Properties

Label 768.4.f.g.383.8
Level $768$
Weight $4$
Character 768.383
Analytic conductor $45.313$
Analytic rank $0$
Dimension $12$
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] = [768,4,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.8
Root \(0.910871i\) of defining polynomial
Character \(\chi\) \(=\) 768.383
Dual form 768.4.f.g.383.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.98169 + 4.80343i) q^{3} +11.9846 q^{5} +22.6995i q^{7} +(-19.1458 + 19.0378i) q^{9} +O(q^{10})\) \(q+(1.98169 + 4.80343i) q^{3} +11.9846 q^{5} +22.6995i q^{7} +(-19.1458 + 19.0378i) q^{9} +61.9410i q^{11} +71.3317i q^{13} +(23.7497 + 57.5672i) q^{15} -74.3305i q^{17} +108.076 q^{19} +(-109.036 + 44.9834i) q^{21} -24.7059 q^{23} +18.6309 q^{25} +(-129.388 - 54.2388i) q^{27} -84.1331 q^{29} -130.377i q^{31} +(-297.529 + 122.748i) q^{33} +272.045i q^{35} -397.468i q^{37} +(-342.637 + 141.357i) q^{39} +104.132i q^{41} +161.650 q^{43} +(-229.455 + 228.160i) q^{45} -72.8162 q^{47} -172.269 q^{49} +(357.041 - 147.300i) q^{51} +136.267 q^{53} +742.339i q^{55} +(214.172 + 519.134i) q^{57} -243.465i q^{59} -358.454i q^{61} +(-432.149 - 434.602i) q^{63} +854.883i q^{65} +449.508 q^{67} +(-48.9593 - 118.673i) q^{69} +329.053 q^{71} +925.687 q^{73} +(36.9207 + 89.4923i) q^{75} -1406.03 q^{77} -55.9579i q^{79} +(4.12647 - 728.988i) q^{81} +928.194i q^{83} -890.823i q^{85} +(-166.725 - 404.127i) q^{87} +853.930i q^{89} -1619.20 q^{91} +(626.256 - 258.366i) q^{93} +1295.25 q^{95} -714.982 q^{97} +(-1179.22 - 1185.91i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 12 q^{5} + 84 q^{15} + 180 q^{19} - 156 q^{21} + 120 q^{23} + 300 q^{25} - 130 q^{27} - 588 q^{29} - 116 q^{33} - 620 q^{39} - 372 q^{43} + 740 q^{45} - 1248 q^{47} - 948 q^{49} - 360 q^{51} + 948 q^{53} + 172 q^{57} + 2744 q^{63} + 2292 q^{67} - 3280 q^{69} + 2040 q^{71} + 216 q^{73} - 2522 q^{75} - 4824 q^{77} - 1076 q^{81} - 4156 q^{87} + 3480 q^{91} + 4180 q^{93} - 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 1.98169 + 4.80343i 0.381376 + 0.924420i
\(4\) 0 0
\(5\) 11.9846 1.07194 0.535968 0.844238i \(-0.319947\pi\)
0.535968 + 0.844238i \(0.319947\pi\)
\(6\) 0 0
\(7\) 22.6995i 1.22566i 0.790215 + 0.612830i \(0.209969\pi\)
−0.790215 + 0.612830i \(0.790031\pi\)
\(8\) 0 0
\(9\) −19.1458 + 19.0378i −0.709105 + 0.705103i
\(10\) 0 0
\(11\) 61.9410i 1.69781i 0.528545 + 0.848905i \(0.322738\pi\)
−0.528545 + 0.848905i \(0.677262\pi\)
\(12\) 0 0
\(13\) 71.3317i 1.52184i 0.648848 + 0.760918i \(0.275251\pi\)
−0.648848 + 0.760918i \(0.724749\pi\)
\(14\) 0 0
\(15\) 23.7497 + 57.5672i 0.408810 + 0.990920i
\(16\) 0 0
\(17\) 74.3305i 1.06046i −0.847854 0.530229i \(-0.822106\pi\)
0.847854 0.530229i \(-0.177894\pi\)
\(18\) 0 0
\(19\) 108.076 1.30496 0.652481 0.757805i \(-0.273728\pi\)
0.652481 + 0.757805i \(0.273728\pi\)
\(20\) 0 0
\(21\) −109.036 + 44.9834i −1.13302 + 0.467437i
\(22\) 0 0
\(23\) −24.7059 −0.223980 −0.111990 0.993709i \(-0.535722\pi\)
−0.111990 + 0.993709i \(0.535722\pi\)
\(24\) 0 0
\(25\) 18.6309 0.149047
\(26\) 0 0
\(27\) −129.388 54.2388i −0.922247 0.386602i
\(28\) 0 0
\(29\) −84.1331 −0.538728 −0.269364 0.963038i \(-0.586814\pi\)
−0.269364 + 0.963038i \(0.586814\pi\)
\(30\) 0 0
\(31\) 130.377i 0.755367i −0.925935 0.377684i \(-0.876721\pi\)
0.925935 0.377684i \(-0.123279\pi\)
\(32\) 0 0
\(33\) −297.529 + 122.748i −1.56949 + 0.647504i
\(34\) 0 0
\(35\) 272.045i 1.31383i
\(36\) 0 0
\(37\) 397.468i 1.76604i −0.469338 0.883019i \(-0.655507\pi\)
0.469338 0.883019i \(-0.344493\pi\)
\(38\) 0 0
\(39\) −342.637 + 141.357i −1.40682 + 0.580391i
\(40\) 0 0
\(41\) 104.132i 0.396649i 0.980136 + 0.198325i \(0.0635500\pi\)
−0.980136 + 0.198325i \(0.936450\pi\)
\(42\) 0 0
\(43\) 161.650 0.573289 0.286644 0.958037i \(-0.407460\pi\)
0.286644 + 0.958037i \(0.407460\pi\)
\(44\) 0 0
\(45\) −229.455 + 228.160i −0.760116 + 0.755825i
\(46\) 0 0
\(47\) −72.8162 −0.225986 −0.112993 0.993596i \(-0.536044\pi\)
−0.112993 + 0.993596i \(0.536044\pi\)
\(48\) 0 0
\(49\) −172.269 −0.502242
\(50\) 0 0
\(51\) 357.041 147.300i 0.980310 0.404433i
\(52\) 0 0
\(53\) 136.267 0.353164 0.176582 0.984286i \(-0.443496\pi\)
0.176582 + 0.984286i \(0.443496\pi\)
\(54\) 0 0
\(55\) 742.339i 1.81995i
\(56\) 0 0
\(57\) 214.172 + 519.134i 0.497681 + 1.20633i
\(58\) 0 0
\(59\) 243.465i 0.537228i −0.963248 0.268614i \(-0.913434\pi\)
0.963248 0.268614i \(-0.0865656\pi\)
\(60\) 0 0
\(61\) 358.454i 0.752383i −0.926542 0.376192i \(-0.877233\pi\)
0.926542 0.376192i \(-0.122767\pi\)
\(62\) 0 0
\(63\) −432.149 434.602i −0.864216 0.869122i
\(64\) 0 0
\(65\) 854.883i 1.63131i
\(66\) 0 0
\(67\) 449.508 0.819645 0.409822 0.912165i \(-0.365591\pi\)
0.409822 + 0.912165i \(0.365591\pi\)
\(68\) 0 0
\(69\) −48.9593 118.673i −0.0854205 0.207052i
\(70\) 0 0
