Properties

Label 384.5.e.c.257.14
Level $384$
Weight $5$
Character 384.257
Analytic conductor $39.694$
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] = [384,5,Mod(257,384)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.257");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.e (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: \(39.6940658242\)
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: \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + \cdots + 21479188203 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 257.14
Root \(-3.05642 - 3.40211i\) of defining polynomial
Character \(\chi\) \(=\) 384.257
Dual form 384.5.e.c.257.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.95805 + 4.20350i) q^{3} -19.3338i q^{5} -67.6413 q^{7} +(45.6611 + 66.9034i) q^{9} +O(q^{10})\) \(q+(7.95805 + 4.20350i) q^{3} -19.3338i q^{5} -67.6413 q^{7} +(45.6611 + 66.9034i) q^{9} -77.9559i q^{11} -79.2239 q^{13} +(81.2697 - 153.859i) q^{15} +309.300i q^{17} +468.110 q^{19} +(-538.293 - 284.331i) q^{21} +617.359i q^{23} +251.204 q^{25} +(82.1450 + 724.357i) q^{27} +939.786i q^{29} +1103.99 q^{31} +(327.688 - 620.377i) q^{33} +1307.77i q^{35} +720.359 q^{37} +(-630.468 - 333.018i) q^{39} +2300.12i q^{41} +2032.68 q^{43} +(1293.50 - 882.804i) q^{45} -2151.47i q^{47} +2174.35 q^{49} +(-1300.14 + 2461.42i) q^{51} +2378.30i q^{53} -1507.19 q^{55} +(3725.25 + 1967.70i) q^{57} -2586.95i q^{59} +3135.05 q^{61} +(-3088.58 - 4525.43i) q^{63} +1531.70i q^{65} -2472.03 q^{67} +(-2595.07 + 4912.97i) q^{69} -7746.80i q^{71} -10238.3 q^{73} +(1999.09 + 1055.94i) q^{75} +5273.04i q^{77} -5314.05 q^{79} +(-2391.12 + 6109.77i) q^{81} +10678.6i q^{83} +5979.95 q^{85} +(-3950.39 + 7478.86i) q^{87} -2024.39i q^{89} +5358.81 q^{91} +(8785.63 + 4640.64i) q^{93} -9050.36i q^{95} +16858.0 q^{97} +(5215.51 - 3559.55i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} - 80 q^{7} - 416 q^{15} - 816 q^{19} + 608 q^{21} - 2000 q^{25} - 280 q^{27} - 592 q^{31} - 496 q^{33} + 2240 q^{37} + 16 q^{39} - 368 q^{43} + 800 q^{45} + 3984 q^{49} + 352 q^{51} - 1920 q^{55} + 560 q^{57} - 3520 q^{61} + 816 q^{63} + 3536 q^{67} - 10784 q^{69} + 3680 q^{73} - 5112 q^{75} + 14448 q^{79} - 624 q^{81} - 11136 q^{85} + 14944 q^{87} + 22944 q^{91} + 13760 q^{93} + 3264 q^{97} + 26976 q^{99}+O(q^{100}) \) 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
\(3\) 7.95805 + 4.20350i 0.884228 + 0.467056i
\(4\) 0 0
\(5\) 19.3338i 0.773353i −0.922216 0.386676i \(-0.873623\pi\)
0.922216 0.386676i \(-0.126377\pi\)
\(6\) 0 0
\(7\) −67.6413 −1.38044 −0.690218 0.723602i \(-0.742485\pi\)
−0.690218 + 0.723602i \(0.742485\pi\)
\(8\) 0 0
\(9\) 45.6611 + 66.9034i 0.563718 + 0.825968i
\(10\) 0 0
\(11\) 77.9559i 0.644264i −0.946695 0.322132i \(-0.895601\pi\)
0.946695 0.322132i \(-0.104399\pi\)
\(12\) 0 0
\(13\) −79.2239 −0.468781 −0.234390 0.972143i \(-0.575309\pi\)
−0.234390 + 0.972143i \(0.575309\pi\)
\(14\) 0 0
\(15\) 81.2697 153.859i 0.361199 0.683820i
\(16\) 0 0
\(17\) 309.300i 1.07024i 0.844775 + 0.535121i \(0.179734\pi\)
−0.844775 + 0.535121i \(0.820266\pi\)
\(18\) 0 0
\(19\) 468.110 1.29670 0.648352 0.761340i \(-0.275458\pi\)
0.648352 + 0.761340i \(0.275458\pi\)
\(20\) 0 0
\(21\) −538.293 284.331i −1.22062 0.644741i
\(22\) 0 0
\(23\) 617.359i 1.16703i 0.812102 + 0.583515i \(0.198323\pi\)
−0.812102 + 0.583515i \(0.801677\pi\)
\(24\) 0 0
\(25\) 251.204 0.401926
\(26\) 0 0
\(27\) 82.1450 + 724.357i 0.112682 + 0.993631i
\(28\) 0 0
\(29\) 939.786i 1.11746i 0.829349 + 0.558731i \(0.188712\pi\)
−0.829349 + 0.558731i \(0.811288\pi\)
\(30\) 0 0
\(31\) 1103.99 1.14880 0.574398 0.818576i \(-0.305236\pi\)
0.574398 + 0.818576i \(0.305236\pi\)
\(32\) 0 0
\(33\) 327.688 620.377i 0.300907 0.569676i
\(34\) 0 0
\(35\) 1307.77i 1.06756i
\(36\) 0 0
\(37\) 720.359 0.526193 0.263097 0.964769i \(-0.415256\pi\)
0.263097 + 0.964769i \(0.415256\pi\)
\(38\) 0 0
\(39\) −630.468 333.018i −0.414509 0.218947i
\(40\) 0 0
\(41\) 2300.12i 1.36830i 0.729340 + 0.684151i \(0.239827\pi\)
−0.729340 + 0.684151i \(0.760173\pi\)
\(42\) 0 0
\(43\) 2032.68 1.09934 0.549669 0.835382i \(-0.314754\pi\)
0.549669 + 0.835382i \(0.314754\pi\)
\(44\) 0 0
\(45\) 1293.50 882.804i 0.638764 0.435952i
\(46\) 0 0
\(47\) 2151.47i 0.973958i −0.873414 0.486979i \(-0.838099\pi\)
0.873414 0.486979i \(-0.161901\pi\)
\(48\) 0 0
\(49\) 2174.35 0.905602
\(50\) 0 0
\(51\) −1300.14 + 2461.42i −0.499863 + 0.946338i
\(52\) 0 0
\(53\) 2378.30i 0.846671i 0.905973 + 0.423335i \(0.139141\pi\)
−0.905973 + 0.423335i \(0.860859\pi\)
\(54\) 0 0
\(55\) −1507.19 −0.498243
\(56\) 0 0
\(57\) 3725.25 + 1967.70i 1.14658 + 0.605634i
\(58\) 0 0
\(59\) 2586.95i 0.743162i −0.928401 0.371581i \(-0.878816\pi\)
0.928401 0.371581i \(-0.121184\pi\)
\(60\) 0 0
