Properties

Label 76.5.h.a.65.2
Level $76$
Weight $5$
Character 76.65
Analytic conductor $7.856$
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] = [76,5,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("76.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.h (of order \(6\), degree \(2\), 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: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.2
Root \(0.500000 + 9.58497i\) of defining polynomial
Character \(\chi\) \(=\) 76.65
Dual form 76.5.h.a.69.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-9.05083 - 5.22550i) q^{3} +(-12.7505 + 22.0845i) q^{5} +27.7589 q^{7} +(14.1116 + 24.4421i) q^{9} +O(q^{10})\) \(q+(-9.05083 - 5.22550i) q^{3} +(-12.7505 + 22.0845i) q^{5} +27.7589 q^{7} +(14.1116 + 24.4421i) q^{9} +143.293 q^{11} +(123.911 - 71.5400i) q^{13} +(230.805 - 133.255i) q^{15} +(43.9489 - 76.1217i) q^{17} +(340.586 - 119.675i) q^{19} +(-251.241 - 145.054i) q^{21} +(350.792 + 607.590i) q^{23} +(-12.6488 - 21.9084i) q^{25} +551.569i q^{27} +(-37.5252 + 21.6652i) q^{29} +111.254i q^{31} +(-1296.92 - 748.776i) q^{33} +(-353.939 + 613.040i) q^{35} -1079.87i q^{37} -1495.33 q^{39} +(80.1784 + 46.2910i) q^{41} +(945.995 - 1638.51i) q^{43} -719.720 q^{45} +(1366.72 + 2367.22i) q^{47} -1630.45 q^{49} +(-795.547 + 459.309i) q^{51} +(2153.80 - 1243.50i) q^{53} +(-1827.05 + 3164.54i) q^{55} +(-3707.95 - 696.575i) q^{57} +(-5032.93 - 2905.77i) q^{59} +(1263.85 + 2189.06i) q^{61} +(391.723 + 678.485i) q^{63} +3648.67i q^{65} +(4300.24 - 2482.74i) q^{67} -7332.25i q^{69} +(3720.00 + 2147.74i) q^{71} +(-1588.81 + 2751.89i) q^{73} +264.385i q^{75} +3977.65 q^{77} +(4934.57 + 2848.97i) q^{79} +(4025.27 - 6971.97i) q^{81} +10463.5 q^{83} +(1120.74 + 1941.17i) q^{85} +452.845 q^{87} +(4822.97 - 2784.54i) q^{89} +(3439.63 - 1985.87i) q^{91} +(581.355 - 1006.94i) q^{93} +(-1699.68 + 9047.57i) q^{95} +(-15618.3 - 9017.22i) q^{97} +(2022.10 + 3502.38i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9} + 6 q^{11} - 93 q^{13} - 741 q^{15} - 483 q^{17} - 533 q^{19} + 972 q^{21} + 531 q^{23} - 217 q^{25} + 2025 q^{29} - 75 q^{33} - 1128 q^{35} - 2250 q^{39} - 1692 q^{41} - 63 q^{43} + 7976 q^{45} - 3471 q^{47} + 420 q^{49} + 6741 q^{51} - 3771 q^{53} - 2014 q^{55} + 7617 q^{57} - 9594 q^{59} + 1229 q^{61} + 1514 q^{63} + 7590 q^{67} + 963 q^{71} - 2838 q^{73} - 15408 q^{77} + 11073 q^{79} + 2086 q^{81} - 14202 q^{83} + 9455 q^{85} - 39510 q^{87} + 6525 q^{89} - 7686 q^{91} - 5316 q^{93} + 1521 q^{95} - 34110 q^{97} + 13220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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