\(71\) 329.053 0.550020 0.275010 0.961441i \(-0.411319\pi\)
0.275010 + 0.961441i \(0.411319\pi\)
\(72\) 0 0
\(73\) 925.687 1.48416 0.742079 0.670313i \(-0.233840\pi\)
0.742079 + 0.670313i \(0.233840\pi\)
\(74\) 0 0
\(75\) 36.9207 + 89.4923i 0.0568431 + 0.137782i
\(76\) 0 0
\(77\) −1406.03 −2.08094
\(78\) 0 0
\(79\) 55.9579i 0.0796931i −0.999206 0.0398465i \(-0.987313\pi\)
0.999206 0.0398465i \(-0.0126869\pi\)
\(80\) 0 0
\(81\) 4.12647 728.988i 0.00566045 0.999984i
\(82\) 0 0
\(83\) 928.194i 1.22750i 0.789500 + 0.613750i \(0.210340\pi\)
−0.789500 + 0.613750i \(0.789660\pi\)
\(84\) 0 0
\(85\) 890.823i 1.13674i
\(86\) 0 0
\(87\) −166.725 404.127i −0.205458 0.498011i
\(88\) 0 0
\(89\) 853.930i 1.01704i 0.861051 + 0.508519i \(0.169807\pi\)
−0.861051 + 0.508519i \(0.830193\pi\)
\(90\) 0 0
\(91\) −1619.20 −1.86525
\(92\) 0 0
\(93\) 626.256 258.366i 0.698277 0.288079i
\(94\) 0 0
\(95\) 1295.25 1.39884
\(96\) 0 0
\(97\) −714.982 −0.748406 −0.374203 0.927347i \(-0.622084\pi\)
−0.374203 + 0.927347i \(0.622084\pi\)
\(98\) 0 0
\(99\) −1179.22 1185.91i −1.19713 1.20393i
\(100\) 0 0
\(101\) −1702.69 −1.67746 −0.838730 0.544547i \(-0.816701\pi\)
−0.838730 + 0.544547i \(0.816701\pi\)
\(102\) 0 0
\(103\) 655.930i 0.627482i −0.949509 0.313741i \(-0.898418\pi\)
0.949509 0.313741i \(-0.101582\pi\)
\(104\) 0 0
\(105\) −1306.75 + 539.108i −1.21453 + 0.501062i
\(106\) 0 0
\(107\) 202.008i 0.182513i −0.995827 0.0912563i \(-0.970912\pi\)
0.995827 0.0912563i \(-0.0290883\pi\)
\(108\) 0 0
\(109\) 588.724i 0.517335i 0.965966 + 0.258667i \(0.0832834\pi\)
−0.965966 + 0.258667i \(0.916717\pi\)
\(110\) 0 0
\(111\) 1909.21 787.657i 1.63256 0.673524i
\(112\) 0 0
\(113\) 2228.76i 1.85544i 0.373280 + 0.927719i \(0.378233\pi\)
−0.373280 + 0.927719i \(0.621767\pi\)
\(114\) 0 0
\(115\) −296.091 −0.240092
\(116\) 0 0
\(117\) −1358.00 1365.71i −1.07305 1.07914i
\(118\) 0 0
\(119\) 1687.27 1.29976
\(120\) 0 0
\(121\) −2505.69 −1.88256
\(122\) 0 0
\(123\) −500.188 + 206.356i −0.366670 + 0.151272i
\(124\) 0 0
\(125\) −1274.79 −0.912167
\(126\) 0 0
\(127\) 1345.93i 0.940409i 0.882558 + 0.470204i \(0.155820\pi\)
−0.882558 + 0.470204i \(0.844180\pi\)
\(128\) 0 0
\(129\) 320.340 + 776.475i 0.218638 + 0.529960i
\(130\) 0 0
\(131\) 679.531i 0.453213i −0.973986 0.226607i \(-0.927237\pi\)
0.973986 0.226607i \(-0.0727632\pi\)
\(132\) 0 0
\(133\) 2453.27i 1.59944i
\(134\) 0 0
\(135\) −1550.66 650.031i −0.988590 0.414413i
\(136\) 0 0
\(137\) 614.487i 0.383205i 0.981473 + 0.191603i \(0.0613685\pi\)
−0.981473 + 0.191603i \(0.938631\pi\)
\(138\) 0 0
\(139\) 3163.89 1.93063 0.965315 0.261088i \(-0.0840813\pi\)
0.965315 + 0.261088i \(0.0840813\pi\)
\(140\) 0 0
\(141\) −144.299 349.768i −0.0861855 0.208906i
\(142\) 0 0
\(143\) −4418.36 −2.58379
\(144\) 0 0
\(145\) −1008.30 −0.577483
\(146\) 0 0
\(147\) −341.383 827.482i −0.191543 0.464283i
\(148\) 0 0
\(149\) 1068.79 0.587640 0.293820 0.955861i \(-0.405073\pi\)
0.293820 + 0.955861i \(0.405073\pi\)
\(150\) 0 0
\(151\) 1343.30i 0.723949i −0.932188 0.361974i \(-0.882103\pi\)
0.932188 0.361974i \(-0.117897\pi\)
\(152\) 0 0
\(153\) 1415.09 + 1423.12i 0.747732 + 0.751977i
\(154\) 0 0
\(155\) 1562.52i 0.809706i
\(156\) 0 0
\(157\) 2626.27i 1.33503i −0.744598 0.667513i \(-0.767359\pi\)
0.744598 0.667513i \(-0.232641\pi\)
\(158\) 0 0
\(159\) 270.038 + 654.548i 0.134688 + 0.326472i
\(160\) 0 0
\(161\) 560.813i 0.274523i
\(162\) 0 0
\(163\) 967.832 0.465070 0.232535 0.972588i \(-0.425298\pi\)
0.232535 + 0.972588i \(0.425298\pi\)
\(164\) 0 0
\(165\) −3565.77 + 1471.08i −1.68239 + 0.694083i
\(166\) 0 0
\(167\) 66.2186 0.0306835 0.0153418 0.999882i \(-0.495116\pi\)
0.0153418 + 0.999882i \(0.495116\pi\)
\(168\) 0 0
\(169\) −2891.22 −1.31598
\(170\) 0 0
\(171\) −2069.20 + 2057.52i −0.925355 + 0.920132i
\(172\) 0 0
\(173\) −3061.46 −1.34543 −0.672713 0.739904i \(-0.734871\pi\)
−0.672713 + 0.739904i \(0.734871\pi\)
\(174\) 0 0
\(175\) 422.913i 0.182681i
\(176\) 0 0
\(177\) 1169.47 482.471i 0.496625 0.204886i
\(178\) 0 0
\(179\) 949.274i 0.396380i −0.980164 0.198190i \(-0.936494\pi\)
0.980164 0.198190i \(-0.0635064\pi\)
\(180\) 0 0
\(181\) 874.923i 0.359296i 0.983731 + 0.179648i \(0.0574958\pi\)
−0.983731 + 0.179648i \(0.942504\pi\)
\(182\) 0 0
\(183\) 1721.81 710.344i 0.695518 0.286941i
\(184\) 0 0
\(185\) 4763.50i 1.89308i
\(186\) 0 0
\(187\) 4604.11 1.80046
\(188\) 0 0
\(189\) 1231.20 2937.04i 0.473843 1.13036i
\(190\) 0 0
\(191\) −683.182 −0.258813 −0.129407 0.991592i \(-0.541307\pi\)
−0.129407 + 0.991592i \(0.541307\pi\)
\(192\) 0 0
\(193\) 3029.86 1.13002 0.565011 0.825083i \(-0.308872\pi\)
0.565011 + 0.825083i \(0.308872\pi\)
\(194\) 0 0
\(195\) −4106.37 + 1694.11i −1.50802 + 0.622142i
\(196\) 0 0
\(197\) 2661.33 0.962496 0.481248 0.876584i \(-0.340184\pi\)