\(61\) 3135.05 0.842529 0.421265 0.906938i \(-0.361586\pi\)
0.421265 + 0.906938i \(0.361586\pi\)
\(62\) 0 0
\(63\) −3088.58 4525.43i −0.778176 1.14019i
\(64\) 0 0
\(65\) 1531.70i 0.362533i
\(66\) 0 0
\(67\) −2472.03 −0.550687 −0.275343 0.961346i \(-0.588792\pi\)
−0.275343 + 0.961346i \(0.588792\pi\)
\(68\) 0 0
\(69\) −2595.07 + 4912.97i −0.545068 + 1.03192i
\(70\) 0 0
\(71\) 7746.80i 1.53676i −0.639994 0.768380i \(-0.721063\pi\)
0.639994 0.768380i \(-0.278937\pi\)
\(72\) 0 0
\(73\) −10238.3 −1.92123 −0.960617 0.277877i \(-0.910369\pi\)
−0.960617 + 0.277877i \(0.910369\pi\)
\(74\) 0 0
\(75\) 1999.09 + 1055.94i 0.355394 + 0.187722i
\(76\) 0 0
\(77\) 5273.04i 0.889365i
\(78\) 0 0
\(79\) −5314.05 −0.851474 −0.425737 0.904847i \(-0.639985\pi\)
−0.425737 + 0.904847i \(0.639985\pi\)
\(80\) 0 0
\(81\) −2391.12 + 6109.77i −0.364445 + 0.931225i
\(82\) 0 0
\(83\) 10678.6i 1.55010i 0.631903 + 0.775048i \(0.282274\pi\)
−0.631903 + 0.775048i \(0.717726\pi\)
\(84\) 0 0
\(85\) 5979.95 0.827675
\(86\) 0 0
\(87\) −3950.39 + 7478.86i −0.521917 + 0.988091i
\(88\) 0 0
\(89\) 2024.39i 0.255572i −0.991802 0.127786i \(-0.959213\pi\)
0.991802 0.127786i \(-0.0407871\pi\)
\(90\) 0 0
\(91\) 5358.81 0.647121
\(92\) 0 0
\(93\) 8785.63 + 4640.64i 1.01580 + 0.536552i
\(94\) 0 0
\(95\) 9050.36i 1.00281i
\(96\) 0 0
\(97\) 16858.0 1.79169 0.895846 0.444365i \(-0.146571\pi\)
0.895846 + 0.444365i \(0.146571\pi\)
\(98\) 0 0
\(99\) 5215.51 3559.55i 0.532141 0.363183i
\(100\) 0 0
\(101\) 19505.3i 1.91210i 0.293209 + 0.956048i \(0.405277\pi\)
−0.293209 + 0.956048i \(0.594723\pi\)
\(102\) 0 0
\(103\) −8458.43 −0.797288 −0.398644 0.917106i \(-0.630519\pi\)
−0.398644 + 0.917106i \(0.630519\pi\)
\(104\) 0 0
\(105\) −5497.19 + 10407.3i −0.498612 + 0.943969i
\(106\) 0 0
\(107\) 11493.8i 1.00392i 0.864892 + 0.501958i \(0.167387\pi\)
−0.864892 + 0.501958i \(0.832613\pi\)
\(108\) 0 0
\(109\) 5163.95 0.434639 0.217319 0.976101i \(-0.430269\pi\)
0.217319 + 0.976101i \(0.430269\pi\)
\(110\) 0 0
\(111\) 5732.65 + 3028.03i 0.465275 + 0.245762i
\(112\) 0 0
\(113\) 4118.50i 0.322539i −0.986910 0.161269i \(-0.948441\pi\)
0.986910 0.161269i \(-0.0515588\pi\)
\(114\) 0 0
\(115\) 11935.9 0.902526
\(116\) 0 0
\(117\) −3617.45 5300.35i −0.264260 0.387198i
\(118\) 0 0
\(119\) 20921.5i 1.47740i
\(120\) 0 0
\(121\) 8563.88 0.584924
\(122\) 0 0
\(123\) −9668.55 + 18304.4i −0.639074 + 1.20989i
\(124\) 0 0
\(125\) 16940.4i 1.08418i
\(126\) 0 0
\(127\) 6955.17 0.431221 0.215611 0.976479i \(-0.430826\pi\)
0.215611 + 0.976479i \(0.430826\pi\)
\(128\) 0 0
\(129\) 16176.1 + 8544.36i 0.972066 + 0.513453i
\(130\) 0 0
\(131\) 12734.3i 0.742048i −0.928623 0.371024i \(-0.879007\pi\)
0.928623 0.371024i \(-0.120993\pi\)
\(132\) 0 0
\(133\) −31663.6 −1.79002
\(134\) 0 0
\(135\) 14004.6 1588.18i 0.768427 0.0871427i
\(136\) 0 0
\(137\) 23048.5i 1.22801i 0.789302 + 0.614006i \(0.210443\pi\)
−0.789302 + 0.614006i \(0.789557\pi\)
\(138\) 0 0
\(139\) −13779.7 −0.713199 −0.356600 0.934257i \(-0.616064\pi\)
−0.356600 + 0.934257i \(0.616064\pi\)
\(140\) 0 0
\(141\) 9043.72 17121.5i 0.454893 0.861200i
\(142\) 0 0
\(143\) 6175.97i 0.302018i
\(144\) 0 0
\(145\) 18169.6 0.864192
\(146\) 0 0
\(147\) 17303.6 + 9139.89i 0.800759 + 0.422967i
\(148\) 0 0
\(149\) 861.178i 0.0387901i 0.999812 + 0.0193950i \(0.00617402\pi\)
−0.999812 + 0.0193950i \(0.993826\pi\)
\(150\) 0 0
\(151\) 11857.7 0.520052 0.260026 0.965602i \(-0.416269\pi\)
0.260026 + 0.965602i \(0.416269\pi\)
\(152\) 0 0
\(153\) −20693.2 + 14123.0i −0.883985 + 0.603314i
\(154\) 0 0
\(155\) 21344.4i 0.888424i
\(156\) 0 0
\(157\) 24128.3 0.978874 0.489437 0.872039i \(-0.337202\pi\)
0.489437 + 0.872039i \(0.337202\pi\)
\(158\) 0 0
\(159\) −9997.18 + 18926.6i −0.395443 + 0.748650i
\(160\) 0 0
\(161\) 41759.0i 1.61101i
\(162\) 0 0
\(163\) −18099.6 −0.681230 −0.340615 0.940203i \(-0.610635\pi\)
−0.340615 + 0.940203i \(0.610635\pi\)
\(164\) 0 0
\(165\) −11994.3 6335.46i −0.440560 0.232707i
\(166\) 0 0
\(167\) 13135.2i 0.470981i 0.971877 + 0.235490i \(0.0756696\pi\)
−0.971877 + 0.235490i \(0.924330\pi\)
\(168\) 0 0
\(169\) −22284.6 −0.780245
\(170\) 0 0
\(171\) 21374.5 + 31318.2i 0.730975 + 1.07104i
\(172\) 0 0
\(173\) 37599.9i 1.25630i 0.778092 + 0.628151i \(0.216188\pi\)
−0.778092 + 0.628151i \(0.783812\pi\)
\(174\) 0 0
\(175\) −16991.8 −0.554833
\(176\) 0 0
\(177\) 10874.2 20587.1i 0.347098 0.657125i
\(178\) 0 0
\(179\) 30293.4i 0.945456i −0.881208 0.472728i \(-0.843269\pi\)
0.881208 0.472728i \(-0.156731\pi\)
\(180\) 0 0
\(181\) −41419.0 −1.26428 −0.632138 0.774855i \(-0.717823\pi\)
−0.632138 + 0.774855i \(0.717823\pi\)
\(182\) 0 0
\(183\) 24948.9 + 13178.2i 0.744988 + 0.393508i
\(184\) 0 0
\(185\) 13927.3i 0.406933i
\(186\) 0 0
\(187\) 24111.8 0.689518
\(188\) 0 0
\(189\) −5556.40 48996.5i −0.155550 1.37164i