\(3\) −9.05083 5.22550i −1.00565 0.580611i −0.0957331 0.995407i \(-0.530520\pi\)
−0.909914 + 0.414796i \(0.863853\pi\)
\(4\) 0 0
\(5\) −12.7505 + 22.0845i −0.510019 + 0.883378i 0.489914 + 0.871771i \(0.337028\pi\)
−0.999933 + 0.0116075i \(0.996305\pi\)
\(6\) 0 0
\(7\) 27.7589 0.566508 0.283254 0.959045i \(-0.408586\pi\)
0.283254 + 0.959045i \(0.408586\pi\)
\(8\) 0 0
\(9\) 14.1116 + 24.4421i 0.174218 + 0.301754i
\(10\) 0 0
\(11\) 143.293 1.18424 0.592119 0.805850i \(-0.298291\pi\)
0.592119 + 0.805850i \(0.298291\pi\)
\(12\) 0 0
\(13\) 123.911 71.5400i 0.733201 0.423314i −0.0863914 0.996261i \(-0.527534\pi\)
0.819592 + 0.572948i \(0.194200\pi\)
\(14\) 0 0
\(15\) 230.805 133.255i 1.02580 0.592245i
\(16\) 0 0
\(17\) 43.9489 76.1217i 0.152072 0.263397i −0.779917 0.625883i \(-0.784739\pi\)
0.931989 + 0.362486i \(0.118072\pi\)
\(18\) 0 0
\(19\) 340.586 119.675i 0.943452 0.331509i
\(20\) 0 0
\(21\) −251.241 145.054i −0.569707 0.328920i
\(22\) 0 0
\(23\) 350.792 + 607.590i 0.663123 + 1.14856i 0.979791 + 0.200026i \(0.0641027\pi\)
−0.316667 + 0.948537i \(0.602564\pi\)
\(24\) 0 0
\(25\) −12.6488 21.9084i −0.0202381 0.0350534i
\(26\) 0 0
\(27\) 551.569i 0.756611i
\(28\) 0 0
\(29\) −37.5252 + 21.6652i −0.0446197 + 0.0257612i −0.522144 0.852857i \(-0.674868\pi\)
0.477524 + 0.878619i \(0.341534\pi\)
\(30\) 0 0
\(31\) 111.254i 0.115769i 0.998323 + 0.0578843i \(0.0184354\pi\)
−0.998323 + 0.0578843i \(0.981565\pi\)
\(32\) 0 0
\(33\) −1296.92 748.776i −1.19093 0.687582i
\(34\) 0 0
\(35\) −353.939 + 613.040i −0.288929 + 0.500440i
\(36\) 0 0
\(37\) 1079.87i 0.788802i −0.918938 0.394401i \(-0.870952\pi\)
0.918938 0.394401i \(-0.129048\pi\)
\(38\) 0 0
\(39\) −1495.33 −0.983122
\(40\) 0 0
\(41\) 80.1784 + 46.2910i 0.0476969 + 0.0275378i 0.523659 0.851928i \(-0.324567\pi\)
−0.475962 + 0.879466i \(0.657900\pi\)
\(42\) 0 0
\(43\) 945.995 1638.51i 0.511625 0.886161i −0.488284 0.872685i \(-0.662377\pi\)
0.999909 0.0134763i \(-0.00428978\pi\)
\(44\) 0 0
\(45\) −719.720 −0.355417
\(46\) 0 0
\(47\) 1366.72 + 2367.22i 0.618703 + 1.07163i 0.989723 + 0.143001i \(0.0456751\pi\)
−0.371019 + 0.928625i \(0.620992\pi\)
\(48\) 0 0
\(49\) −1630.45 −0.679069
\(50\) 0 0
\(51\) −795.547 + 459.309i −0.305862 + 0.176589i
\(52\) 0 0
\(53\) 2153.80 1243.50i 0.766749 0.442683i −0.0649646 0.997888i \(-0.520693\pi\)
0.831714 + 0.555205i \(0.187360\pi\)
\(54\) 0 0
\(55\) −1827.05 + 3164.54i −0.603984 + 1.04613i
\(56\) 0 0
\(57\) −3707.95 696.575i −1.14126 0.214397i
\(58\) 0 0