0.481248 + 0.876584i \(0.340184\pi\)
\(198\) 0 0
\(199\) 2975.37i 1.05989i 0.848031 + 0.529946i \(0.177788\pi\)
−0.848031 + 0.529946i \(0.822212\pi\)
\(200\) 0 0
\(201\) 890.784 + 2159.18i 0.312592 + 0.757696i
\(202\) 0 0
\(203\) 1909.78i 0.660298i
\(204\) 0 0
\(205\) 1247.98i 0.425183i
\(206\) 0 0
\(207\) 473.015 470.345i 0.158825 0.157929i
\(208\) 0 0
\(209\) 6694.32i 2.21558i
\(210\) 0 0
\(211\) −2601.38 −0.848751 −0.424376 0.905486i \(-0.639506\pi\)
−0.424376 + 0.905486i \(0.639506\pi\)
\(212\) 0 0
\(213\) 652.080 + 1580.58i 0.209764 + 0.508450i
\(214\) 0 0
\(215\) 1937.31 0.614529
\(216\) 0 0
\(217\) 2959.50 0.925824
\(218\) 0 0
\(219\) 1834.42 + 4446.47i 0.566022 + 1.37199i
\(220\) 0 0
\(221\) 5302.13 1.61384
\(222\) 0 0
\(223\) 4031.63i 1.21066i 0.795973 + 0.605332i \(0.206960\pi\)
−0.795973 + 0.605332i \(0.793040\pi\)
\(224\) 0 0
\(225\) −356.705 + 354.691i −0.105690 + 0.105094i
\(226\) 0 0
\(227\) 4990.76i 1.45924i 0.683851 + 0.729622i \(0.260304\pi\)
−0.683851 + 0.729622i \(0.739696\pi\)
\(228\) 0 0
\(229\) 5091.07i 1.46912i −0.678546 0.734558i \(-0.737390\pi\)
0.678546 0.734558i \(-0.262610\pi\)
\(230\) 0 0
\(231\) −2786.31 6753.78i −0.793619 1.92366i
\(232\) 0 0
\(233\) 229.872i 0.0646326i −0.999478 0.0323163i \(-0.989712\pi\)
0.999478 0.0323163i \(-0.0102884\pi\)
\(234\) 0 0
\(235\) −872.674 −0.242243
\(236\) 0 0
\(237\) 268.790 110.891i 0.0736699 0.0303930i
\(238\) 0 0
\(239\) −1523.39 −0.412300 −0.206150 0.978520i \(-0.566093\pi\)
−0.206150 + 0.978520i \(0.566093\pi\)
\(240\) 0 0
\(241\) 1736.76 0.464210 0.232105 0.972691i \(-0.425439\pi\)
0.232105 + 0.972691i \(0.425439\pi\)
\(242\) 0 0
\(243\) 3509.82 1424.80i 0.926564 0.376137i
\(244\) 0 0
\(245\) −2064.58 −0.538371
\(246\) 0 0
\(247\) 7709.23i 1.98594i
\(248\) 0 0
\(249\) −4458.51 + 1839.39i −1.13473 + 0.468139i
\(250\) 0 0
\(251\) 1246.85i 0.313547i 0.987635 + 0.156774i \(0.0501093\pi\)
−0.987635 + 0.156774i \(0.949891\pi\)
\(252\) 0 0
\(253\) 1530.31i 0.380275i
\(254\) 0 0
\(255\) 4279.00 1765.33i 1.05083 0.433527i
\(256\) 0 0
\(257\) 6505.35i 1.57896i −0.613777 0.789479i \(-0.710351\pi\)
0.613777 0.789479i \(-0.289649\pi\)
\(258\) 0 0
\(259\) 9022.35 2.16456
\(260\) 0 0
\(261\) 1610.80 1601.71i 0.382015 0.379859i
\(262\) 0 0
\(263\) −7974.40 −1.86967 −0.934834 0.355086i \(-0.884451\pi\)
−0.934834 + 0.355086i \(0.884451\pi\)
\(264\) 0 0
\(265\) 1633.11 0.378569
\(266\) 0 0
\(267\) −4101.79 + 1692.22i −0.940171 + 0.387874i
\(268\) 0 0
\(269\) −569.640 −0.129114 −0.0645568 0.997914i \(-0.520563\pi\)
−0.0645568 + 0.997914i \(0.520563\pi\)
\(270\) 0 0
\(271\) 6729.09i 1.50835i 0.656673 + 0.754175i \(0.271963\pi\)
−0.656673 + 0.754175i \(0.728037\pi\)
\(272\) 0 0
\(273\) −3208.74 7777.70i −0.711362 1.72428i
\(274\) 0 0
\(275\) 1154.02i 0.253054i
\(276\) 0 0
\(277\) 3054.27i 0.662502i 0.943543 + 0.331251i \(0.107471\pi\)
−0.943543 + 0.331251i \(0.892529\pi\)
\(278\) 0 0
\(279\) 2482.09 + 2496.18i 0.532612 + 0.535635i
\(280\) 0 0
\(281\) 673.673i 0.143018i 0.997440 + 0.0715088i \(0.0227814\pi\)
−0.997440 + 0.0715088i \(0.977219\pi\)
\(282\) 0 0
\(283\) 1972.26 0.414270 0.207135 0.978312i \(-0.433586\pi\)
0.207135 + 0.978312i \(0.433586\pi\)
\(284\) 0 0
\(285\) 2566.77 + 6221.62i 0.533482 + 1.29311i
\(286\) 0 0
\(287\) −2363.74 −0.486157
\(288\) 0 0
\(289\) −612.028 −0.124573
\(290\) 0 0
\(291\) −1416.87 3434.36i −0.285424 0.691842i
\(292\) 0 0
\(293\) 3641.01 0.725974 0.362987 0.931794i \(-0.381757\pi\)
0.362987 + 0.931794i \(0.381757\pi\)
\(294\) 0 0
\(295\) 2917.84i 0.575874i
\(296\) 0 0
\(297\) 3359.61 8014.40i 0.656377 1.56580i
\(298\) 0 0
\(299\) 1762.32i 0.340861i
\(300\) 0 0
\(301\) 3669.38i 0.702657i
\(302\) 0 0
\(303\) −3374.19 8178.72i −0.639743 1.55068i
\(304\) 0 0
\(305\) 4295.94i 0.806507i
\(306\) 0 0
\(307\) 2847.42 0.529351 0.264676 0.964337i \(-0.414735\pi\)
0.264676 + 0.964337i \(0.414735\pi\)
\(308\) 0 0
\(309\) 3150.71 1299.85i 0.580057 0.239307i
\(310\) 0 0
\(311\) 8784.69 1.60172 0.800859 0.598853i \(-0.204377\pi\)
0.800859 + 0.598853i \(0.204377\pi\)
\(312\) 0 0
\(313\) 4790.21 0.865043 0.432522 0.901624i \(-0.357624\pi\)
0.432522 + 0.901624i \(0.357624\pi\)
\(314\) 0 0
\(315\) −5179.13 5208.53i −0.926385 0.931643i
\(316\) 0 0
\(317\) −6390.34 −1.13223 −0.566116 0.824326i \(-0.691554\pi\)
−0.566116 + 0.824326i \(0.691554\pi\)
\(318\) 0 0
\(319\) 5211.29i 0.914659i
\(320\) 0 0
\(321\) 970.331 400.316i 0.168718 0.0696059i
\(322\) 0 0
\(323\) 8033.33i 1.38386i
\(324\) 0 0
\(325\) 1328.98i 0.226826i
\(326\) 0 0
\(327\) −2827.89 + 1166.67i −0.478235 + 0.197299i
\(328\) 0 0
\(329\) 1652.89i 0.276982i
\(330\) 0 0
\(331\) −3472.47 −0.576629 −0.288315 0.957536i \(-0.593095\pi\)
−0.288315 + 0.957536i \(0.593095\pi\)
\(332\) 0 0
\(333\) 7566.91 + 7609.87i 1.24524 + 1.25231i
