(530\) 0 0
\(531\) −649691. + 4.39075e6i −0.0999932 + 0.675776i
\(532\) 0 0
\(533\) 555606. 0.0847128
\(534\) 0 0
\(535\) 7.35282e6i 1.11063i
\(536\) 0 0
\(537\) 4.57578e6 + 336701.i 0.684745 + 0.0503859i
\(538\) 0 0
\(539\) 995768.i 0.147634i
\(540\) 0 0
\(541\) 9.20507e6i 1.35218i 0.736820 + 0.676089i \(0.236327\pi\)
−0.736820 + 0.676089i \(0.763673\pi\)
\(542\) 0 0
\(543\) 6.87667e6 + 506009.i 1.00087 + 0.0736476i
\(544\) 0 0
\(545\) 5.59068e6i 0.806257i
\(546\) 0 0
\(547\) 5.73723e6 0.819850 0.409925 0.912119i \(-0.365555\pi\)
0.409925 + 0.912119i \(0.365555\pi\)
\(548\) 0 0
\(549\) −194196. + 1.31242e6i −0.0274985 + 0.185841i
\(550\) 0 0
\(551\) −1.00248e7 −1.40669
\(552\) 0 0
\(553\) −3.14766e6 −0.437699
\(554\) 0 0
\(555\) −1.22907e7 904391.i −1.69373 0.124630i
\(556\) 0 0
\(557\) −6.94218e6 −0.948108 −0.474054 0.880496i \(-0.657210\pi\)
−0.474054 + 0.880496i \(0.657210\pi\)
\(558\) 0 0
\(559\) 7.09766e6i 0.960695i
\(560\) 0 0
\(561\) −374396. + 5.08805e6i −0.0502255 + 0.682565i
\(562\) 0 0
\(563\) 4.89782e6i 0.651226i 0.945503 + 0.325613i \(0.105571\pi\)
−0.945503 + 0.325613i \(0.894429\pi\)
\(564\) 0 0
\(565\) 8.35272e6i 1.10080i
\(566\) 0 0
\(567\) 1.71840e6 5.67952e6i 0.224474 0.741914i
\(568\) 0 0
\(569\) 6.24106e6i 0.808123i 0.914732 + 0.404062i \(0.132402\pi\)
−0.914732 + 0.404062i \(0.867598\pi\)
\(570\) 0 0
\(571\) −3.53464e6 −0.453685 −0.226842 0.973931i \(-0.572840\pi\)
−0.226842 + 0.973931i \(0.572840\pi\)
\(572\) 0 0
\(573\) 356047. 4.83869e6i 0.0453024 0.615660i
\(574\) 0 0
\(575\) 7.16100e6 0.903241
\(576\) 0 0
\(577\) 2.23801e6 0.279848 0.139924 0.990162i \(-0.455314\pi\)
0.139924 + 0.990162i \(0.455314\pi\)
\(578\) 0 0
\(579\) 735371. 9.99371e6i 0.0911612 1.23888i
\(580\) 0 0
\(581\) 4.68396e6 0.575669
\(582\) 0 0
\(583\) 718741.i 0.0875793i
\(584\) 0 0
\(585\) −972056. + 6.56937e6i −0.117436 + 0.793659i
\(586\) 0 0
\(587\) 1.22958e7i 1.47287i −0.676510 0.736433i \(-0.736509\pi\)
0.676510 0.736433i \(-0.263491\pi\)
\(588\) 0 0
\(589\) 8.89464e6i 1.05643i
\(590\) 0 0
\(591\) −308279. + 4.18951e6i −0.0363057 + 0.493395i
\(592\) 0 0
\(593\) 1.06803e7i 1.24723i −0.781733 0.623614i \(-0.785664\pi\)
0.781733 0.623614i \(-0.214336\pi\)
\(594\) 0 0
\(595\) 1.58835e7 1.83930
\(596\) 0 0
\(597\) 1.05032e7 + 772861.i 1.20611 + 0.0887494i
\(598\) 0 0
\(599\) 4.78335e6 0.544710 0.272355 0.962197i \(-0.412198\pi\)
0.272355 + 0.962197i \(0.412198\pi\)
\(600\) 0 0