\(190\) 0 0
\(191\) 26217.3i 0.718655i 0.933211 + 0.359328i \(0.116994\pi\)
−0.933211 + 0.359328i \(0.883006\pi\)
\(192\) 0 0
\(193\) −24871.7 −0.667716 −0.333858 0.942623i \(-0.608351\pi\)
−0.333858 + 0.942623i \(0.608351\pi\)
\(194\) 0 0
\(195\) −6438.51 + 12189.4i −0.169323 + 0.320561i
\(196\) 0 0
\(197\) 44861.5i 1.15595i 0.816053 + 0.577977i \(0.196158\pi\)
−0.816053 + 0.577977i \(0.803842\pi\)
\(198\) 0 0
\(199\) −1782.37 −0.0450081 −0.0225041 0.999747i \(-0.507164\pi\)
−0.0225041 + 0.999747i \(0.507164\pi\)
\(200\) 0 0
\(201\) −19672.6 10391.2i −0.486932 0.257201i
\(202\) 0 0
\(203\) 63568.4i 1.54258i
\(204\) 0 0
\(205\) 44470.0 1.05818
\(206\) 0 0
\(207\) −41303.4 + 28189.3i −0.963929 + 0.657876i
\(208\) 0 0
\(209\) 36492.0i 0.835420i
\(210\) 0 0
\(211\) −27516.8 −0.618064 −0.309032 0.951052i \(-0.600005\pi\)
−0.309032 + 0.951052i \(0.600005\pi\)
\(212\) 0 0
\(213\) 32563.7 61649.5i 0.717753 1.35885i
\(214\) 0 0
\(215\) 39299.4i 0.850176i
\(216\) 0 0
\(217\) −74675.5 −1.58584
\(218\) 0 0
\(219\) −81476.5 43036.5i −1.69881 0.897323i
\(220\) 0 0
\(221\) 24504.0i 0.501709i
\(222\) 0 0
\(223\) 58322.1 1.17280 0.586399 0.810023i \(-0.300545\pi\)
0.586399 + 0.810023i \(0.300545\pi\)
\(224\) 0 0
\(225\) 11470.2 + 16806.4i 0.226573 + 0.331978i
\(226\) 0 0
\(227\) 96363.0i 1.87007i −0.354552 0.935036i \(-0.615367\pi\)
0.354552 0.935036i \(-0.384633\pi\)
\(228\) 0 0
\(229\) −63619.0 −1.21315 −0.606577 0.795025i \(-0.707458\pi\)
−0.606577 + 0.795025i \(0.707458\pi\)
\(230\) 0 0
\(231\) −22165.2 + 41963.1i −0.415383 + 0.786401i
\(232\) 0 0
\(233\) 104862.i 1.93155i −0.259380 0.965775i \(-0.583518\pi\)
0.259380 0.965775i \(-0.416482\pi\)
\(234\) 0 0
\(235\) −41596.2 −0.753213
\(236\) 0 0
\(237\) −42289.4 22337.6i −0.752897 0.397686i
\(238\) 0 0
\(239\) 54132.9i 0.947688i −0.880609 0.473844i \(-0.842866\pi\)
0.880609 0.473844i \(-0.157134\pi\)
\(240\) 0 0
\(241\) 18650.0 0.321103 0.160552 0.987027i \(-0.448673\pi\)
0.160552 + 0.987027i \(0.448673\pi\)
\(242\) 0 0
\(243\) −44711.1 + 38570.7i −0.757186 + 0.653199i
\(244\) 0 0
\(245\) 42038.5i 0.700350i
\(246\) 0 0
\(247\) −37085.6 −0.607870
\(248\) 0 0
\(249\) −44887.6 + 84980.9i −0.723981 + 1.37064i
\(250\) 0 0
\(251\) 1998.43i 0.0317206i 0.999874 + 0.0158603i \(0.00504871\pi\)
−0.999874 + 0.0158603i \(0.994951\pi\)
\(252\) 0 0
\(253\) 48126.8 0.751875
\(254\) 0 0
\(255\) 47588.7 + 25136.7i 0.731853 + 0.386570i
\(256\) 0 0
\(257\) 68298.2i 1.03405i −0.855969 0.517027i \(-0.827039\pi\)
0.855969 0.517027i \(-0.172961\pi\)
\(258\) 0 0
\(259\) −48726.0 −0.726376
\(260\) 0 0
\(261\) −62874.8 + 42911.7i −0.922987 + 0.629933i
\(262\) 0 0
\(263\) 84260.2i 1.21818i 0.793102 + 0.609089i \(0.208465\pi\)
−0.793102 + 0.609089i \(0.791535\pi\)
\(264\) 0 0
\(265\) 45981.6 0.654775
\(266\) 0 0
\(267\) 8509.52 16110.2i 0.119367 0.225984i
\(268\) 0 0
\(269\) 84312.3i 1.16516i −0.812773 0.582581i \(-0.802043\pi\)
0.812773 0.582581i \(-0.197957\pi\)
\(270\) 0 0
\(271\) 97886.0 1.33285 0.666426 0.745571i \(-0.267823\pi\)
0.666426 + 0.745571i \(0.267823\pi\)
\(272\) 0 0
\(273\) 42645.7 + 22525.8i 0.572203 + 0.302242i
\(274\) 0 0
\(275\) 19582.8i 0.258946i
\(276\) 0 0
\(277\) 104604. 1.36329 0.681646 0.731682i \(-0.261264\pi\)
0.681646 + 0.731682i \(0.261264\pi\)
\(278\) 0 0
\(279\) 50409.6 + 73860.8i 0.647596 + 0.948868i
\(280\) 0 0
\(281\) 76354.4i 0.966988i −0.875348 0.483494i \(-0.839367\pi\)
0.875348 0.483494i \(-0.160633\pi\)
\(282\) 0 0
\(283\) 129472. 1.61660 0.808300 0.588771i \(-0.200388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(284\) 0 0
\(285\) 38043.2 72023.2i 0.468368 0.886713i
\(286\) 0 0
\(287\) 155583.i 1.88885i
\(288\) 0 0
\(289\) −12145.5 −0.145418
\(290\) 0 0
\(291\) 134157. + 70862.8i 1.58426 + 0.836820i
\(292\) 0 0
\(293\) 100946.i 1.17585i 0.808915 + 0.587926i \(0.200055\pi\)
−0.808915 + 0.587926i \(0.799945\pi\)
\(294\) 0 0
\(295\) −50015.6 −0.574726
\(296\) 0 0
\(297\) 56467.9 6403.69i 0.640161 0.0725968i
\(298\) 0 0
\(299\) 48909.6i 0.547081i
\(300\) 0 0
\(301\) −137493. −1.51757
\(302\) 0 0
\(303\) −81990.6 + 155224.i −0.893056 + 1.69073i
\(304\) 0 0
\(305\) 60612.5i 0.651572i
\(306\) 0 0
\(307\) 11808.0 0.125285 0.0626426 0.998036i \(-0.480047\pi\)
0.0626426 + 0.998036i \(0.480047\pi\)
\(308\) 0 0
\(309\) −67312.6 35555.0i −0.704985 0.372378i
\(310\) 0 0
\(311\) 36362.0i 0.375947i 0.982174 + 0.187974i \(0.0601920\pi\)
−0.982174 + 0.187974i \(0.939808\pi\)
\(312\) 0 0
\(313\) −66566.3 −0.679463 −0.339731 0.940522i \(-0.610336\pi\)
−0.339731 + 0.940522i \(0.610336\pi\)
\(314\) 0 0
\(315\) −87493.9 + 59714.0i −0.881773 + 0.601804i
\(316\) 0 0
\(317\) 105833.i 1.05318i 0.850119 + 0.526591i \(0.176530\pi\)
−0.850119 + 0.526591i \(0.823470\pi\)
\(318\) 0 0
\(319\) 73261.8 0.719940