\(334\) 0 0
\(335\) 5387.18 0.878607
\(336\) 0 0
\(337\) 672.232 0.108661 0.0543305 0.998523i \(-0.482698\pi\)
0.0543305 + 0.998523i \(0.482698\pi\)
\(338\) 0 0
\(339\) −10705.7 + 4416.71i −1.71520 + 0.707619i
\(340\) 0 0
\(341\) 8075.68 1.28247
\(342\) 0 0
\(343\) 3875.51i 0.610082i
\(344\) 0 0
\(345\) −586.759 1422.25i −0.0915653 0.221946i
\(346\) 0 0
\(347\) 606.106i 0.0937679i −0.998900 0.0468839i \(-0.985071\pi\)
0.998900 0.0468839i \(-0.0149291\pi\)
\(348\) 0 0
\(349\) 7881.87i 1.20890i 0.796643 + 0.604451i \(0.206607\pi\)
−0.796643 + 0.604451i \(0.793393\pi\)
\(350\) 0 0
\(351\) 3868.95 9229.44i 0.588345 1.40351i
\(352\) 0 0
\(353\) 528.013i 0.0796128i −0.999207 0.0398064i \(-0.987326\pi\)
0.999207 0.0398064i \(-0.0126741\pi\)
\(354\) 0 0
\(355\) 3943.58 0.589587
\(356\) 0 0
\(357\) 3343.64 + 8104.67i 0.495698 + 1.20153i
\(358\) 0 0
\(359\) −9239.79 −1.35838 −0.679188 0.733964i \(-0.737668\pi\)
−0.679188 + 0.733964i \(0.737668\pi\)
\(360\) 0 0
\(361\) 4821.37 0.702926
\(362\) 0 0
\(363\) −4965.49 12035.9i −0.717963 1.74028i
\(364\) 0 0
\(365\) 11094.0 1.59092
\(366\) 0 0
\(367\) 8258.02i 1.17456i −0.809382 0.587282i \(-0.800198\pi\)
0.809382 0.587282i \(-0.199802\pi\)
\(368\) 0 0
\(369\) −1982.43 1993.69i −0.279678 0.281266i
\(370\) 0 0
\(371\) 3093.19i 0.432859i
\(372\) 0 0
\(373\) 2343.62i 0.325330i −0.986681 0.162665i \(-0.947991\pi\)
0.986681 0.162665i \(-0.0520090\pi\)
\(374\) 0 0
\(375\) −2526.24 6123.37i −0.347878 0.843226i
\(376\) 0 0
\(377\) 6001.36i 0.819856i
\(378\) 0 0
\(379\) 778.226 0.105474 0.0527372 0.998608i \(-0.483205\pi\)
0.0527372 + 0.998608i \(0.483205\pi\)
\(380\) 0 0
\(381\) −6465.07 + 2667.21i −0.869333 + 0.358649i
\(382\) 0 0
\(383\) 9755.66 1.30154 0.650772 0.759274i \(-0.274446\pi\)
0.650772 + 0.759274i \(0.274446\pi\)
\(384\) 0 0
\(385\) −16850.8 −2.23063
\(386\) 0 0
\(387\) −3094.93 + 3077.46i −0.406522 + 0.404227i
\(388\) 0 0
\(389\) 3566.20 0.464816 0.232408 0.972618i \(-0.425340\pi\)
0.232408 + 0.972618i \(0.425340\pi\)
\(390\) 0 0
\(391\) 1836.40i 0.237521i
\(392\) 0 0
\(393\) 3264.08 1346.62i 0.418960 0.172845i
\(394\) 0 0
\(395\) 670.634i 0.0854259i
\(396\) 0 0
\(397\) 8958.31i 1.13251i −0.824232 0.566253i \(-0.808393\pi\)
0.824232 0.566253i \(-0.191607\pi\)
\(398\) 0 0
\(399\) −11784.1 + 4861.61i −1.47855 + 0.609987i
\(400\) 0 0
\(401\) 12946.7i 1.61229i −0.591716 0.806146i \(-0.701549\pi\)
0.591716 0.806146i \(-0.298451\pi\)
\(402\) 0 0
\(403\) 9300.01 1.14955
\(404\) 0 0
\(405\) 49.4541 8736.64i 0.00606764 1.07192i
\(406\) 0 0
\(407\) 24619.6 2.99840
\(408\) 0 0
\(409\) 7979.27 0.964669 0.482335 0.875987i \(-0.339789\pi\)
0.482335 + 0.875987i \(0.339789\pi\)
\(410\) 0 0
\(411\) −2951.64 + 1217.72i −0.354243 + 0.146145i
\(412\) 0 0
\(413\) 5526.55 0.658459
\(414\) 0 0
\(415\) 11124.0i 1.31580i
\(416\) 0 0
\(417\) 6269.83 + 15197.5i 0.736295 + 1.78471i
\(418\) 0 0
\(419\) 8671.29i 1.01103i 0.862819 + 0.505514i \(0.168697\pi\)
−0.862819 + 0.505514i \(0.831303\pi\)
\(420\) 0 0
\(421\) 2908.89i 0.336747i −0.985723 0.168373i \(-0.946149\pi\)
0.985723 0.168373i \(-0.0538514\pi\)
\(422\) 0 0
\(423\) 1394.13 1386.26i 0.160248 0.159343i
\(424\) 0 0
\(425\) 1384.85i 0.158059i
\(426\) 0 0
\(427\) 8136.75 0.922166
\(428\) 0 0
\(429\) −8755.80 21223.3i −0.985394 2.38851i
\(430\) 0 0
\(431\) 4968.52 0.555279 0.277639 0.960685i \(-0.410448\pi\)
0.277639 + 0.960685i \(0.410448\pi\)
\(432\) 0 0
\(433\) 9208.60 1.02203 0.511013 0.859573i \(-0.329271\pi\)
0.511013 + 0.859573i \(0.329271\pi\)
\(434\) 0 0
\(435\) −1998.14 4843.31i −0.220238 0.533837i
\(436\) 0 0
\(437\) −2670.11 −0.292285
\(438\) 0 0
\(439\) 16534.4i 1.79760i 0.438361 + 0.898799i \(0.355559\pi\)
−0.438361 + 0.898799i \(0.644441\pi\)
\(440\) 0 0
\(441\) 3298.23 3279.62i 0.356142 0.354132i
\(442\) 0 0
\(443\) 9024.44i 0.967865i 0.875105 + 0.483933i \(0.160792\pi\)
−0.875105 + 0.483933i \(0.839208\pi\)
\(444\) 0 0
\(445\) 10234.0i 1.09020i
\(446\) 0 0
\(447\) 2118.00 + 5133.83i 0.224111 + 0.543226i
\(448\) 0 0
\(449\) 8971.07i 0.942920i −0.881888 0.471460i \(-0.843727\pi\)
0.881888 0.471460i \(-0.156273\pi\)
\(450\) 0 0
\(451\) −6450.01 −0.673435
\(452\) 0 0
\(453\) 6452.45 2662.00i 0.669233 0.276096i
\(454\) 0 0
\(455\) −19405.5 −1.99943
\(456\) 0 0
\(457\) −11467.8 −1.17383 −0.586915 0.809648i \(-0.699658\pi\)
−0.586915 + 0.809648i \(0.699658\pi\)
\(458\) 0 0
\(459\) −4031.60 + 9617.45i −0.409976 + 0.978005i
\(460\) 0 0
\(461\) 14270.2 1.44171 0.720855 0.693086i \(-0.243749\pi\)
0.720855 + 0.693086i \(0.243749\pi\)
\(462\) 0 0
\(463\) 557.295i 0.0559389i −0.999609 0.0279694i \(-0.991096\pi\)
0.999609 0.0279694i \(-0.00890411\pi\)
\(464\) 0 0
\(465\) 7505.44 3096.42i 0.748508 0.308802i
\(466\) 0 0
\(467\) 9659.10i 0.957109i −0.878058 0.478554i \(-0.841161\pi\)