\(601\) −894944. −0.101067 −0.0505335 0.998722i \(-0.516092\pi\)
−0.0505335 + 0.998722i \(0.516092\pi\)
\(602\) 0 0
\(603\) −5.33700e6 789705.i −0.597729 0.0884447i
\(604\) 0 0
\(605\) 9.96534e6 1.10689
\(606\) 0 0
\(607\) 1.19696e7i 1.31858i 0.751888 + 0.659291i \(0.229144\pi\)
−0.751888 + 0.659291i \(0.770856\pi\)
\(608\) 0 0
\(609\) 9.01819e6 + 663589.i 0.985318 + 0.0725030i
\(610\) 0 0
\(611\) 4.74849e6i 0.514579i
\(612\) 0 0
\(613\) 4.89530e6i 0.526172i −0.964772 0.263086i \(-0.915260\pi\)
0.964772 0.263086i \(-0.0847403\pi\)
\(614\) 0 0
\(615\) −1.62405e6 119503.i −0.173145 0.0127406i
\(616\) 0 0
\(617\) 2.29935e6i 0.243160i 0.992582 + 0.121580i \(0.0387961\pi\)
−0.992582 + 0.121580i \(0.961204\pi\)
\(618\) 0 0
\(619\) 2.34773e6 0.246275 0.123138 0.992390i \(-0.460704\pi\)
0.123138 + 0.992390i \(0.460704\pi\)
\(620\) 0 0
\(621\) 2.94488e6 1.31474e7i 0.306435 1.36807i
\(622\) 0 0
\(623\) 1.00283e7 1.03516
\(624\) 0 0
\(625\) −1.20038e7 −1.22919
\(626\) 0 0
\(627\) −4.00722e6 294865.i −0.407074 0.0299539i
\(628\) 0 0
\(629\) −2.43193e7 −2.45089
\(630\) 0 0
\(631\) 1.32907e7i 1.32885i −0.747356 0.664424i \(-0.768677\pi\)
0.747356 0.664424i \(-0.231323\pi\)
\(632\) 0 0
\(633\) 1.02710e6 1.39583e7i 0.101884 1.38460i
\(634\) 0 0
\(635\) 1.01417e7i 0.998103i
\(636\) 0 0
\(637\) 2.55778e6i 0.249755i
\(638\) 0 0
\(639\) 2.47847e6 + 366734.i 0.240121 + 0.0355303i
\(640\) 0 0
\(641\) 1.68877e7i 1.62340i −0.584078 0.811698i \(-0.698544\pi\)
0.584078 0.811698i \(-0.301456\pi\)
\(642\) 0 0
\(643\) −1.77198e7 −1.69017 −0.845086 0.534631i \(-0.820451\pi\)
−0.845086 + 0.534631i \(0.820451\pi\)
\(644\) 0 0
\(645\) −1.52661e6 + 2.07466e7i −0.144487 + 1.96358i
\(646\) 0 0
\(647\) −2.19463e6 −0.206111 −0.103056 0.994676i \(-0.532862\pi\)
−0.103056 + 0.994676i \(0.532862\pi\)
\(648\) 0 0
\(649\) −2.71107e6 −0.252656
\(650\) 0 0
\(651\) 588777. 8.00149e6i 0.0544501 0.739977i
\(652\) 0 0
\(653\) −1.02224e7 −0.938141 −0.469071 0.883161i \(-0.655411\pi\)
−0.469071 + 0.883161i \(0.655411\pi\)
\(654\) 0 0
\(655\) 1.22723e7i 1.11769i
\(656\) 0 0
\(657\) −1.47344e7 2.18022e6i −1.33174 0.197055i
\(658\) 0 0
\(659\) 1.14626e7i 1.02818i 0.857736 + 0.514090i \(0.171870\pi\)
−0.857736 + 0.514090i \(0.828130\pi\)
\(660\) 0 0
\(661\) 1.12139e6i 0.0998278i −0.998754 0.0499139i \(-0.984105\pi\)
0.998754 0.0499139i \(-0.0158947\pi\)
\(662\) 0 0
\(663\) −961692. + 1.30694e7i −0.0849674 + 1.15471i
\(664\) 0 0
\(665\) 1.25094e7i 1.09694i