\(320\) 0 0
\(321\) −48314.4 + 91468.5i −0.468885 + 0.887691i
\(322\) 0 0
\(323\) 144787.i 1.38779i
\(324\) 0 0
\(325\) −19901.3 −0.188415
\(326\) 0 0
\(327\) 41094.9 + 21706.7i 0.384320 + 0.203001i
\(328\) 0 0
\(329\) 145528.i 1.34449i
\(330\) 0 0
\(331\) 58819.5 0.536865 0.268432 0.963298i \(-0.413494\pi\)
0.268432 + 0.963298i \(0.413494\pi\)
\(332\) 0 0
\(333\) 32892.4 + 48194.4i 0.296624 + 0.434619i
\(334\) 0 0
\(335\) 47793.8i 0.425875i
\(336\) 0 0
\(337\) 19308.9 0.170020 0.0850098 0.996380i \(-0.472908\pi\)
0.0850098 + 0.996380i \(0.472908\pi\)
\(338\) 0 0
\(339\) 17312.1 32775.2i 0.150644 0.285198i
\(340\) 0 0
\(341\) 86062.8i 0.740127i
\(342\) 0 0
\(343\) 15330.9 0.130310
\(344\) 0 0
\(345\) 94986.5 + 50172.6i 0.798039 + 0.421530i
\(346\) 0 0
\(347\) 49750.9i 0.413182i −0.978427 0.206591i \(-0.933763\pi\)
0.978427 0.206591i \(-0.0662370\pi\)
\(348\) 0 0
\(349\) −149030. −1.22355 −0.611777 0.791030i \(-0.709545\pi\)
−0.611777 + 0.791030i \(0.709545\pi\)
\(350\) 0 0
\(351\) −6507.85 57386.4i −0.0528230 0.465795i
\(352\) 0 0
\(353\) 98810.0i 0.792960i 0.918043 + 0.396480i \(0.129768\pi\)
−0.918043 + 0.396480i \(0.870232\pi\)
\(354\) 0 0
\(355\) −149775. −1.18846
\(356\) 0 0
\(357\) 87943.4 166494.i 0.690028 1.30636i
\(358\) 0 0
\(359\) 233349.i 1.81058i −0.424798 0.905288i \(-0.639655\pi\)
0.424798 0.905288i \(-0.360345\pi\)
\(360\) 0 0
\(361\) 88806.4 0.681444
\(362\) 0 0
\(363\) 68151.7 + 35998.3i 0.517206 + 0.273192i
\(364\) 0 0
\(365\) 197944.i 1.48579i
\(366\) 0 0
\(367\) −187106. −1.38917 −0.694587 0.719409i \(-0.744413\pi\)
−0.694587 + 0.719409i \(0.744413\pi\)
\(368\) 0 0
\(369\) −153886. + 105026.i −1.13017 + 0.771336i
\(370\) 0 0
\(371\) 160871.i 1.16877i
\(372\) 0 0
\(373\) 33982.5 0.244252 0.122126 0.992515i \(-0.461029\pi\)
0.122126 + 0.992515i \(0.461029\pi\)
\(374\) 0 0
\(375\) 71208.8 134812.i 0.506374 0.958665i
\(376\) 0 0
\(377\) 74453.5i 0.523845i
\(378\) 0 0
\(379\) 150006. 1.04431 0.522156 0.852850i \(-0.325128\pi\)
0.522156 + 0.852850i \(0.325128\pi\)
\(380\) 0 0
\(381\) 55349.6 + 29236.1i 0.381298 + 0.201405i
\(382\) 0 0
\(383\) 71272.7i 0.485876i −0.970042 0.242938i \(-0.921889\pi\)
0.970042 0.242938i \(-0.0781112\pi\)
\(384\) 0 0
\(385\) 101948. 0.687792
\(386\) 0 0
\(387\) 92814.3 + 135993.i 0.619717 + 0.908018i
\(388\) 0 0
\(389\) 90246.1i 0.596388i 0.954505 + 0.298194i \(0.0963843\pi\)
−0.954505 + 0.298194i \(0.903616\pi\)
\(390\) 0 0
\(391\) −190949. −1.24901
\(392\) 0 0
\(393\) 53528.6 101340.i 0.346578 0.656139i
\(394\) 0 0
\(395\) 102741.i 0.658489i
\(396\) 0 0
\(397\) 145813. 0.925158 0.462579 0.886578i \(-0.346924\pi\)
0.462579 + 0.886578i \(0.346924\pi\)
\(398\) 0 0
\(399\) −251981. 133098.i −1.58278 0.836038i
\(400\) 0 0
\(401\) 108331.i 0.673695i 0.941559 + 0.336848i \(0.109361\pi\)
−0.941559 + 0.336848i \(0.890639\pi\)
\(402\) 0 0
\(403\) −87462.6 −0.538533
\(404\) 0 0
\(405\) 118125. + 46229.5i 0.720165 + 0.281844i
\(406\) 0 0
\(407\) 56156.2i 0.339007i
\(408\) 0 0
\(409\) 294079. 1.75800 0.878998 0.476825i \(-0.158213\pi\)
0.878998 + 0.476825i \(0.158213\pi\)
\(410\) 0 0
\(411\) −96884.6 + 183421.i −0.573550 + 1.08584i
\(412\) 0 0
\(413\) 174985.i 1.02589i
\(414\) 0 0
\(415\) 206458. 1.19877
\(416\) 0 0
\(417\) −109660. 57923.1i −0.630631 0.333104i
\(418\) 0 0
\(419\) 17811.7i 0.101456i −0.998713 0.0507281i \(-0.983846\pi\)
0.998713 0.0507281i \(-0.0161542\pi\)
\(420\) 0 0
\(421\) 349782. 1.97348 0.986741 0.162304i \(-0.0518924\pi\)
0.986741 + 0.162304i \(0.0518924\pi\)
\(422\) 0 0
\(423\) 143941. 98238.6i 0.804457 0.549037i
\(424\) 0 0
\(425\) 77697.3i 0.430158i
\(426\) 0 0
\(427\) −212059. −1.16306
\(428\) 0 0
\(429\) −25960.7 + 49148.7i −0.141059 + 0.267053i
\(430\) 0 0
\(431\) 149894.i 0.806920i 0.914997 + 0.403460i \(0.132193\pi\)
−0.914997 + 0.403460i \(0.867807\pi\)
\(432\) 0 0
\(433\) −248766. −1.32683 −0.663415 0.748252i \(-0.730893\pi\)
−0.663415 + 0.748252i \(0.730893\pi\)
\(434\) 0 0
\(435\) 144595. + 76376.1i 0.764143 + 0.403626i
\(436\) 0 0
\(437\) 288992.i 1.51329i
\(438\) 0 0
\(439\) −120416. −0.624822 −0.312411 0.949947i \(-0.601137\pi\)
−0.312411 + 0.949947i \(0.601137\pi\)
\(440\) 0 0
\(441\) 99283.3 + 145471.i 0.510504 + 0.747998i
\(442\) 0 0
\(443\) 168777.i 0.860014i 0.902826 + 0.430007i \(0.141489\pi\)
−0.902826 + 0.430007i \(0.858511\pi\)
\(444\) 0 0
\(445\) −39139.1 −0.197647
\(446\) 0 0
\(447\) −3619.97 + 6853.30i −0.0181171 + 0.0342993i
\(448\) 0 0
\(449\) 1278.50i 0.00634173i 0.999995 + 0.00317087i \(0.00100932\pi\)
−0.999995 + 0.00317087i \(0.998991\pi\)
\(450\) 0 0
\(451\) 179308. 0.881548
\(452\) 0 0
\(453\) 94364.2 + 49843.9i 0.459844 + 0.242893i
\(454\) 0 0
\(455\) 103606.i 0.500453i
\(456\) 0 0
\(457\) −202856. −0.971306 −0.485653 0.874152i \(-0.661418\pi\)