0.878058 0.478554i \(-0.158839\pi\)
\(468\) 0 0
\(469\) 10203.6i 1.00461i
\(470\) 0 0
\(471\) 12615.1 5204.44i 1.23412 0.509146i
\(472\) 0 0
\(473\) 10012.8i 0.973336i
\(474\) 0 0
\(475\) 2013.55 0.194501
\(476\) 0 0
\(477\) −2608.94 + 2594.22i −0.250430 + 0.249017i
\(478\) 0 0
\(479\) 4353.23 0.415249 0.207625 0.978209i \(-0.433427\pi\)
0.207625 + 0.978209i \(0.433427\pi\)
\(480\) 0 0
\(481\) 28352.1 2.68762
\(482\) 0 0
\(483\) 2693.82 1111.35i 0.253775 0.104696i
\(484\) 0 0
\(485\) −8568.78 −0.802244
\(486\) 0 0
\(487\) 12368.5i 1.15087i −0.817849 0.575433i \(-0.804833\pi\)
0.817849 0.575433i \(-0.195167\pi\)
\(488\) 0 0
\(489\) 1917.94 + 4648.91i 0.177367 + 0.429920i
\(490\) 0 0
\(491\) 19510.2i 1.79324i 0.442798 + 0.896621i \(0.353986\pi\)
−0.442798 + 0.896621i \(0.646014\pi\)
\(492\) 0 0
\(493\) 6253.66i 0.571299i
\(494\) 0 0
\(495\) −14132.5 14212.7i −1.28325 1.29053i
\(496\) 0 0
\(497\) 7469.36i 0.674138i
\(498\) 0 0
\(499\) 1269.62 0.113899 0.0569497 0.998377i \(-0.481863\pi\)
0.0569497 + 0.998377i \(0.481863\pi\)
\(500\) 0 0
\(501\) 131.224 + 318.076i 0.0117020 + 0.0283645i
\(502\) 0 0
\(503\) −14901.0 −1.32088 −0.660442 0.750877i \(-0.729631\pi\)
−0.660442 + 0.750877i \(0.729631\pi\)
\(504\) 0 0
\(505\) −20406.0 −1.79813
\(506\) 0 0
\(507\) −5729.48 13887.8i −0.501884 1.21652i
\(508\) 0 0
\(509\) 1853.05 0.161365 0.0806826 0.996740i \(-0.474290\pi\)
0.0806826 + 0.996740i \(0.474290\pi\)
\(510\) 0 0
\(511\) 21012.7i 1.81907i
\(512\) 0 0
\(513\) −13983.7 5861.90i −1.20350 0.504501i
\(514\) 0 0
\(515\) 7861.07i 0.672621i
\(516\) 0 0
\(517\) 4510.31i 0.383681i
\(518\) 0 0
\(519\) −6066.85 14705.5i −0.513112 1.24374i
\(520\) 0 0
\(521\) 1482.78i 0.124686i 0.998055 + 0.0623432i \(0.0198573\pi\)
−0.998055 + 0.0623432i \(0.980143\pi\)
\(522\) 0 0
\(523\) 16235.8 1.35744 0.678721 0.734396i \(-0.262535\pi\)
0.678721 + 0.734396i \(0.262535\pi\)
\(524\) 0 0
\(525\) −2031.43 + 838.082i −0.168874 + 0.0696703i
\(526\) 0 0
\(527\) −9690.99 −0.801036
\(528\) 0 0
\(529\) −11556.6 −0.949833
\(530\) 0 0
\(531\) 4635.03 + 4661.35i 0.378801 + 0.380951i
\(532\) 0 0
\(533\) −7427.88 −0.603635
\(534\) 0 0
\(535\) 2420.99i 0.195642i
\(536\) 0 0
\(537\) 4559.77 1881.16i 0.366422 0.151170i
\(538\) 0 0
\(539\) 10670.5i 0.852712i
\(540\) 0 0
\(541\) 13039.6i 1.03626i −0.855301 0.518131i \(-0.826628\pi\)
0.855301 0.518131i \(-0.173372\pi\)
\(542\) 0 0
\(543\) −4202.63 + 1733.82i −0.332140 + 0.137027i
\(544\) 0 0
\(545\) 7055.63i 0.554550i
\(546\) 0 0
\(547\) −18647.7 −1.45762 −0.728811 0.684715i \(-0.759926\pi\)
−0.728811 + 0.684715i \(0.759926\pi\)
\(548\) 0 0
\(549\) 6824.17 + 6862.91i 0.530507 + 0.533519i
\(550\) 0 0
\(551\) −9092.75 −0.703020
\(552\) 0 0
\(553\) 1270.22 0.0976766
\(554\) 0 0
\(555\) 22881.1 9439.77i 1.75000 0.721975i
\(556\) 0 0
\(557\) 6780.36 0.515786 0.257893 0.966173i \(-0.416972\pi\)
0.257893 + 0.966173i \(0.416972\pi\)
\(558\) 0 0
\(559\) 11530.8i 0.872451i
\(560\) 0 0
\(561\) 9123.90 + 22115.5i 0.686651 + 1.66438i
\(562\) 0 0
\(563\) 5028.45i 0.376419i −0.982129 0.188209i \(-0.939732\pi\)
0.982129 0.188209i \(-0.0602683\pi\)
\(564\) 0 0
\(565\) 26710.9i 1.98891i
\(566\) 0 0
\(567\) 16547.7 + 93.6689i 1.22564 + 0.00693779i
\(568\) 0 0
\(569\) 3264.38i 0.240510i −0.992743 0.120255i \(-0.961629\pi\)
0.992743 0.120255i \(-0.0383712\pi\)
\(570\) 0 0
\(571\) 3622.18 0.265470 0.132735 0.991152i \(-0.457624\pi\)
0.132735 + 0.991152i \(0.457624\pi\)
\(572\) 0 0
\(573\) −1353.85 3281.62i −0.0987050 0.239252i
\(574\) 0 0
\(575\) −460.294 −0.0333836
\(576\) 0 0
\(577\) −9604.14 −0.692939 −0.346469 0.938061i \(-0.612620\pi\)
−0.346469 + 0.938061i \(0.612620\pi\)
\(578\) 0 0
\(579\) 6004.24 + 14553.7i 0.430963 + 1.04462i
\(580\) 0 0
\(581\) −21069.6 −1.50450
\(582\) 0 0
\(583\) 8440.51i 0.599606i
\(584\) 0 0
\(585\) −16275.1 16367.5i −1.15024 1.15677i
\(586\) 0 0
\(587\) 8942.98i 0.628818i 0.949288 + 0.314409i \(0.101806\pi\)
−0.949288 + 0.314409i \(0.898194\pi\)
\(588\) 0 0
\(589\) 14090.6i 0.985726i
\(590\) 0 0
\(591\) 5273.92 + 12783.5i 0.367073 + 0.889751i
\(592\) 0 0
\(593\) 9534.33i 0.660250i 0.943937 + 0.330125i \(0.107091\pi\)
−0.943937 + 0.330125i \(0.892909\pi\)
\(594\) 0 0
\(595\) 20221.3 1.39326
\(596\) 0 0
\(597\) −14292.0 + 5896.25i −0.979786 + 0.404217i
\(598\) 0 0
\(599\) 5691.25 0.388210 0.194105 0.980981i \(-0.437820\pi\)
0.194105 + 0.980981i \(0.437820\pi\)
\(600\) 0 0
\(601\) −13566.3 −0.920769 −0.460385 0.887720i \(-0.652288\pi\)
−0.460385 + 0.887720i \(0.652288\pi\)
\(602\) 0 0
\(603\) −8606.21 + 8557.64i −0.581214 + 0.577934i
\(604\) 0 0
\(605\) −30029.7 −2.01799
\(606\) 0 0
\(607\) 813.549i 0.0544002i −0.999630 0.0272001i \(-0.991341\pi\)
0.999630 0.0272001i \(-0.00865913\pi\)
\(608\) 0 0
\(609\) 9173.50 3784.59i 0.610393 0.251821i