\(666\) 0 0
\(667\) 2.05319e7 1.78696
\(668\) 0 0
\(669\) −1.63279e7 1.20146e6i −1.41047 0.103787i
\(670\) 0 0
\(671\) −810353. −0.0694813
\(672\) 0 0
\(673\) 7.05752e6 0.600640 0.300320 0.953838i \(-0.402907\pi\)
0.300320 + 0.953838i \(0.402907\pi\)
\(674\) 0 0
\(675\) 1.66695e6 7.44206e6i 0.140819 0.628686i
\(676\) 0 0
\(677\) 3.09408e6 0.259454 0.129727 0.991550i \(-0.458590\pi\)
0.129727 + 0.991550i \(0.458590\pi\)
\(678\) 0 0
\(679\) 1.16180e7i 0.967067i
\(680\) 0 0
\(681\) 365989. + 26930.7i 0.0302413 + 0.00222526i
\(682\) 0 0
\(683\) 8.19357e6i 0.672080i −0.941848 0.336040i \(-0.890912\pi\)
0.941848 0.336040i \(-0.109088\pi\)
\(684\) 0 0
\(685\) 9.90270e6i 0.806357i
\(686\) 0 0
\(687\) 1.83950e7 + 1.35357e6i 1.48699 + 0.109418i
\(688\) 0 0
\(689\) 1.84619e6i 0.148159i
\(690\) 0 0
\(691\) 7.08153e6 0.564199 0.282099 0.959385i \(-0.408969\pi\)
0.282099 + 0.959385i \(0.408969\pi\)
\(692\) 0 0
\(693\) 3.58532e6 + 530512.i 0.283592 + 0.0419626i
\(694\) 0 0
\(695\) −2.15071e7 −1.68896
\(696\) 0 0
\(697\) −3.21346e6 −0.250548
\(698\) 0 0
\(699\) 1.51231e6 + 111281.i 0.117070 + 0.00861445i
\(700\) 0 0
\(701\) 57414.8 0.00441295 0.00220647 0.999998i \(-0.499298\pi\)
0.00220647 + 0.999998i \(0.499298\pi\)
\(702\) 0 0
\(703\) 1.91532e7i 1.46169i
\(704\) 0 0
\(705\) −1.02133e6 + 1.38799e7i −0.0773917 + 1.05175i
\(706\) 0 0
\(707\) 1.28498e7i 0.966824i
\(708\) 0 0
\(709\) 1.30024e7i 0.971419i −0.874120 0.485710i \(-0.838561\pi\)
0.874120 0.485710i \(-0.161439\pi\)
\(710\) 0 0
\(711\) −1.11414e6 + 7.52960e6i −0.0826544 + 0.558596i
\(712\) 0 0
\(713\) 1.82172e7i 1.34201i
\(714\) 0 0
\(715\) −4.05625e6 −0.296729
\(716\) 0 0
\(717\) 1.50103e6 2.03991e7i 0.109042 1.48188i
\(718\) 0 0
\(719\) −1.03530e7 −0.746868 −0.373434 0.927657i \(-0.621820\pi\)
−0.373434 + 0.927657i \(0.621820\pi\)
\(720\) 0 0
\(721\) −1.09957e7 −0.787746
\(722\) 0 0
\(723\) −336684. + 4.57554e6i −0.0239539 + 0.325534i
\(724\) 0 0
\(725\) 1.16221e7 0.821180
\(726\) 0 0
\(727\) 2.30522e7i 1.61762i 0.588072 + 0.808809i \(0.299887\pi\)
−0.588072 + 0.808809i \(0.700113\pi\)
\(728\) 0 0
\(729\) −1.29779e7 6.12093e6i −0.904451 0.426578i
\(730\) 0 0
\(731\) 4.10508e7i 2.84137i
\(732\) 0 0
\(733\) 1.18210e7i 0.812634i −0.913732 0.406317i \(-0.866813\pi\)
0.913732 0.406317i \(-0.133187\pi\)
\(734\) 0 0
\(735\) −550141. + 7.47643e6i −0.0375626 + 0.510477i
\(736\) 0 0
\(737\) 3.29533e6i 0.223476i
\(738\) 0 0