−0.485653 + 0.874152i \(0.661418\pi\)
\(458\) 0 0
\(459\) −224044. + 25407.5i −1.06343 + 0.120597i
\(460\) 0 0
\(461\) 65727.6i 0.309276i 0.987971 + 0.154638i \(0.0494211\pi\)
−0.987971 + 0.154638i \(0.950579\pi\)
\(462\) 0 0
\(463\) −266165. −1.24162 −0.620812 0.783960i \(-0.713197\pi\)
−0.620812 + 0.783960i \(0.713197\pi\)
\(464\) 0 0
\(465\) 89721.2 169860.i 0.414944 0.785569i
\(466\) 0 0
\(467\) 83490.5i 0.382828i 0.981509 + 0.191414i \(0.0613072\pi\)
−0.981509 + 0.191414i \(0.938693\pi\)
\(468\) 0 0
\(469\) 167212. 0.760187
\(470\) 0 0
\(471\) 192014. + 101423.i 0.865547 + 0.457189i
\(472\) 0 0
\(473\) 158459.i 0.708264i
\(474\) 0 0
\(475\) 117591. 0.521179
\(476\) 0 0
\(477\) −159116. + 108596.i −0.699323 + 0.477283i
\(478\) 0 0
\(479\) 203102.i 0.885205i −0.896718 0.442603i \(-0.854055\pi\)
0.896718 0.442603i \(-0.145945\pi\)
\(480\) 0 0
\(481\) −57069.6 −0.246669
\(482\) 0 0
\(483\) 175534. 332320.i 0.752432 1.42450i
\(484\) 0 0
\(485\) 325930.i 1.38561i
\(486\) 0 0
\(487\) −360891. −1.52166 −0.760831 0.648950i \(-0.775208\pi\)
−0.760831 + 0.648950i \(0.775208\pi\)
\(488\) 0 0
\(489\) −144038. 76081.7i −0.602362 0.318172i
\(490\) 0 0
\(491\) 102193.i 0.423895i 0.977281 + 0.211947i \(0.0679805\pi\)
−0.977281 + 0.211947i \(0.932019\pi\)
\(492\) 0 0
\(493\) −290676. −1.19596
\(494\) 0 0
\(495\) −68819.8 100836.i −0.280868 0.411533i
\(496\) 0 0
\(497\) 524004.i 2.12140i
\(498\) 0 0
\(499\) −76271.0 −0.306308 −0.153154 0.988202i \(-0.548943\pi\)
−0.153154 + 0.988202i \(0.548943\pi\)
\(500\) 0 0
\(501\) −55213.8 + 104530.i −0.219974 + 0.416454i
\(502\) 0 0
\(503\) 285141.i 1.12700i −0.826116 0.563500i \(-0.809454\pi\)
0.826116 0.563500i \(-0.190546\pi\)
\(504\) 0 0
\(505\) 377112. 1.47873
\(506\) 0 0
\(507\) −177342. 93673.3i −0.689914 0.364418i
\(508\) 0 0
\(509\) 169808.i 0.655425i −0.944777 0.327713i \(-0.893722\pi\)
0.944777 0.327713i \(-0.106278\pi\)
\(510\) 0 0
\(511\) 692529. 2.65214
\(512\) 0 0
\(513\) 38452.9 + 339079.i 0.146115 + 1.28845i
\(514\) 0 0
\(515\) 163534.i 0.616585i
\(516\) 0 0
\(517\) −167720. −0.627486
\(518\) 0 0
\(519\) −158051. + 299222.i −0.586763 + 1.11086i
\(520\) 0 0
\(521\) 169037.i 0.622738i 0.950289 + 0.311369i \(0.100787\pi\)
−0.950289 + 0.311369i \(0.899213\pi\)
\(522\) 0 0
\(523\) 467508. 1.70917 0.854586 0.519309i \(-0.173811\pi\)
0.854586 + 0.519309i \(0.173811\pi\)
\(524\) 0 0
\(525\) −135221. 71424.9i −0.490598 0.259138i
\(526\) 0 0
\(527\) 341465.i 1.22949i
\(528\) 0 0
\(529\) −101291. −0.361960
\(530\) 0 0
\(531\) 173076. 118123.i 0.613828 0.418934i
\(532\) 0 0
\(533\) 182224.i 0.641434i
\(534\) 0 0
\(535\) 222220. 0.776381
\(536\) 0 0
\(537\) 127338. 241076.i 0.441581 0.835999i
\(538\) 0 0
\(539\) 169504.i 0.583447i
\(540\) 0 0
\(541\) −228195. −0.779673 −0.389837 0.920884i \(-0.627468\pi\)
−0.389837 + 0.920884i \(0.627468\pi\)
\(542\) 0 0
\(543\) −329614. 174105.i −1.11791 0.590488i
\(544\) 0 0
\(545\) 99838.8i 0.336129i
\(546\) 0 0
\(547\) −190129. −0.635440 −0.317720 0.948185i \(-0.602917\pi\)
−0.317720 + 0.948185i \(0.602917\pi\)
\(548\) 0 0
\(549\) 143150. + 209745.i 0.474948 + 0.695902i
\(550\) 0 0
\(551\) 439923.i 1.44902i
\(552\) 0 0
\(553\) 359449. 1.17540
\(554\) 0 0
\(555\) 58543.4 110834.i 0.190060 0.359821i
\(556\) 0 0
\(557\) 183226.i 0.590579i −0.955408 0.295289i \(-0.904584\pi\)
0.955408 0.295289i \(-0.0954160\pi\)
\(558\) 0 0
\(559\) −161037. −0.515349
\(560\) 0 0
\(561\) 191883. + 101354.i 0.609691 + 0.322044i
\(562\) 0 0
\(563\) 339876.i 1.07227i 0.844133 + 0.536134i \(0.180116\pi\)
−0.844133 + 0.536134i \(0.819884\pi\)
\(564\) 0 0
\(565\) −79626.3 −0.249436
\(566\) 0 0
\(567\) 161739. 413273.i 0.503093 1.28550i
\(568\) 0 0
\(569\) 263919.i 0.815165i −0.913168 0.407582i \(-0.866372\pi\)
0.913168 0.407582i \(-0.133628\pi\)
\(570\) 0 0
\(571\) 168475. 0.516730 0.258365 0.966047i \(-0.416816\pi\)
0.258365 + 0.966047i \(0.416816\pi\)
\(572\) 0 0
\(573\) −110204. + 208638.i −0.335652 + 0.635455i
\(574\) 0 0
\(575\) 155083.i 0.469060i
\(576\) 0 0
\(577\) −452005. −1.35766 −0.678831 0.734295i \(-0.737513\pi\)
−0.678831 + 0.734295i \(0.737513\pi\)
\(578\) 0 0
\(579\) −197931. 104548.i −0.590413 0.311860i
\(580\) 0 0
\(581\) 722315.i 2.13981i
\(582\) 0 0
\(583\) 185402. 0.545479
\(584\) 0 0
\(585\) −102476. + 69939.2i −0.299440 + 0.204366i
\(586\) 0 0
\(587\) 4585.69i 0.0133085i −0.999978 0.00665424i \(-0.997882\pi\)
0.999978 0.00665424i \(-0.00211813\pi\)
\(588\) 0 0
\(589\) 516791. 1.48965
\(590\) 0 0
\(591\) −188575. + 357010.i −0.539895 + 1.02213i
\(592\) 0 0
\(593\) 277349.i 0.788710i −0.918958 0.394355i \(-0.870968\pi\)
0.918958 0.394355i \(-0.129032\pi\)
\(594\) 0 0
\(595\) −404492. −1.14255
\(596\) 0 0
\(597\) −14184.2 7492.18i −0.0397974 0.0210213i
\(598\) 0 0