\(59\) −5032.93 2905.77i −1.44583 0.834750i −0.447600 0.894234i \(-0.647721\pi\)
−0.998229 + 0.0594837i \(0.981055\pi\)
\(60\) 0 0
\(61\) 1263.85 + 2189.06i 0.339654 + 0.588299i 0.984368 0.176126i \(-0.0563565\pi\)
−0.644713 + 0.764425i \(0.723023\pi\)
\(62\) 0 0
\(63\) 391.723 + 678.485i 0.0986957 + 0.170946i
\(64\) 0 0
\(65\) 3648.67i 0.863591i
\(66\) 0 0
\(67\) 4300.24 2482.74i 0.957950 0.553073i 0.0624086 0.998051i \(-0.480122\pi\)
0.895542 + 0.444978i \(0.146788\pi\)
\(68\) 0 0
\(69\) 7332.25i 1.54007i
\(70\) 0 0
\(71\) 3720.00 + 2147.74i 0.737949 + 0.426055i 0.821323 0.570463i \(-0.193236\pi\)
−0.0833740 + 0.996518i \(0.526570\pi\)
\(72\) 0 0
\(73\) −1588.81 + 2751.89i −0.298143 + 0.516399i −0.975711 0.219061i \(-0.929701\pi\)
0.677568 + 0.735460i \(0.263034\pi\)
\(74\) 0 0
\(75\) 264.385i 0.0470018i
\(76\) 0 0
\(77\) 3977.65 0.670880
\(78\) 0 0
\(79\) 4934.57 + 2848.97i 0.790669 + 0.456493i 0.840198 0.542280i \(-0.182439\pi\)
−0.0495288 + 0.998773i \(0.515772\pi\)
\(80\) 0 0
\(81\) 4025.27 6971.97i 0.613514 1.06264i
\(82\) 0 0
\(83\) 10463.5 1.51888 0.759439 0.650579i \(-0.225474\pi\)
0.759439 + 0.650579i \(0.225474\pi\)
\(84\) 0 0
\(85\) 1120.74 + 1941.17i 0.155119 + 0.268674i
\(86\) 0 0
\(87\) 452.845 0.0598289
\(88\) 0 0
\(89\) 4822.97 2784.54i 0.608884 0.351539i −0.163645 0.986519i \(-0.552325\pi\)
0.772528 + 0.634980i \(0.218992\pi\)
\(90\) 0 0
\(91\) 3439.63 1985.87i 0.415364 0.239810i
\(92\) 0 0
\(93\) 581.355 1006.94i 0.0672165 0.116422i
\(94\) 0 0
\(95\) −1699.68 + 9047.57i −0.188330 + 1.00250i
\(96\) 0 0
\(97\) −15618.3 9017.22i −1.65993 0.958361i −0.972745 0.231878i \(-0.925513\pi\)
−0.687185 0.726482i \(-0.741154\pi\)
\(98\) 0 0
\(99\) 2022.10 + 3502.38i 0.206315 + 0.357349i
\(100\) 0 0
\(101\) −48.5674 84.1212i −0.00476104 0.00824636i 0.863635 0.504118i \(-0.168182\pi\)
−0.868396 + 0.495871i \(0.834849\pi\)
\(102\) 0 0
\(103\) 1061.32i 0.100040i 0.998748 + 0.0500198i \(0.0159285\pi\)
−0.998748 + 0.0500198i \(0.984072\pi\)
\(104\) 0 0
\(105\) 6406.87 3699.01i 0.581122 0.335511i
\(106\) 0 0
\(107\) 86.6771i 0.00757072i 0.999993 + 0.00378536i \(0.00120492\pi\)
−0.999993 + 0.00378536i \(0.998795\pi\)
\(108\) 0 0
\(109\) −17989.3 10386.1i −1.51412 0.874179i −0.999863 0.0165449i \(-0.994733\pi\)
−0.514260 0.857634i \(-0.671933\pi\)
\(110\) 0 0
\(111\) −5642.86 + 9773.72i −0.457987 + 0.793257i
\(112\) 0 0
\(113\) 18503.4i 1.44909i 0.689227 + 0.724546i \(0.257950\pi\)
−0.689227 + 0.724546i \(0.742050\pi\)
\(114\) 0 0
\(115\) −17891.1 −1.35282
\(116\) 0 0
\(117\) 3497.17 + 2019.09i 0.255473 + 0.147498i
\(118\) 0 0
\(119\) 1219.97 2113.05i 0.0861500 0.149216i