\(739\) −1.69579e7 −1.14225 −0.571123 0.820864i \(-0.693492\pi\)
−0.571123 + 0.820864i \(0.693492\pi\)
\(740\) 0 0
\(741\) −1.02931e7 757403.i −0.688655 0.0506736i
\(742\) 0 0
\(743\) −1.53213e7 −1.01817 −0.509087 0.860715i \(-0.670017\pi\)
−0.509087 + 0.860715i \(0.670017\pi\)
\(744\) 0 0
\(745\) −2.75169e7 −1.81639
\(746\) 0 0
\(747\) 1.65793e6 1.12046e7i 0.108709 0.734676i
\(748\) 0 0
\(749\) 1.03077e7 0.671364
\(750\) 0 0
\(751\) 2.38132e7i 1.54070i 0.637620 + 0.770351i \(0.279919\pi\)
−0.637620 + 0.770351i \(0.720081\pi\)
\(752\) 0 0
\(753\) −1.06217e7 781582.i −0.682665 0.0502328i
\(754\) 0 0
\(755\) 3.45154e7i 2.20367i
\(756\) 0 0
\(757\) 9.77331e6i 0.619872i −0.950757 0.309936i \(-0.899692\pi\)
0.950757 0.309936i \(-0.100308\pi\)
\(758\) 0 0
\(759\) 8.20720e6 + 603914.i 0.517119 + 0.0380514i
\(760\) 0 0
\(761\) 9.77488e6i 0.611857i 0.952054 + 0.305928i \(0.0989668\pi\)
−0.952054 + 0.305928i \(0.901033\pi\)
\(762\) 0 0
\(763\) 7.83742e6 0.487374
\(764\) 0 0
\(765\) 5.62209e6 3.79953e7i 0.347332 2.34734i
\(766\) 0 0
\(767\) −6.96379e6 −0.427422
\(768\) 0 0
\(769\) −6.19002e6 −0.377465 −0.188732 0.982029i \(-0.560438\pi\)
−0.188732 + 0.982029i \(0.560438\pi\)
\(770\) 0 0
\(771\) −6.23013e6 458434.i −0.377452 0.0277742i
\(772\) 0 0
\(773\) 1.56139e7 0.939860 0.469930 0.882704i \(-0.344279\pi\)
0.469930 + 0.882704i \(0.344279\pi\)
\(774\) 0 0
\(775\) 1.03118e7i 0.616710i
\(776\) 0 0
\(777\) −1.26784e6 + 1.72300e7i −0.0753377 + 1.02384i
\(778\) 0 0
\(779\) 2.53084e6i 0.149424i
\(780\) 0 0
\(781\) 1.53033e6i 0.0897753i
\(782\) 0 0
\(783\) 4.77945e6 2.13378e7i 0.278595 1.24378i
\(784\) 0 0
\(785\) 4.25061e7i 2.46194i
\(786\) 0 0
\(787\) −2.87295e7 −1.65345 −0.826725 0.562607i \(-0.809798\pi\)
−0.826725 + 0.562607i \(0.809798\pi\)
\(788\) 0 0
\(789\) −1.53047e6 + 2.07992e7i −0.0875252 + 1.18947i
\(790\) 0 0
\(791\) −1.17094e7 −0.665419
\(792\) 0 0
\(793\) −2.08151e6 −0.117543
\(794\) 0 0
\(795\) 397090. 5.39646e6i 0.0222829 0.302825i
\(796\) 0 0
\(797\) −1.60767e7 −0.896503 −0.448251 0.893907i \(-0.647953\pi\)
−0.448251 + 0.893907i \(0.647953\pi\)
\(798\) 0 0
\(799\) 2.74639e7i 1.52193i
\(800\) 0 0
\(801\) 3.54959e6 2.39889e7i 0.195477 1.32108i
\(802\) 0 0
\(803\) 9.09776e6i 0.497904i
\(804\) 0 0
\(805\) 2.56206e7i 1.39348i
\(806\) 0 0
\(807\) 2.55854e6 3.47706e7i 0.138296 1.87944i
\(808\) 0 0
\(809\) 1.95204e7i 1.04862i 0.851528 + 0.524309i \(0.175676\pi\)
−0.851528 + 0.524309i \(0.824324\pi\)
\(810\) 0 0