\(599\) 354316.i 0.987501i 0.869604 + 0.493751i \(0.164374\pi\)
−0.869604 + 0.493751i \(0.835626\pi\)
\(600\) 0 0
\(601\) −286642. −0.793582 −0.396791 0.917909i \(-0.629876\pi\)
−0.396791 + 0.917909i \(0.629876\pi\)
\(602\) 0 0
\(603\) −112876. 165387.i −0.310432 0.454849i
\(604\) 0 0
\(605\) 165572.i 0.452353i
\(606\) 0 0
\(607\) 690944. 1.87528 0.937639 0.347611i \(-0.113007\pi\)
0.937639 + 0.347611i \(0.113007\pi\)
\(608\) 0 0
\(609\) 267210. 505880.i 0.720473 1.36400i
\(610\) 0 0
\(611\) 170448.i 0.456572i
\(612\) 0 0
\(613\) 139902. 0.372309 0.186154 0.982520i \(-0.440398\pi\)
0.186154 + 0.982520i \(0.440398\pi\)
\(614\) 0 0
\(615\) 353895. + 186930.i 0.935672 + 0.494229i
\(616\) 0 0
\(617\) 280737.i 0.737444i −0.929540 0.368722i \(-0.879795\pi\)
0.929540 0.368722i \(-0.120205\pi\)
\(618\) 0 0
\(619\) 403456. 1.05297 0.526483 0.850185i \(-0.323510\pi\)
0.526483 + 0.850185i \(0.323510\pi\)
\(620\) 0 0
\(621\) −447188. + 50713.0i −1.15960 + 0.131503i
\(622\) 0 0
\(623\) 136932.i 0.352801i
\(624\) 0 0
\(625\) −170519. −0.436530
\(626\) 0 0
\(627\) 153394. 290405.i 0.390188 0.738702i
\(628\) 0 0
\(629\) 222807.i 0.563154i
\(630\) 0 0
\(631\) 124703. 0.313198 0.156599 0.987662i \(-0.449947\pi\)
0.156599 + 0.987662i \(0.449947\pi\)
\(632\) 0 0
\(633\) −218980. 115667.i −0.546509 0.288670i
\(634\) 0 0
\(635\) 134470.i 0.333486i
\(636\) 0 0
\(637\) −172261. −0.424529
\(638\) 0 0
\(639\) 518287. 353728.i 1.26931 0.866298i
\(640\) 0 0
\(641\) 25834.8i 0.0628766i 0.999506 + 0.0314383i \(0.0100088\pi\)
−0.999506 + 0.0314383i \(0.989991\pi\)
\(642\) 0 0
\(643\) −621594. −1.50343 −0.751717 0.659485i \(-0.770774\pi\)
−0.751717 + 0.659485i \(0.770774\pi\)
\(644\) 0 0
\(645\) 165195. 312747.i 0.397080 0.751750i
\(646\) 0 0
\(647\) 667951.i 1.59564i 0.602893 + 0.797822i \(0.294015\pi\)
−0.602893 + 0.797822i \(0.705985\pi\)
\(648\) 0 0
\(649\) −201668. −0.478792
\(650\) 0 0
\(651\) −594272. 313899.i −1.40224 0.740675i
\(652\) 0 0
\(653\) 330534.i 0.775157i −0.921837 0.387579i \(-0.873312\pi\)
0.921837 0.387579i \(-0.126688\pi\)
\(654\) 0 0
\(655\) −246202. −0.573865
\(656\) 0 0
\(657\) −467490. 684974.i −1.08303 1.58688i
\(658\) 0 0
\(659\) 273975.i 0.630869i 0.948947 + 0.315435i \(0.102150\pi\)
−0.948947 + 0.315435i \(0.897850\pi\)
\(660\) 0 0
\(661\) −841184. −1.92525 −0.962627 0.270831i \(-0.912702\pi\)
−0.962627 + 0.270831i \(0.912702\pi\)
\(662\) 0 0
\(663\) 103002. 195004.i 0.234326 0.443625i
\(664\) 0 0
\(665\) 612179.i 1.38431i
\(666\) 0 0
\(667\) −580185. −1.30411
\(668\) 0 0
\(669\) 464130. + 245157.i 1.03702 + 0.547762i
\(670\) 0 0
\(671\) 244396.i 0.542811i
\(672\) 0 0
\(673\) 365221. 0.806354 0.403177 0.915122i \(-0.367906\pi\)
0.403177 + 0.915122i \(0.367906\pi\)
\(674\) 0 0
\(675\) 20635.1 + 181961.i 0.0452897 + 0.399366i
\(676\) 0 0
\(677\) 204902.i 0.447062i −0.974697 0.223531i \(-0.928242\pi\)
0.974697 0.223531i \(-0.0717585\pi\)
\(678\) 0 0
\(679\) −1.14030e6 −2.47332
\(680\) 0 0
\(681\) 405062. 766861.i 0.873429 1.65357i
\(682\) 0 0
\(683\) 651055.i 1.39565i −0.716268 0.697825i \(-0.754151\pi\)
0.716268 0.697825i \(-0.245849\pi\)
\(684\) 0 0
\(685\) 445616. 0.949686
\(686\) 0 0
\(687\) −506283. 267423.i −1.07270 0.566610i
\(688\) 0 0
\(689\) 188418.i 0.396903i
\(690\) 0 0
\(691\) 81069.8 0.169786 0.0848932 0.996390i \(-0.472945\pi\)
0.0848932 + 0.996390i \(0.472945\pi\)
\(692\) 0 0
\(693\) −352784. + 240773.i −0.734586 + 0.501350i
\(694\) 0 0
\(695\) 266415.i 0.551554i
\(696\) 0 0
\(697\) −711426. −1.46442
\(698\) 0 0
\(699\) 440788. 834497.i 0.902142 1.70793i
\(700\) 0 0
\(701\) 202875.i 0.412851i −0.978462 0.206425i \(-0.933817\pi\)
0.978462 0.206425i \(-0.0661831\pi\)
\(702\) 0 0
\(703\) 337207. 0.682317
\(704\) 0 0
\(705\) −331024. 174850.i −0.666011 0.351792i
\(706\) 0 0
\(707\) 1.31936e6i 2.63953i
\(708\) 0 0
\(709\) 309617. 0.615932 0.307966 0.951397i \(-0.400352\pi\)
0.307966 + 0.951397i \(0.400352\pi\)
\(710\) 0 0
\(711\) −242645. 355528.i −0.479991 0.703290i
\(712\) 0 0
\(713\) 681560.i 1.34068i
\(714\) 0 0
\(715\) 119405. 0.233567
\(716\) 0 0
\(717\) 227548. 430792.i 0.442623 0.837972i
\(718\) 0 0
\(719\) 722490.i 1.39757i 0.715331 + 0.698785i \(0.246276\pi\)
−0.715331 + 0.698785i \(0.753724\pi\)
\(720\) 0 0
\(721\) 572140. 1.10061
\(722\) 0 0
\(723\) 148418. + 78395.3i 0.283928 + 0.149973i
\(724\) 0 0
\(725\) 236078.i 0.449137i
\(726\) 0 0
\(727\) −187688. −0.355113 −0.177557 0.984111i \(-0.556819\pi\)
−0.177557 + 0.984111i \(0.556819\pi\)
\(728\) 0 0
\(729\) −517945. + 119005.i −0.974606 + 0.223928i
\(730\) 0 0
\(731\) 628707.i 1.17656i
\(732\) 0 0
\(733\) −35683.0 −0.0664130 −0.0332065 0.999449i \(-0.510572\pi\)
−0.0332065 + 0.999449i \(0.510572\pi\)
\(734\) 0 0
\(735\) 176709. 334544.i 0.327102 0.619269i
\(736\) 0 0
\(737\) 192710.i 0.354787i