\(120\) 0 0
\(121\) 5891.84 0.402421
\(122\) 0 0
\(123\) −483.787 837.944i −0.0319775 0.0553866i
\(124\) 0 0
\(125\) −15293.0 −0.978750
\(126\) 0 0
\(127\) 16075.2 9281.03i 0.996665 0.575425i 0.0894051 0.995995i \(-0.471503\pi\)
0.907260 + 0.420571i \(0.138170\pi\)
\(128\) 0 0
\(129\) −17124.1 + 9886.59i −1.02903 + 0.594111i
\(130\) 0 0
\(131\) −7020.17 + 12159.3i −0.409077 + 0.708542i −0.994787 0.101979i \(-0.967483\pi\)
0.585710 + 0.810521i \(0.300816\pi\)
\(132\) 0 0
\(133\) 9454.29 3322.04i 0.534473 0.187803i
\(134\) 0 0
\(135\) −12181.1 7032.76i −0.668373 0.385886i
\(136\) 0 0
\(137\) 8900.54 + 15416.2i 0.474215 + 0.821364i 0.999564 0.0295224i \(-0.00939864\pi\)
−0.525349 + 0.850887i \(0.676065\pi\)
\(138\) 0 0
\(139\) −18227.9 31571.7i −0.943426 1.63406i −0.758872 0.651240i \(-0.774249\pi\)
−0.184554 0.982822i \(-0.559084\pi\)
\(140\) 0 0
\(141\) 28567.1i 1.43690i
\(142\) 0 0
\(143\) 17755.5 10251.2i 0.868284 0.501304i
\(144\) 0 0
\(145\) 1104.96i 0.0525548i
\(146\) 0 0
\(147\) 14756.9 + 8519.89i 0.682904 + 0.394275i
\(148\) 0 0
\(149\) −10085.2 + 17468.1i −0.454268 + 0.786816i −0.998646 0.0520247i \(-0.983433\pi\)
0.544378 + 0.838840i \(0.316766\pi\)
\(150\) 0 0
\(151\) 1839.49i 0.0806759i −0.999186 0.0403379i \(-0.987157\pi\)
0.999186 0.0403379i \(-0.0128434\pi\)
\(152\) 0 0
\(153\) 2480.76 0.105975
\(154\) 0 0
\(155\) −2456.97 1418.54i −0.102267 0.0590441i
\(156\) 0 0
\(157\) −13219.3 + 22896.5i −0.536301 + 0.928901i 0.462798 + 0.886464i \(0.346845\pi\)
−0.999099 + 0.0424369i \(0.986488\pi\)
\(158\) 0 0
\(159\) −25991.5 −1.02811
\(160\) 0 0
\(161\) 9737.59 + 16866.0i 0.375664 + 0.650670i
\(162\) 0 0
\(163\) −13879.3 −0.522386 −0.261193 0.965287i \(-0.584116\pi\)
−0.261193 + 0.965287i \(0.584116\pi\)
\(164\) 0 0
\(165\) 33072.6 19094.5i 1.21479 0.701359i
\(166\) 0 0
\(167\) −13289.5 + 7672.69i −0.476514 + 0.275115i −0.718962 0.695049i \(-0.755383\pi\)
0.242449 + 0.970164i \(0.422049\pi\)
\(168\) 0 0
\(169\) −4044.56 + 7005.39i −0.141611 + 0.245278i
\(170\) 0 0
\(171\) 7731.34 + 6635.83i 0.264401 + 0.226936i
\(172\) 0 0
\(173\) 14804.8 + 8547.56i 0.494664 + 0.285594i 0.726507 0.687159i \(-0.241142\pi\)
−0.231843 + 0.972753i \(0.574476\pi\)
\(174\) 0 0
\(175\) −351.116 608.152i −0.0114650 0.0198580i
\(176\) 0 0
\(177\) 30368.1 + 52599.2i 0.969330 + 1.67893i
\(178\) 0 0
\(179\) 30559.4i 0.953758i −0.878969 0.476879i \(-0.841768\pi\)
0.878969 0.476879i \(-0.158232\pi\)
\(180\) 0 0
\(181\) 48767.9 28156.2i 1.48860 0.859441i 0.488681 0.872463i \(-0.337478\pi\)
0.999915 + 0.0130215i \(0.00414497\pi\)