\(610\) 0 0
\(611\) 5194.11i 0.343913i
\(612\) 0 0
\(613\) 25165.4i 1.65811i −0.559167 0.829055i \(-0.688879\pi\)
0.559167 0.829055i \(-0.311121\pi\)
\(614\) 0 0
\(615\) −5994.56 + 2473.10i −0.393047 + 0.162154i
\(616\) 0 0
\(617\) 24993.6i 1.63080i 0.578898 + 0.815400i \(0.303483\pi\)
−0.578898 + 0.815400i \(0.696517\pi\)
\(618\) 0 0
\(619\) 7970.28 0.517532 0.258766 0.965940i \(-0.416684\pi\)
0.258766 + 0.965940i \(0.416684\pi\)
\(620\) 0 0
\(621\) 3196.64 + 1340.02i 0.206565 + 0.0865911i
\(622\) 0 0
\(623\) −19383.8 −1.24654
\(624\) 0 0
\(625\) −17606.8 −1.12683
\(626\) 0 0
\(627\) −32155.7 + 13266.0i −2.04813 + 0.844968i
\(628\) 0 0
\(629\) −29544.0 −1.87281
\(630\) 0 0
\(631\) 4608.28i 0.290734i 0.989378 + 0.145367i \(0.0464362\pi\)
−0.989378 + 0.145367i \(0.953564\pi\)
\(632\) 0 0
\(633\) −5155.12 12495.5i −0.323693 0.784603i
\(634\) 0 0
\(635\) 16130.4i 1.00806i
\(636\) 0 0
\(637\) 12288.2i 0.764330i
\(638\) 0 0
\(639\) −6300.00 + 6264.44i −0.390022 + 0.387821i
\(640\) 0 0
\(641\) 4200.53i 0.258832i −0.991590 0.129416i \(-0.958690\pi\)
0.991590 0.129416i \(-0.0413102\pi\)
\(642\) 0 0
\(643\) −3546.92 −0.217538 −0.108769 0.994067i \(-0.534691\pi\)
−0.108769 + 0.994067i \(0.534691\pi\)
\(644\) 0 0
\(645\) 3839.15 + 9305.75i 0.234366 + 0.568083i
\(646\) 0 0
\(647\) −2851.58 −0.173272 −0.0866362 0.996240i \(-0.527612\pi\)
−0.0866362 + 0.996240i \(0.527612\pi\)
\(648\) 0 0
\(649\) 15080.5 0.912112
\(650\) 0 0
\(651\) 5864.79 + 14215.7i 0.353087 + 0.855850i
\(652\) 0 0
\(653\) −33080.9 −1.98247 −0.991237 0.132098i \(-0.957829\pi\)
−0.991237 + 0.132098i \(0.957829\pi\)
\(654\) 0 0
\(655\) 8143.92i 0.485816i
\(656\) 0 0
\(657\) −17723.1 + 17623.0i −1.05242 + 1.04648i
\(658\) 0 0
\(659\) 11761.6i 0.695244i 0.937635 + 0.347622i \(0.113011\pi\)
−0.937635 + 0.347622i \(0.886989\pi\)
\(660\) 0 0
\(661\) 2478.98i 0.145871i 0.997337 + 0.0729357i \(0.0232368\pi\)
−0.997337 + 0.0729357i \(0.976763\pi\)
\(662\) 0 0
\(663\) 10507.1 + 25468.4i 0.615481 + 1.49187i
\(664\) 0 0
\(665\) 29401.5i 1.71450i
\(666\) 0 0
\(667\) 2078.58 0.120664
\(668\) 0 0
\(669\) −19365.6 + 7989.42i −1.11916 + 0.461718i
\(670\) 0 0
\(671\) 22203.0 1.27740
\(672\) 0 0
\(673\) 5752.56 0.329487 0.164744 0.986336i \(-0.447320\pi\)
0.164744 + 0.986336i \(0.447320\pi\)
\(674\) 0 0
\(675\) −2410.61 1010.52i −0.137458 0.0576221i
\(676\) 0 0
\(677\) 25177.1 1.42930 0.714649 0.699483i \(-0.246586\pi\)
0.714649 + 0.699483i \(0.246586\pi\)
\(678\) 0 0
\(679\) 16229.8i 0.917292i
\(680\) 0 0
\(681\) −23972.8 + 9890.12i −1.34895 + 0.556520i
\(682\) 0 0
\(683\) 33891.4i 1.89871i −0.314203 0.949356i \(-0.601737\pi\)
0.314203 0.949356i \(-0.398263\pi\)
\(684\) 0 0
\(685\) 7364.38i 0.410772i
\(686\) 0 0
\(687\) 24454.6 10088.9i 1.35808 0.560285i
\(688\) 0 0
\(689\) 9720.15i 0.537457i
\(690\) 0 0
\(691\) −1194.54 −0.0657634 −0.0328817 0.999459i \(-0.510468\pi\)
−0.0328817 + 0.999459i \(0.510468\pi\)
\(692\) 0 0
\(693\) 26919.7 26767.7i 1.47560 1.46728i
\(694\) 0 0
\(695\) 37918.0 2.06951
\(696\) 0 0
\(697\) 7740.15 0.420630
\(698\) 0 0
\(699\) 1104.17 455.534i 0.0597477 0.0246493i
\(700\) 0 0
\(701\) 23289.7 1.25484 0.627419 0.778682i \(-0.284111\pi\)
0.627419 + 0.778682i \(0.284111\pi\)
\(702\) 0 0
\(703\) 42956.7i 2.30461i
\(704\) 0 0
\(705\) −1729.37 4191.83i −0.0923854 0.223934i
\(706\) 0 0
\(707\) 38650.2i 2.05600i
\(708\) 0 0
\(709\) 11916.8i 0.631236i −0.948886 0.315618i \(-0.897788\pi\)
0.948886 0.315618i \(-0.102212\pi\)
\(710\) 0 0
\(711\) 1065.31 + 1071.36i 0.0561918 + 0.0565108i
\(712\) 0 0
\(713\) 3221.08i 0.169187i
\(714\) 0 0
\(715\) −52952.3 −2.76966
\(716\) 0 0
\(717\) −3018.87 7317.48i −0.157241 0.381138i
\(718\) 0 0
\(719\) −5520.22 −0.286327 −0.143164 0.989699i \(-0.545728\pi\)
−0.143164 + 0.989699i \(0.545728\pi\)
\(720\) 0 0
\(721\) 14889.3 0.769080
\(722\) 0 0
\(723\) 3441.72 + 8342.41i 0.177038 + 0.429125i
\(724\) 0 0
\(725\) −1567.48 −0.0802961
\(726\) 0 0
\(727\) 28589.0i 1.45847i −0.684264 0.729234i \(-0.739876\pi\)
0.684264 0.729234i \(-0.260124\pi\)
\(728\) 0 0
\(729\) 13799.3 + 14035.7i 0.701078 + 0.713085i
\(730\) 0 0
\(731\) 12015.5i 0.607949i
\(732\) 0 0
\(733\) 23340.5i 1.17613i −0.808814 0.588064i \(-0.799890\pi\)
0.808814 0.588064i \(-0.200110\pi\)
\(734\) 0 0
\(735\) −4091.34 9917.05i −0.205322 0.497681i
\(736\) 0 0
\(737\) 27843.0i 1.39160i
\(738\) 0 0
\(739\) 8818.16 0.438946 0.219473 0.975619i \(-0.429566\pi\)
0.219473 + 0.975619i \(0.429566\pi\)
\(740\) 0 0
\(741\) −37030.7 + 15277.3i −1.83584 + 0.757388i
\(742\) 0 0
\(743\) −25836.1 −1.27569 −0.637843 0.770166i \(-0.720173\pi\)
−0.637843 + 0.770166i \(0.720173\pi\)
\(744\) 0 0
\(745\) 12809.0 0.629912
\(746\) 0 0
\(747\) −17670.8 17771.1i −0.865514 0.870427i
\(748\) 0 0
\(749\) 4585.49 0.223698