\(738\) 0 0
\(739\) 136210. 0.249413 0.124707 0.992194i \(-0.460201\pi\)
0.124707 + 0.992194i \(0.460201\pi\)
\(740\) 0 0
\(741\) −295129. 155889.i −0.537496 0.283909i
\(742\) 0 0
\(743\) 73802.1i 0.133688i −0.997763 0.0668438i \(-0.978707\pi\)
0.997763 0.0668438i \(-0.0212929\pi\)
\(744\) 0 0
\(745\) 16649.9 0.0299984
\(746\) 0 0
\(747\) −714435. + 487597.i −1.28033 + 0.873816i
\(748\) 0 0
\(749\) 777459.i 1.38584i
\(750\) 0 0
\(751\) −231199. −0.409926 −0.204963 0.978770i \(-0.565707\pi\)
−0.204963 + 0.978770i \(0.565707\pi\)
\(752\) 0 0
\(753\) −8400.41 + 15903.6i −0.0148153 + 0.0280483i
\(754\) 0 0
\(755\) 229255.i 0.402183i
\(756\) 0 0
\(757\) 26453.5 0.0461627 0.0230813 0.999734i \(-0.492652\pi\)
0.0230813 + 0.999734i \(0.492652\pi\)
\(758\) 0 0
\(759\) 382995. + 202301.i 0.664829 + 0.351168i
\(760\) 0 0
\(761\) 254385.i 0.439260i 0.975583 + 0.219630i \(0.0704851\pi\)
−0.975583 + 0.219630i \(0.929515\pi\)
\(762\) 0 0
\(763\) −349296. −0.599991
\(764\) 0 0
\(765\) 273051. + 400079.i 0.466575 + 0.683632i
\(766\) 0 0
\(767\) 204948.i 0.348380i
\(768\) 0 0
\(769\) −87292.0 −0.147612 −0.0738060 0.997273i \(-0.523515\pi\)
−0.0738060 + 0.997273i \(0.523515\pi\)
\(770\) 0 0
\(771\) 287092. 543521.i 0.482961 0.914339i
\(772\) 0 0
\(773\) 318085.i 0.532334i 0.963927 + 0.266167i \(0.0857573\pi\)
−0.963927 + 0.266167i \(0.914243\pi\)
\(774\) 0 0
\(775\) 277327. 0.461731
\(776\) 0 0
\(777\) −387764. 204820.i −0.642282 0.339258i
\(778\) 0 0
\(779\) 1.07671e6i 1.77428i
\(780\) 0 0
\(781\) −603909. −0.990078
\(782\) 0 0
\(783\) −680740. + 77198.7i −1.11035 + 0.125918i
\(784\) 0 0
\(785\) 466491.i 0.757014i
\(786\) 0 0
\(787\) −11698.3 −0.0188874 −0.00944372 0.999955i \(-0.503006\pi\)
−0.00944372 + 0.999955i \(0.503006\pi\)
\(788\) 0 0
\(789\) −354188. + 670547.i −0.568958 + 1.07715i
\(790\) 0 0
\(791\) 278581.i 0.445244i
\(792\) 0 0
\(793\) −248371. −0.394961
\(794\) 0 0
\(795\) 365924. + 193284.i 0.578970 + 0.305817i
\(796\) 0 0
\(797\) 210654.i 0.331629i 0.986157 + 0.165814i \(0.0530253\pi\)
−0.986157 + 0.165814i \(0.946975\pi\)
\(798\) 0 0
\(799\) 665450. 1.04237
\(800\) 0 0
\(801\) 135438. 92435.8i 0.211094 0.144071i
\(802\) 0 0
\(803\) 798132.i 1.23778i
\(804\) 0 0
\(805\) −807361. −1.24588
\(806\) 0 0
\(807\) 354407. 670961.i 0.544196 1.03027i
\(808\) 0 0
\(809\) 732334.i 1.11895i −0.828846 0.559477i \(-0.811002\pi\)
0.828846 0.559477i \(-0.188998\pi\)
\(810\) 0 0
\(811\) 440995. 0.670490 0.335245 0.942131i \(-0.391181\pi\)
0.335245 + 0.942131i \(0.391181\pi\)
\(812\) 0 0
\(813\) 778982. + 411464.i 1.17855 + 0.622517i
\(814\) 0 0
\(815\) 349934.i 0.526831i
\(816\) 0 0
\(817\) 951518. 1.42552
\(818\) 0 0
\(819\) 244689. + 358523.i 0.364794 + 0.534501i
\(820\) 0 0
\(821\) 813364.i 1.20670i −0.797477 0.603349i \(-0.793833\pi\)
0.797477 0.603349i \(-0.206167\pi\)
\(822\) 0 0
\(823\) 350711. 0.517785 0.258892 0.965906i \(-0.416642\pi\)
0.258892 + 0.965906i \(0.416642\pi\)
\(824\) 0 0
\(825\) 82316.4 155841.i 0.120942 0.228967i
\(826\) 0 0
\(827\) 95232.7i 0.139244i −0.997573 0.0696218i \(-0.977821\pi\)
0.997573 0.0696218i \(-0.0221793\pi\)
\(828\) 0 0
\(829\) −160042. −0.232876 −0.116438 0.993198i \(-0.537148\pi\)
−0.116438 + 0.993198i \(0.537148\pi\)
\(830\) 0 0
\(831\) 832444. + 439703.i 1.20546 + 0.636733i
\(832\) 0 0
\(833\) 672527.i 0.969214i
\(834\) 0 0
\(835\) 253953. 0.364234
\(836\) 0 0
\(837\) 90687.5 + 799685.i 0.129448 + 1.14148i
\(838\) 0 0
\(839\) 35266.1i 0.0500995i 0.999686 + 0.0250498i \(0.00797442\pi\)
−0.999686 + 0.0250498i \(0.992026\pi\)
\(840\) 0 0
\(841\) −175916. −0.248721
\(842\) 0 0
\(843\) 320956. 607632.i 0.451638 0.855038i
\(844\) 0 0
\(845\) 430846.i 0.603404i
\(846\) 0 0
\(847\) −579272. −0.807450
\(848\) 0 0
\(849\) 1.03034e6 + 544235.i 1.42944 + 0.755043i
\(850\) 0 0
\(851\) 444720.i 0.614084i
\(852\) 0 0
\(853\) 668061. 0.918159 0.459080 0.888395i \(-0.348179\pi\)
0.459080 + 0.888395i \(0.348179\pi\)
\(854\) 0 0
\(855\) 605500. 413250.i 0.828289 0.565302i
\(856\) 0 0
\(857\) 409636.i 0.557746i 0.960328 + 0.278873i \(0.0899608\pi\)
−0.960328 + 0.278873i \(0.910039\pi\)
\(858\) 0 0
\(859\) −96808.2 −0.131198 −0.0655988 0.997846i \(-0.520896\pi\)
−0.0655988 + 0.997846i \(0.520896\pi\)
\(860\) 0 0
\(861\) 653993. 1.23814e6i 0.882200 1.67018i
\(862\) 0 0
\(863\) 627704.i 0.842817i −0.906871 0.421408i \(-0.861536\pi\)
0.906871 0.421408i \(-0.138464\pi\)
\(864\) 0 0
\(865\) 726949. 0.971564
\(866\) 0 0
\(867\) −96654.4 51053.6i −0.128583 0.0679185i
\(868\) 0 0
\(869\) 414261.i 0.548574i
\(870\) 0 0
\(871\) 195844. 0.258151
\(872\) 0 0
\(873\) 769757. + 1.12786e6i 1.01001 + 1.47988i
\(874\) 0 0
\(875\) 1.14587e6i 1.49664i
\(876\) 0 0
\(877\) 1.29296e6 1.68107 0.840533 0.541761i \(-0.182242\pi\)
0.840533 + 0.541761i \(0.182242\pi\)