\(182\) 0 0
\(183\) 26417.1i 0.788828i
\(184\) 0 0
\(185\) 23848.3 + 13768.9i 0.696811 + 0.402304i
\(186\) 0 0
\(187\) 6297.56 10907.7i 0.180090 0.311925i
\(188\) 0 0
\(189\) 15310.9i 0.428626i
\(190\) 0 0
\(191\) −20263.1 −0.555443 −0.277721 0.960662i \(-0.589579\pi\)
−0.277721 + 0.960662i \(0.589579\pi\)
\(192\) 0 0
\(193\) 902.539 + 521.081i 0.0242299 + 0.0139891i 0.512066 0.858946i \(-0.328880\pi\)
−0.487836 + 0.872935i \(0.662214\pi\)
\(194\) 0 0
\(195\) 19066.1 33023.5i 0.501410 0.868468i
\(196\) 0 0
\(197\) 59986.7 1.54569 0.772845 0.634594i \(-0.218833\pi\)
0.772845 + 0.634594i \(0.218833\pi\)
\(198\) 0 0
\(199\) −24312.1 42109.9i −0.613928 1.06335i −0.990572 0.136996i \(-0.956255\pi\)
0.376644 0.926358i \(-0.377078\pi\)
\(200\) 0 0
\(201\) −51894.3 −1.28448
\(202\) 0 0
\(203\) −1041.66 + 601.401i −0.0252774 + 0.0145939i
\(204\) 0 0
\(205\) −2044.62 + 1180.46i −0.0486526 + 0.0280896i
\(206\) 0 0
\(207\) −9900.51 + 17148.2i −0.231056 + 0.400200i
\(208\) 0 0
\(209\) 48803.6 17148.6i 1.11727 0.392586i
\(210\) 0 0
\(211\) 30887.6 + 17833.0i 0.693777 + 0.400552i 0.805025 0.593240i \(-0.202152\pi\)
−0.111248 + 0.993793i \(0.535485\pi\)
\(212\) 0 0
\(213\) −22446.1 38877.7i −0.494744 0.856923i
\(214\) 0 0
\(215\) 24123.8 + 41783.6i 0.521877 + 0.903918i
\(216\) 0 0
\(217\) 3088.27i 0.0655838i
\(218\) 0 0
\(219\) 28760.0 16604.6i 0.599654 0.346211i
\(220\) 0 0
\(221\) 12576.4i 0.257497i
\(222\) 0 0
\(223\) 31201.2 + 18014.0i 0.627425 + 0.362244i 0.779754 0.626086i \(-0.215344\pi\)
−0.152329 + 0.988330i \(0.548677\pi\)
\(224\) 0 0
\(225\) 356.991 618.326i 0.00705167 0.0122139i
\(226\) 0 0
\(227\) 92863.0i 1.80215i 0.433663 + 0.901075i \(0.357221\pi\)
−0.433663 + 0.901075i \(0.642779\pi\)
\(228\) 0 0
\(229\) −56402.1 −1.07553 −0.537767 0.843094i \(-0.680732\pi\)
−0.537767 + 0.843094i \(0.680732\pi\)
\(230\) 0 0
\(231\) −36001.0 20785.2i −0.674669 0.389520i
\(232\) 0 0
\(233\) −23210.6 + 40202.0i −0.427538 + 0.740518i −0.996654 0.0817394i \(-0.973952\pi\)
0.569115 + 0.822258i \(0.307286\pi\)
\(234\) 0 0
\(235\) −69705.1 −1.26220
\(236\) 0 0
\(237\) −29774.6 51571.1i −0.530090 0.918142i
\(238\) 0 0
\(239\) −35290.9 −0.617828 −0.308914 0.951090i \(-0.599965\pi\)
−0.308914 + 0.951090i \(0.599965\pi\)
\(240\) 0 0
\(241\) −96639.8 + 55795.0i −1.66388 + 0.960642i −0.693045 + 0.720894i \(0.743731\pi\)
−0.970835 + 0.239748i \(0.922935\pi\)
\(242\) 0 0
\(243\) −34172.5 + 19729.5i −0.578714 + 0.334121i
\(244\) 0 0
\(245\) 20788.9 36007.5i 0.346338 0.599875i
\(246\) 0 0
\(247\) 33640.8 39194.5i 0.551407 0.642439i
\(248\) 0 0