\(811\) 8.99009e6 0.479968 0.239984 0.970777i \(-0.422858\pi\)
0.239984 + 0.970777i \(0.422858\pi\)
\(812\) 0 0
\(813\) 1.64957e7 + 1.21381e6i 0.875276 + 0.0644058i
\(814\) 0 0
\(815\) −3.46101e7 −1.82519
\(816\) 0 0
\(817\) −3.23305e7 −1.69456
\(818\) 0 0
\(819\) 9.20941e6 + 1.36270e6i 0.479758 + 0.0709888i
\(820\) 0 0
\(821\) 3.64087e6 0.188516 0.0942578 0.995548i \(-0.469952\pi\)
0.0942578 + 0.995548i \(0.469952\pi\)
\(822\) 0 0
\(823\) 1.60705e7i 0.827048i 0.910493 + 0.413524i \(0.135702\pi\)
−0.910493 + 0.413524i \(0.864298\pi\)
\(824\) 0 0
\(825\) 4.64568e6 + 341845.i 0.237637 + 0.0174862i
\(826\) 0 0
\(827\) 6.68243e6i 0.339759i −0.985465 0.169879i \(-0.945662\pi\)
0.985465 0.169879i \(-0.0543378\pi\)
\(828\) 0 0
\(829\) 2.43981e7i 1.23302i 0.787348 + 0.616509i \(0.211454\pi\)
−0.787348 + 0.616509i \(0.788546\pi\)
\(830\) 0 0
\(831\) 6.59017e6 + 484928.i 0.331051 + 0.0243598i
\(832\) 0 0
\(833\) 1.47934e7i 0.738680i
\(834\) 0 0
\(835\) 5.09373e7 2.52825
\(836\) 0 0
\(837\) −1.89322e7 4.24062e6i −0.934086 0.209226i
\(838\) 0 0
\(839\) 1.26116e7 0.618535 0.309267 0.950975i \(-0.399916\pi\)
0.309267 + 0.950975i \(0.399916\pi\)
\(840\) 0 0
\(841\) 1.28115e7 0.624611
\(842\) 0 0
\(843\) 1.82136e7 + 1.34022e6i 0.882729 + 0.0649542i
\(844\) 0 0
\(845\) 1.61960e7 0.780307
\(846\) 0 0
\(847\) 1.39701e7i 0.669101i
\(848\) 0 0
\(849\) 694457. 9.43769e6i 0.0330656 0.449362i
\(850\) 0 0
\(851\) 3.92279e7i 1.85682i
\(852\) 0 0
\(853\) 2.53134e7i 1.19118i −0.803289 0.595590i \(-0.796918\pi\)
0.803289 0.595590i \(-0.203082\pi\)
\(854\) 0 0
\(855\) 2.99241e7 + 4.42781e6i 1.39993 + 0.207145i
\(856\) 0 0
\(857\) 4.08243e7i 1.89874i −0.314155 0.949372i \(-0.601721\pi\)
0.314155 0.949372i \(-0.398279\pi\)
\(858\) 0 0
\(859\) −3.83562e7 −1.77359 −0.886793 0.462167i \(-0.847072\pi\)
−0.886793 + 0.462167i \(0.847072\pi\)
\(860\) 0 0
\(861\) −167528. + 2.27671e6i −0.00770157 + 0.104664i
\(862\) 0 0
\(863\) −9.90424e6 −0.452683 −0.226342 0.974048i \(-0.572677\pi\)
−0.226342 + 0.974048i \(0.572677\pi\)
\(864\) 0 0
\(865\) 1.49313e7 0.678510
\(866\) 0 0
\(867\) 3.93789e6 5.35160e7i 0.177916 2.41789i
\(868\) 0 0
\(869\) −4.64915e6 −0.208845
\(870\) 0 0
\(871\) 8.46455e6i 0.378058i
\(872\) 0 0
\(873\) −2.77917e7 4.11228e6i −1.23418 0.182620i
\(874\) 0 0
\(875\) 8.00770e6i 0.353580i
\(876\) 0 0
\(877\) 5.33103e6i 0.234052i 0.993129 + 0.117026i \(0.0373361\pi\)
−0.993129 + 0.117026i \(0.962664\pi\)
\(878\) 0 0