\(878\) 0 0
\(879\) −424326. + 803331.i −0.549189 + 1.03972i
\(880\) 0 0
\(881\) 1.13499e6i 1.46231i 0.682213 + 0.731153i \(0.261018\pi\)
−0.682213 + 0.731153i \(0.738982\pi\)
\(882\) 0 0
\(883\) −216522. −0.277703 −0.138851 0.990313i \(-0.544341\pi\)
−0.138851 + 0.990313i \(0.544341\pi\)
\(884\) 0 0
\(885\) −398026. 210241.i −0.508189 0.268429i
\(886\) 0 0
\(887\) 1.24588e6i 1.58354i −0.610821 0.791769i \(-0.709160\pi\)
0.610821 0.791769i \(-0.290840\pi\)
\(888\) 0 0
\(889\) −470457. −0.595273
\(890\) 0 0
\(891\) 476292. + 186402.i 0.599954 + 0.234799i
\(892\) 0 0
\(893\) 1.00713e6i 1.26294i
\(894\) 0 0
\(895\) −585686. −0.731171
\(896\) 0 0
\(897\) 205592. 389225.i 0.255518 0.483744i
\(898\) 0 0
\(899\) 1.03752e6i 1.28374i
\(900\) 0 0
\(901\) −735608. −0.906143
\(902\) 0 0
\(903\) −1.09418e6 577952.i −1.34187 0.708788i
\(904\) 0 0
\(905\) 800787.i 0.977732i
\(906\) 0 0
\(907\) 71744.4 0.0872115 0.0436057 0.999049i \(-0.486115\pi\)
0.0436057 + 0.999049i \(0.486115\pi\)
\(908\) 0 0
\(909\) −1.30497e6 + 890634.i −1.57933 + 1.07788i
\(910\) 0 0
\(911\) 463492.i 0.558477i −0.960222 0.279239i \(-0.909918\pi\)
0.960222 0.279239i \(-0.0900821\pi\)
\(912\) 0 0
\(913\) 832461. 0.998670
\(914\) 0 0
\(915\) 254785. 482357.i 0.304321 0.576138i
\(916\) 0 0
\(917\) 861364.i 1.02435i
\(918\) 0 0
\(919\) 540851. 0.640393 0.320196 0.947351i \(-0.396251\pi\)
0.320196 + 0.947351i \(0.396251\pi\)
\(920\) 0 0
\(921\) 93968.7 + 49635.0i 0.110781 + 0.0585152i
\(922\) 0 0
\(923\) 613732.i 0.720403i
\(924\) 0 0
\(925\) 180957. 0.211491
\(926\) 0 0
\(927\) −386222. 565898.i −0.449446 0.658534i
\(928\) 0 0
\(929\) 818980.i 0.948947i −0.880270 0.474474i \(-0.842638\pi\)
0.880270 0.474474i \(-0.157362\pi\)
\(930\) 0 0
\(931\) 1.01784e6 1.17430
\(932\) 0 0
\(933\) −152848. + 289371.i −0.175588 + 0.332423i
\(934\) 0 0
\(935\) 466172.i 0.533241i
\(936\) 0 0
\(937\) −388827. −0.442871 −0.221435 0.975175i \(-0.571074\pi\)
−0.221435 + 0.975175i \(0.571074\pi\)
\(938\) 0 0
\(939\) −529738. 279812.i −0.600800 0.317347i
\(940\) 0 0
\(941\) 165913.i 0.187371i −0.995602 0.0936855i \(-0.970135\pi\)
0.995602 0.0936855i \(-0.0298648\pi\)
\(942\) 0 0
\(943\) −1.42000e6 −1.59685
\(944\) 0 0
\(945\) −947289. + 107426.i −1.06076 + 0.120295i
\(946\) 0 0
\(947\) 764662.i 0.852647i −0.904571 0.426324i \(-0.859809\pi\)
0.904571 0.426324i \(-0.140191\pi\)
\(948\) 0 0
\(949\) 811115. 0.900637
\(950\) 0 0
\(951\) −444870. + 842225.i −0.491894 + 0.931252i
\(952\) 0 0
\(953\) 1.05066e6i 1.15685i 0.815737 + 0.578424i \(0.196332\pi\)
−0.815737 + 0.578424i \(0.803668\pi\)
\(954\) 0 0
\(955\) 506880. 0.555774
\(956\) 0 0
\(957\) 583021. + 307956.i 0.636591 + 0.336252i
\(958\) 0 0
\(959\) 1.55903e6i 1.69519i
\(960\) 0 0
\(961\) 295279. 0.319732
\(962\) 0 0
\(963\) −768977. + 524822.i −0.829202 + 0.565925i
\(964\) 0 0
\(965\) 480865.i 0.516379i
\(966\) 0 0
\(967\) 201023. 0.214977 0.107488 0.994206i \(-0.465719\pi\)
0.107488 + 0.994206i \(0.465719\pi\)
\(968\) 0 0
\(969\) −608611. + 1.15222e6i −0.648175 + 1.22712i
\(970\) 0 0
\(971\) 1.06992e6i 1.13478i −0.823448 0.567392i \(-0.807952\pi\)
0.823448 0.567392i \(-0.192048\pi\)
\(972\) 0 0
\(973\) 932079. 0.984525
\(974\) 0 0
\(975\) −158376. 83655.3i −0.166602 0.0880004i
\(976\) 0 0
\(977\) 1.04258e6i 1.09224i −0.837706 0.546121i \(-0.816104\pi\)
0.837706 0.546121i \(-0.183896\pi\)
\(978\) 0 0
\(979\) −157813. −0.164656
\(980\) 0 0
\(981\) 235792. + 345485.i 0.245014 + 0.358998i
\(982\) 0 0
\(983\) 201986.i 0.209033i −0.994523 0.104516i \(-0.966671\pi\)
0.994523 0.104516i \(-0.0333294\pi\)
\(984\) 0 0
\(985\) 867343. 0.893961
\(986\) 0 0
\(987\) −611729. + 1.15812e6i −0.627950 + 1.18883i
\(988\) 0 0
\(989\) 1.25489e6i 1.28296i
\(990\) 0 0
\(991\) 905070. 0.921584 0.460792 0.887508i \(-0.347565\pi\)
0.460792 + 0.887508i \(0.347565\pi\)
\(992\) 0 0
\(993\) 468088. + 247248.i 0.474711 + 0.250746i
\(994\) 0 0
\(995\) 34459.9i 0.0348071i
\(996\) 0 0
\(997\) −772571. −0.777227 −0.388614 0.921401i \(-0.627046\pi\)
−0.388614 + 0.921401i \(0.627046\pi\)
\(998\) 0 0
\(999\) 59173.9 + 521797.i 0.0592924 + 0.522842i
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 384.5.e.c.257.14 yes 16
3.2 odd 2 inner 384.5.e.c.257.13 yes 16
4.3 odd 2 384.5.e.b.257.3 yes 16
8.3 odd 2 384.5.e.d.257.14 yes 16
8.5 even 2 384.5.e.a.257.3 16
12.11 even 2 384.5.e.b.257.4 yes 16
24.5 odd 2 384.5.e.a.257.4 yes 16
24.11 even 2 384.5.e.d.257.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.e.a.257.3 16 8.5 even 2
384.5.e.a.257.4 yes 16 24.5 odd 2
384.5.e.b.257.3 yes 16 4.3 odd 2
384.5.e.b.257.4 yes 16 12.11 even 2
384.5.e.c.257.13 yes 16 3.2 odd 2 inner
384.5.e.c.257.14 yes 16 1.1 even 1 trivial
384.5.e.d.257.13 yes 16 24.11 even 2
384.5.e.d.257.14 yes 16 8.3 odd 2