\(249\) −94703.8 54677.2i −1.52746 0.881877i
\(250\) 0 0
\(251\) 18656.9 + 32314.8i 0.296137 + 0.512924i 0.975249 0.221111i \(-0.0709682\pi\)
−0.679112 + 0.734035i \(0.737635\pi\)
\(252\) 0 0
\(253\) 50266.0 + 87063.3i 0.785296 + 1.36017i
\(254\) 0 0
\(255\) 23425.6i 0.360256i
\(256\) 0 0
\(257\) −54521.2 + 31477.8i −0.825466 + 0.476583i −0.852298 0.523057i \(-0.824792\pi\)
0.0268315 + 0.999640i \(0.491458\pi\)
\(258\) 0 0
\(259\) 29976.0i 0.446862i
\(260\) 0 0
\(261\) −1059.08 611.462i −0.0155471 0.00897612i
\(262\) 0 0
\(263\) 43980.5 76176.4i 0.635841 1.10131i −0.350495 0.936565i \(-0.613987\pi\)
0.986336 0.164745i \(-0.0526800\pi\)
\(264\) 0 0
\(265\) 63420.6i 0.903106i
\(266\) 0 0
\(267\) −58202.5 −0.816430
\(268\) 0 0
\(269\) −51001.2 29445.6i −0.704816 0.406926i 0.104323 0.994544i \(-0.466733\pi\)
−0.809139 + 0.587618i \(0.800066\pi\)
\(270\) 0 0
\(271\) 44150.5 76470.9i 0.601170 1.04126i −0.391475 0.920189i \(-0.628035\pi\)
0.992644 0.121068i \(-0.0386318\pi\)
\(272\) 0 0
\(273\) −41508.6 −0.556946
\(274\) 0 0
\(275\) −1812.48 3139.31i −0.0239667 0.0415116i
\(276\) 0 0
\(277\) 73761.7 0.961327 0.480664 0.876905i \(-0.340396\pi\)
0.480664 + 0.876905i \(0.340396\pi\)
\(278\) 0 0
\(279\) −2719.27 + 1569.97i −0.0349336 + 0.0201689i
\(280\) 0 0
\(281\) 51830.2 29924.2i 0.656403 0.378975i −0.134502 0.990913i \(-0.542943\pi\)
0.790905 + 0.611939i \(0.209610\pi\)
\(282\) 0 0
\(283\) −70780.2 + 122595.i −0.883769 + 1.53073i −0.0366506 + 0.999328i \(0.511669\pi\)
−0.847118 + 0.531404i \(0.821664\pi\)
\(284\) 0 0
\(285\) 62661.5 73006.3i 0.771456 0.898816i
\(286\) 0 0
\(287\) 2225.66 + 1284.99i 0.0270206 + 0.0156004i
\(288\) 0 0
\(289\) 37897.5 + 65640.4i 0.453748 + 0.785915i
\(290\) 0 0
\(291\) 94238.9 + 163227.i 1.11287 + 1.92755i
\(292\) 0 0
\(293\) 171279.i 1.99512i −0.0697935 0.997561i \(-0.522234\pi\)
0.0697935 0.997561i \(-0.477766\pi\)
\(294\) 0 0
\(295\) 128344. 74099.7i 1.47480 0.851476i
\(296\) 0 0
\(297\) 79035.9i 0.896007i
\(298\) 0 0
\(299\) 86933.9 + 50191.3i 0.972404 + 0.561418i
\(300\) 0 0
\(301\) 26259.8 45483.2i 0.289840 0.502017i
\(302\) 0 0
\(303\) 1015.15i 0.0110572i
\(304\) 0 0
\(305\) −64458.9 −0.692921
\(306\) 0 0
\(307\) 23997.7 + 13855.1i 0.254620 + 0.147005i 0.621878 0.783114i \(-0.286370\pi\)
−0.367258 + 0.930119i \(0.619703\pi\)
\(308\) 0 0
\(309\) 5545.93 9605.83i 0.0580841 0.100605i
\(310\) 0 0
\(311\) −189674. −1.96105 −0.980524 0.196401i \(-0.937075\pi\)
−0.980524 + 0.196401i \(0.937075\pi\)
\(312\) 0 0
\(313\) −79050.5 136919.i −0.806893 1.39758i −0.915006 0.403441i \(-0.867814\pi\)