\(750\) 0 0
\(751\) 19166.2i 0.931272i 0.884976 + 0.465636i \(0.154174\pi\)
−0.884976 + 0.465636i \(0.845826\pi\)
\(752\) 0 0
\(753\) −5989.14 + 2470.86i −0.289849 + 0.119579i
\(754\) 0 0
\(755\) 16098.9i 0.776027i
\(756\) 0 0
\(757\) 18644.0i 0.895150i −0.894247 0.447575i \(-0.852288\pi\)
0.894247 0.447575i \(-0.147712\pi\)
\(758\) 0 0
\(759\) 7350.73 3032.59i 0.351534 0.145028i
\(760\) 0 0
\(761\) 5444.32i 0.259338i −0.991557 0.129669i \(-0.958608\pi\)
0.991557 0.129669i \(-0.0413915\pi\)
\(762\) 0 0
\(763\) −13363.8 −0.634077
\(764\) 0 0
\(765\) 16959.3 + 17055.5i 0.801522 + 0.806071i
\(766\) 0 0
\(767\) 17366.8 0.817573
\(768\) 0 0
\(769\) −5426.95 −0.254487 −0.127244 0.991871i \(-0.540613\pi\)
−0.127244 + 0.991871i \(0.540613\pi\)
\(770\) 0 0
\(771\) 31248.0 12891.6i 1.45962 0.602176i
\(772\) 0 0
\(773\) −27580.1 −1.28330 −0.641648 0.766999i \(-0.721749\pi\)
−0.641648 + 0.766999i \(0.721749\pi\)
\(774\) 0 0
\(775\) 2429.04i 0.112586i
\(776\) 0 0
\(777\) 17879.5 + 43338.2i 0.825511 + 2.00096i
\(778\) 0 0
\(779\) 11254.1i 0.517612i
\(780\) 0 0
\(781\) 20381.9i 0.933831i
\(782\) 0 0
\(783\) 10885.8 + 4563.28i 0.496840 + 0.208274i
\(784\) 0 0
\(785\) 31474.8i 1.43106i
\(786\) 0 0
\(787\) −25227.6 −1.14265 −0.571325 0.820724i \(-0.693571\pi\)
−0.571325 + 0.820724i \(0.693571\pi\)
\(788\) 0 0
\(789\) −15802.8 38304.4i −0.713045 1.72836i
\(790\) 0 0
\(791\) −50591.9 −2.27413
\(792\) 0 0
\(793\) 25569.2 1.14500
\(794\) 0 0
\(795\) 3236.30 + 7844.50i 0.144377 + 0.349957i
\(796\) 0 0
\(797\) 11553.8 0.513497 0.256748 0.966478i \(-0.417349\pi\)
0.256748 + 0.966478i \(0.417349\pi\)
\(798\) 0 0
\(799\) 5412.47i 0.239649i
\(800\) 0 0
\(801\) −16256.9 16349.2i −0.717116 0.721187i
\(802\) 0 0
\(803\) 57338.0i 2.51982i
\(804\) 0 0
\(805\) 6721.12i 0.294271i
\(806\) 0 0
\(807\) −1128.85 2736.22i −0.0492408 0.119355i
\(808\) 0 0
\(809\) 23754.7i 1.03235i −0.856483 0.516175i \(-0.827355\pi\)
0.856483 0.516175i \(-0.172645\pi\)
\(810\) 0 0
\(811\) 8303.89 0.359543 0.179771 0.983708i \(-0.442464\pi\)
0.179771 + 0.983708i \(0.442464\pi\)
\(812\) 0 0
\(813\) −32322.7 + 13334.9i −1.39435 + 0.575248i
\(814\) 0 0
\(815\) 11599.1 0.498526
\(816\) 0 0
\(817\) 17470.5 0.748120
\(818\) 0 0
\(819\) 31000.9 30825.9i 1.32266 1.31519i
\(820\) 0 0
\(821\) 38021.6 1.61628 0.808139 0.588992i \(-0.200475\pi\)
0.808139 + 0.588992i \(0.200475\pi\)
\(822\) 0 0
\(823\) 16842.2i 0.713345i 0.934230 + 0.356672i \(0.116089\pi\)
−0.934230 + 0.356672i \(0.883911\pi\)
\(824\) 0 0
\(825\) −5543.25 + 2286.90i −0.233929 + 0.0965088i
\(826\) 0 0
\(827\) 17470.7i 0.734600i −0.930102 0.367300i \(-0.880282\pi\)
0.930102 0.367300i \(-0.119718\pi\)
\(828\) 0 0
\(829\) 19906.7i 0.834001i 0.908906 + 0.417001i \(0.136919\pi\)
−0.908906 + 0.417001i \(0.863081\pi\)
\(830\) 0 0
\(831\) −14671.0 + 6052.60i −0.612430 + 0.252662i
\(832\) 0 0
\(833\) 12804.8i 0.532607i
\(834\) 0 0
\(835\) 793.604 0.0328908
\(836\) 0 0
\(837\) −7071.49 + 16869.2i −0.292027 + 0.696635i
\(838\) 0 0
\(839\) 18413.2 0.757682 0.378841 0.925462i \(-0.376323\pi\)
0.378841 + 0.925462i \(0.376323\pi\)
\(840\) 0 0
\(841\) −17310.6 −0.709772
\(842\) 0 0
\(843\) −3235.94 + 1335.01i −0.132208 + 0.0545435i
\(844\) 0 0
\(845\) −34650.1 −1.41065
\(846\) 0 0
\(847\) 56878.0i 2.30738i
\(848\) 0 0
\(849\) 3908.39 + 9473.59i 0.157992 + 0.382960i
\(850\) 0 0
\(851\) 9819.81i 0.395557i
\(852\) 0 0
\(853\) 23193.1i 0.930969i −0.885056 0.465485i \(-0.845880\pi\)
0.885056 0.465485i \(-0.154120\pi\)
\(854\) 0 0
\(855\) −24798.6 + 24658.6i −0.991922 + 0.986323i
\(856\) 0 0
\(857\) 16140.3i 0.643341i −0.946852 0.321671i \(-0.895756\pi\)
0.946852 0.321671i \(-0.104244\pi\)
\(858\) 0 0
\(859\) −2325.65 −0.0923751 −0.0461875 0.998933i \(-0.514707\pi\)
−0.0461875 + 0.998933i \(0.514707\pi\)
\(860\) 0 0
\(861\) −4684.19 11354.0i −0.185408 0.449413i
\(862\) 0 0
\(863\) −8816.27 −0.347751 −0.173876 0.984768i \(-0.555629\pi\)
−0.173876 + 0.984768i \(0.555629\pi\)
\(864\) 0 0
\(865\) −36690.4 −1.44221
\(866\) 0 0
\(867\) −1212.85 2939.83i −0.0475092 0.115158i
\(868\) 0 0
\(869\) 3466.09 0.135304
\(870\) 0 0
\(871\) 32064.2i 1.24736i
\(872\) 0 0
\(873\) 13688.9 13611.7i 0.530699 0.527703i
\(874\) 0 0
\(875\) 28937.2i 1.11801i
\(876\) 0 0
\(877\) 14921.3i 0.574524i 0.957852 + 0.287262i \(0.0927451\pi\)
−0.957852 + 0.287262i \(0.907255\pi\)
\(878\) 0 0
\(879\) 7215.34 + 17489.3i 0.276869 + 0.671105i
\(880\) 0 0
\(881\) 16861.0i 0.644793i −0.946605 0.322397i \(-0.895512\pi\)
0.946605 0.322397i \(-0.104488\pi\)
\(882\) 0 0
\(883\) 14458.8 0.551049 0.275524 0.961294i \(-0.411149\pi\)
0.275524 + 0.961294i \(0.411149\pi\)
\(884\) 0 0
\(885\) 14015.6 5782.23i 0.532350 0.219624i
\(886\) 0 0
\(887\) 19882.3 0.752629 0.376314 0.926492i \(-0.377191\pi\)
0.376314 + 0.926492i \(0.377191\pi\)