\(879\) 752213. 1.02226e7i 0.0328374 0.446261i
\(880\) 0 0
\(881\) 2.41429e7i 1.04797i −0.851726 0.523987i \(-0.824444\pi\)
0.851726 0.523987i \(-0.175556\pi\)
\(882\) 0 0
\(883\) 3.51549e6 0.151734 0.0758672 0.997118i \(-0.475828\pi\)
0.0758672 + 0.997118i \(0.475828\pi\)
\(884\) 0 0
\(885\) 2.03553e7 + 1.49781e6i 0.873613 + 0.0642834i
\(886\) 0 0
\(887\) 3.61885e7 1.54441 0.772204 0.635375i \(-0.219154\pi\)
0.772204 + 0.635375i \(0.219154\pi\)
\(888\) 0 0
\(889\) 1.42173e7 0.603342
\(890\) 0 0
\(891\) 2.53810e6 8.38875e6i 0.107106 0.354000i
\(892\) 0 0
\(893\) −2.16298e7 −0.907662
\(894\) 0 0
\(895\) 2.10982e7i 0.880415i
\(896\) 0 0
\(897\) 2.10814e7 + 1.55124e6i 0.874820 + 0.0643722i
\(898\) 0 0
\(899\) 2.95659e7i 1.22009i
\(900\) 0 0
\(901\) 1.06778e7i 0.438199i
\(902\) 0 0
\(903\) 2.90841e7 + 2.14011e6i 1.18696 + 0.0873406i
\(904\) 0 0
\(905\) 3.17073e7i 1.28688i
\(906\) 0 0
\(907\) −4.36597e7 −1.76223 −0.881115 0.472902i \(-0.843206\pi\)
−0.881115 + 0.472902i \(0.843206\pi\)
\(908\) 0 0
\(909\) −3.07383e7 4.54829e6i −1.23387 0.182574i
\(910\) 0 0
\(911\) −3.48211e7 −1.39010 −0.695051 0.718960i \(-0.744618\pi\)
−0.695051 + 0.718960i \(0.744618\pi\)
\(912\) 0 0
\(913\) 6.91830e6 0.274677
\(914\) 0 0
\(915\) 6.08430e6 + 447703.i 0.240247 + 0.0176782i
\(916\) 0 0
\(917\) −1.72041e7 −0.675631
\(918\) 0 0
\(919\) 2.48938e7i 0.972305i 0.873874 + 0.486152i \(0.161600\pi\)
−0.873874 + 0.486152i \(0.838400\pi\)
\(920\) 0 0
\(921\) −2.04758e6 + 2.78267e7i −0.0795411 + 1.08097i
\(922\) 0 0
\(923\) 3.93087e6i 0.151875i
\(924\) 0 0
\(925\) 2.22049e7i 0.853286i
\(926\) 0 0
\(927\) −3.89203e6 + 2.63032e7i −0.148757 + 1.00533i
\(928\) 0 0
\(929\) 7.53134e6i 0.286308i 0.989700 + 0.143154i \(0.0457243\pi\)
−0.989700 + 0.143154i \(0.954276\pi\)
\(930\) 0 0
\(931\) −1.16509e7 −0.440541
\(932\) 0 0
\(933\) −2.10339e6 + 2.85851e7i −0.0791070 + 1.07507i
\(934\) 0 0
\(935\) 2.34602e7 0.877612
\(936\) 0 0
\(937\) −3.35884e7 −1.24980 −0.624899 0.780706i \(-0.714860\pi\)
−0.624899 + 0.780706i \(0.714860\pi\)
\(938\) 0 0
\(939\) 2.83664e6 3.85500e7i 0.104988 1.42679i
\(940\) 0 0
\(941\) −1.89573e7 −0.697915 −0.348958 0.937138i \(-0.613464\pi\)
−0.348958 + 0.937138i \(0.613464\pi\)
\(942\) 0 0
\(943\) 5.18343e6i 0.189818i
\(944\) 0 0
\(945\) −2.66262e7 5.96401e6i −0.969906 0.217250i
\(946\) 0 0
\(947\) 3.39256e7i 1.22929i 0.788806 + 0.614643i \(0.210700\pi\)
−0.788806 + 0.614643i \(0.789300\pi\)
\(948\) 0 0