0.108113 0.994139i \(-0.465519\pi\)
\(314\) 0 0
\(315\) −19978.6 −0.201347
\(316\) 0 0
\(317\) 39512.7 22812.7i 0.393204 0.227017i −0.290343 0.956923i \(-0.593769\pi\)
0.683548 + 0.729906i \(0.260436\pi\)
\(318\) 0 0
\(319\) −5377.09 + 3104.46i −0.0528404 + 0.0305074i
\(320\) 0 0
\(321\) 452.931 784.500i 0.00439564 0.00761347i
\(322\) 0 0
\(323\) 5858.52 31185.6i 0.0561543 0.298915i
\(324\) 0 0
\(325\) −3134.65 1809.79i −0.0296771 0.0171341i
\(326\) 0 0
\(327\) 108545. + 188006.i 1.01512 + 1.75823i
\(328\) 0 0
\(329\) 37938.5 + 65711.4i 0.350500 + 0.607084i
\(330\) 0 0
\(331\) 51241.9i 0.467702i −0.972272 0.233851i \(-0.924867\pi\)
0.972272 0.233851i \(-0.0751328\pi\)
\(332\) 0 0
\(333\) 26394.3 15238.7i 0.238024 0.137423i
\(334\) 0 0
\(335\) 126625.i 1.12831i
\(336\) 0 0
\(337\) 7928.49 + 4577.52i 0.0698121 + 0.0403060i 0.534500 0.845169i \(-0.320500\pi\)
−0.464688 + 0.885475i \(0.653833\pi\)
\(338\) 0 0
\(339\) 96689.7 167471.i 0.841358 1.45727i
\(340\) 0 0
\(341\) 15941.8i 0.137098i
\(342\) 0 0
\(343\) −111908. −0.951205
\(344\) 0 0
\(345\) 161929. + 93489.6i 1.36046 + 0.785462i
\(346\) 0 0
\(347\) 76816.1 133049.i 0.637960 1.10498i −0.347920 0.937524i \(-0.613112\pi\)
0.985880 0.167455i \(-0.0535548\pi\)
\(348\) 0 0
\(349\) 168335. 1.38205 0.691023 0.722832i \(-0.257160\pi\)
0.691023 + 0.722832i \(0.257160\pi\)
\(350\) 0 0
\(351\) 39459.2 + 68345.4i 0.320283 + 0.554747i
\(352\) 0 0
\(353\) −130340. −1.04599 −0.522994 0.852336i \(-0.675185\pi\)
−0.522994 + 0.852336i \(0.675185\pi\)
\(354\) 0 0
\(355\) −94863.5 + 54769.5i −0.752736 + 0.434592i
\(356\) 0 0
\(357\) −22083.5 + 12749.9i −0.173273 + 0.100039i
\(358\) 0 0
\(359\) −18849.9 + 32649.0i −0.146258 + 0.253327i −0.929842 0.367960i \(-0.880056\pi\)
0.783583 + 0.621287i \(0.213390\pi\)
\(360\) 0 0
\(361\) 101677. 81519.2i 0.780203 0.625526i
\(362\) 0 0
\(363\) −53326.0 30787.8i −0.404693 0.233650i
\(364\) 0 0
\(365\) −40516.0 70175.8i −0.304117 0.526747i
\(366\) 0 0
\(367\) 48448.8 + 83915.8i 0.359709 + 0.623034i 0.987912 0.155015i \(-0.0495427\pi\)
−0.628203 + 0.778049i \(0.716209\pi\)
\(368\) 0 0
\(369\) 2612.97i 0.0191903i
\(370\) 0 0
\(371\) 59787.0 34518.0i 0.434369 0.250783i
\(372\) 0 0
\(373\) 125723.i 0.903644i −0.892108 0.451822i \(-0.850774\pi\)
0.892108 0.451822i \(-0.149226\pi\)
\(374\) 0 0
\(375\) 138414. + 79913.4i 0.984278 + 0.568273i
\(376\) 0 0
\(377\) −3099.85 + 5369.10i −0.0218101 + 0.0377762i
\(378\) 0 0
\(379\) 7410.05i 0.0515873i −0.999667 0.0257936i \(-0.991789\pi\)
0.999667 0.0257936i \(-0.00821128\pi\)
\(380\) 0 0
\(381\) −193992. −1.33639
\(382\) 0 0