\(888\) 0 0
\(889\) −30552.0 −1.15262
\(890\) 0 0
\(891\) 45154.3 + 255.598i 1.69778 + 0.00961038i
\(892\) 0 0
\(893\) −7869.67 −0.294903
\(894\) 0 0
\(895\) 11376.7i 0.424895i
\(896\) 0 0
\(897\) 8465.15 3492.35i 0.315098 0.129996i
\(898\) 0 0
\(899\) 10969.0i 0.406938i
\(900\) 0 0
\(901\) 10128.8i 0.374516i
\(902\) 0 0
\(903\) −17625.6 + 7271.57i −0.649550 + 0.267976i
\(904\) 0 0
\(905\) 10485.6i 0.385142i
\(906\) 0 0
\(907\) −29281.5 −1.07197 −0.535985 0.844227i \(-0.680060\pi\)
−0.535985 + 0.844227i \(0.680060\pi\)
\(908\) 0 0
\(909\) 32599.3 32415.3i 1.18950 1.18278i
\(910\) 0 0
\(911\) −37513.5 −1.36430 −0.682150 0.731212i \(-0.738955\pi\)
−0.682150 + 0.731212i \(0.738955\pi\)
\(912\) 0 0
\(913\) −57493.3 −2.08406
\(914\) 0 0
\(915\) 20635.2 8513.20i 0.745551 0.307582i
\(916\) 0 0
\(917\) 15425.1 0.555485
\(918\) 0 0
\(919\) 22016.4i 0.790266i 0.918624 + 0.395133i \(0.129302\pi\)
−0.918624 + 0.395133i \(0.870698\pi\)
\(920\) 0 0
\(921\) 5642.69 + 13677.4i 0.201882 + 0.489343i
\(922\) 0 0
\(923\) 23471.9i 0.837041i
\(924\) 0 0
\(925\) 7405.20i 0.263223i
\(926\) 0 0
\(927\) 12487.4 + 12558.3i 0.442440 + 0.444951i
\(928\) 0 0
\(929\) 37944.4i 1.34006i 0.742334 + 0.670030i \(0.233719\pi\)
−0.742334 + 0.670030i \(0.766281\pi\)
\(930\) 0 0
\(931\) −18618.1 −0.655407
\(932\) 0 0
\(933\) 17408.5 + 42196.6i 0.610856 + 1.48066i
\(934\) 0 0
\(935\) 55178.5 1.92998
\(936\) 0 0
\(937\) −30773.5 −1.07292 −0.536460 0.843925i \(-0.680239\pi\)
−0.536460 + 0.843925i \(0.680239\pi\)
\(938\) 0 0
\(939\) 9492.69 + 23009.4i 0.329906 + 0.799663i
\(940\) 0 0
\(941\) 4127.42 0.142986 0.0714931 0.997441i \(-0.477224\pi\)
0.0714931 + 0.997441i \(0.477224\pi\)
\(942\) 0 0
\(943\) 2572.66i 0.0888414i
\(944\) 0 0
\(945\) 14755.4 35199.3i 0.507929 1.21167i
\(946\) 0 0
\(947\) 11313.0i 0.388196i 0.980982 + 0.194098i \(0.0621780\pi\)
−0.980982 + 0.194098i \(0.937822\pi\)
\(948\) 0 0
\(949\) 66030.9i 2.25864i
\(950\) 0 0
\(951\) −12663.7 30695.5i −0.431805 1.04666i
\(952\) 0 0
\(953\) 25633.6i 0.871304i 0.900115 + 0.435652i \(0.143482\pi\)
−0.900115 + 0.435652i \(0.856518\pi\)
\(954\) 0 0
\(955\) −8187.67 −0.277431
\(956\) 0 0
\(957\) 25032.1 10327.1i 0.845529 0.348829i
\(958\) 0 0
\(959\) −13948.6 −0.469679
\(960\) 0 0
\(961\) 12792.9 0.429420
\(962\) 0 0
\(963\) 3845.78 + 3867.61i 0.128690 + 0.129421i
\(964\) 0 0
\(965\) 36311.7 1.21131
\(966\) 0 0
\(967\) 50706.5i 1.68626i −0.537712 0.843129i \(-0.680711\pi\)
0.537712 0.843129i \(-0.319289\pi\)
\(968\) 0 0
\(969\) 38587.5 15919.5i 1.27927 0.527770i
\(970\) 0 0
\(971\) 48466.7i 1.60182i 0.598782 + 0.800912i \(0.295652\pi\)
−0.598782 + 0.800912i \(0.704348\pi\)
\(972\) 0 0
\(973\) 71818.8i 2.36630i
\(974\) 0 0
\(975\) −6383.64 + 2633.61i −0.209682 + 0.0865058i
\(976\) 0 0
\(977\) 33563.4i 1.09907i −0.835472 0.549533i \(-0.814806\pi\)
0.835472 0.549533i \(-0.185194\pi\)
\(978\) 0 0
\(979\) −52893.3 −1.72674
\(980\) 0 0
\(981\) −11208.0 11271.6i −0.364774 0.366845i
\(982\) 0 0
\(983\) 25587.6 0.830231 0.415115 0.909769i \(-0.363741\pi\)
0.415115 + 0.909769i \(0.363741\pi\)
\(984\) 0 0
\(985\) 31895.0 1.03173
\(986\) 0 0
\(987\) 7939.56 3275.52i 0.256048 0.105634i
\(988\) 0 0
\(989\) −3993.71 −0.128405
\(990\) 0 0
\(991\) 22594.0i 0.724239i 0.932132 + 0.362119i \(0.117947\pi\)
−0.932132 + 0.362119i \(0.882053\pi\)
\(992\) 0 0
\(993\) −6881.35 16679.8i −0.219912 0.533047i
\(994\) 0 0
\(995\) 35658.7i 1.13614i
\(996\) 0 0
\(997\) 5714.27i 0.181517i 0.995873 + 0.0907586i \(0.0289292\pi\)
−0.995873 + 0.0907586i \(0.971071\pi\)
\(998\) 0 0
\(999\) −21558.2 + 51427.5i −0.682754 + 1.62872i
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 768.4.f.g.383.8 12
3.2 odd 2 768.4.f.h.383.7 12
4.3 odd 2 768.4.f.e.383.5 12
8.3 odd 2 768.4.f.h.383.8 12
8.5 even 2 768.4.f.f.383.5 12
12.11 even 2 768.4.f.f.383.6 12
16.3 odd 4 384.4.c.c.383.11 yes 12
16.5 even 4 384.4.c.d.383.11 yes 12
16.11 odd 4 384.4.c.a.383.2 yes 12
16.13 even 4 384.4.c.b.383.2 yes 12
24.5 odd 2 768.4.f.e.383.6 12
24.11 even 2 inner 768.4.f.g.383.7 12
48.5 odd 4 384.4.c.a.383.1 12
48.11 even 4 384.4.c.d.383.12 yes 12
48.29 odd 4 384.4.c.c.383.12 yes 12
48.35 even 4 384.4.c.b.383.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.c.a.383.1 12 48.5 odd 4
384.4.c.a.383.2 yes 12 16.11 odd 4
384.4.c.b.383.1 yes 12 48.35 even 4
384.4.c.b.383.2 yes 12 16.13 even 4
384.4.c.c.383.11 yes 12 16.3 odd 4
384.4.c.c.383.12 yes 12 48.29 odd 4
384.4.c.d.383.11 yes 12 16.5 even 4
384.4.c.d.383.12 yes 12 48.11 even 4
768.4.f.e.383.5 12 4.3 odd 2
768.4.f.e.383.6 12 24.5 odd 2
768.4.f.f.383.5 12 8.5 even 2
768.4.f.f.383.6 12 12.11 even 2
768.4.f.g.383.7 12 24.11 even 2 inner
768.4.f.g.383.8 12 1.1 even 1 trivial
768.4.f.h.383.7 12 3.2 odd 2
768.4.f.h.383.8 12 8.3 odd 2