\(949\) 2.33689e7i 0.842313i
\(950\) 0 0
\(951\) 701804. 9.53753e6i 0.0251631 0.341967i
\(952\) 0 0
\(953\) 2.08981e7i 0.745373i 0.927957 + 0.372687i \(0.121563\pi\)
−0.927957 + 0.372687i \(0.878437\pi\)
\(954\) 0 0
\(955\) −2.23104e7 −0.791588
\(956\) 0 0
\(957\) 1.33200e7 + 980133.i 0.470138 + 0.0345944i
\(958\) 0 0
\(959\) −1.38823e7 −0.487434
\(960\) 0 0
\(961\) 2.39649e6 0.0837082
\(962\) 0 0
\(963\) 3.64850e6 2.46573e7i 0.126779 0.856802i
\(964\) 0 0
\(965\) −4.60794e7 −1.59290
\(966\) 0 0
\(967\) 9.33064e6i 0.320882i −0.987045 0.160441i \(-0.948708\pi\)
0.987045 0.160441i \(-0.0512916\pi\)
\(968\) 0 0
\(969\) 5.95324e7 + 4.38060e6i 2.03678 + 0.149873i
\(970\) 0 0
\(971\) 2.63366e7i 0.896419i 0.893929 + 0.448209i \(0.147938\pi\)
−0.893929 + 0.448209i \(0.852062\pi\)
\(972\) 0 0
\(973\) 3.01502e7i 1.02096i
\(974\) 0 0
\(975\) 1.19331e7 + 878080.i 0.402015 + 0.0295816i
\(976\) 0 0
\(977\) 2.77880e7i 0.931366i −0.884952 0.465683i \(-0.845809\pi\)
0.884952 0.465683i \(-0.154191\pi\)
\(978\) 0 0
\(979\) 1.48119e7 0.493918
\(980\) 0 0
\(981\) 2.77412e6 1.87481e7i 0.0920349 0.621992i
\(982\) 0 0
\(983\) −6.72065e6 −0.221834 −0.110917 0.993830i \(-0.535379\pi\)
−0.110917 + 0.993830i \(0.535379\pi\)
\(984\) 0 0
\(985\) 1.93172e7 0.634385
\(986\) 0 0
\(987\) 1.94579e7 + 1.43178e6i 0.635774 + 0.0467824i
\(988\) 0 0
\(989\) 6.62163e7 2.15266
\(990\) 0 0
\(991\) 1.58678e7i 0.513254i 0.966510 + 0.256627i \(0.0826112\pi\)
−0.966510 + 0.256627i \(0.917389\pi\)
\(992\) 0 0
\(993\) 784079. 1.06557e7i 0.0252340 0.342931i
\(994\) 0 0
\(995\) 4.84286e7i 1.55076i
\(996\) 0 0
\(997\) 2.44639e7i 0.779451i 0.920931 + 0.389725i \(0.127430\pi\)
−0.920931 + 0.389725i \(0.872570\pi\)
\(998\) 0 0
\(999\) 4.07675e7 + 9.13152e6i 1.29241 + 0.289487i
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.191.10 yes 20
3.2 odd 2 384.6.f.f.191.9 yes 20
4.3 odd 2 384.6.f.f.191.11 yes 20
8.3 odd 2 384.6.f.f.191.10 yes 20
8.5 even 2 inner 384.6.f.e.191.11 yes 20
12.11 even 2 inner 384.6.f.e.191.12 yes 20
24.5 odd 2 384.6.f.f.191.12 yes 20
24.11 even 2 inner 384.6.f.e.191.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.f.e.191.9 20 24.11 even 2 inner
384.6.f.e.191.10 yes 20 1.1 even 1 trivial
384.6.f.e.191.11 yes 20 8.5 even 2 inner
384.6.f.e.191.12 yes 20 12.11 even 2 inner
384.6.f.f.191.9 yes 20 3.2 odd 2
384.6.f.f.191.10 yes 20 8.3 odd 2
384.6.f.f.191.11 yes 20 4.3 odd 2
384.6.f.f.191.12 yes 20 24.5 odd 2