\(383\) −105519. 60921.6i −0.719340 0.415311i 0.0951699 0.995461i \(-0.469661\pi\)
−0.814510 + 0.580150i \(0.802994\pi\)
\(384\) 0 0
\(385\) −50716.9 + 87844.2i −0.342161 + 0.592641i
\(386\) 0 0
\(387\) 53398.2 0.356537
\(388\) 0 0
\(389\) −129529. 224351.i −0.855990 1.48262i −0.875724 0.482812i \(-0.839615\pi\)
0.0197344 0.999805i \(-0.493718\pi\)
\(390\) 0 0
\(391\) 61667.6 0.403370
\(392\) 0 0
\(393\) 127077. 73367.7i 0.822774 0.475029i
\(394\) 0 0
\(395\) −125836. + 72651.5i −0.806512 + 0.465640i
\(396\) 0 0
\(397\) −109875. + 190310.i −0.697139 + 1.20748i 0.272315 + 0.962208i \(0.412211\pi\)
−0.969454 + 0.245273i \(0.921122\pi\)
\(398\) 0 0
\(399\) −102928. 19336.1i −0.646531 0.121457i
\(400\) 0 0
\(401\) −200338. 115665.i −1.24588 0.719307i −0.275593 0.961275i \(-0.588874\pi\)
−0.970284 + 0.241967i \(0.922207\pi\)
\(402\) 0 0
\(403\) 7959.08 + 13785.5i 0.0490064 + 0.0848816i
\(404\) 0 0
\(405\) 102648. + 177792.i 0.625807 + 1.08393i
\(406\) 0 0
\(407\) 154738.i 0.934130i
\(408\) 0 0
\(409\) 36379.7 21003.8i 0.217477 0.125560i −0.387305 0.921952i \(-0.626594\pi\)
0.604781 + 0.796392i \(0.293261\pi\)
\(410\) 0 0
\(411\) 186039.i 1.10134i
\(412\) 0 0
\(413\) −139709. 80660.8i −0.819073 0.472892i
\(414\) 0 0
\(415\) −133415. + 231082.i −0.774656 + 1.34174i
\(416\) 0 0
\(417\) 381000.i 2.19105i
\(418\) 0 0
\(419\) −17406.8 −0.0991494 −0.0495747 0.998770i \(-0.515787\pi\)
−0.0495747 + 0.998770i \(0.515787\pi\)
\(420\) 0 0
\(421\) −161802. 93416.2i −0.912891 0.527058i −0.0315305 0.999503i \(-0.510038\pi\)
−0.881360 + 0.472445i \(0.843371\pi\)
\(422\) 0 0
\(423\) −38573.2 + 66810.8i −0.215578 + 0.373393i
\(424\) 0 0
\(425\) −2223.60 −0.0123106
\(426\) 0 0
\(427\) 35083.2 + 60765.8i 0.192417 + 0.333276i
\(428\) 0 0
\(429\) −214270. −1.16425
\(430\) 0 0
\(431\) 9163.54 5290.57i 0.0493297 0.0284805i −0.475132 0.879914i \(-0.657600\pi\)
0.524462 + 0.851434i \(0.324266\pi\)
\(432\) 0 0
\(433\) −188173. + 108642.i −1.00365 + 0.579458i −0.909326 0.416083i \(-0.863402\pi\)
−0.0943244 + 0.995542i \(0.530069\pi\)
\(434\) 0 0
\(435\) −5773.99 + 10000.8i −0.0305139 + 0.0528516i
\(436\) 0 0
\(437\) 192188. + 164956.i 1.00638 + 0.863782i
\(438\) 0 0
\(439\) −27000.5 15588.7i −0.140101 0.0808875i 0.428311 0.903631i \(-0.359109\pi\)
−0.568412 + 0.822744i \(0.692442\pi\)
\(440\) 0 0
\(441\) −23008.3 39851.5i −0.118306 0.204912i
\(442\) 0 0
\(443\) −84484.4 146331.i −0.430496 0.745641i 0.566420 0.824117i \(-0.308328\pi\)
−0.996916 + 0.0784760i \(0.974995\pi\)
\(444\) 0 0
\(445\) 142017.i 0.717166i
\(446\) 0 0
\(447\) 182559. 105400.i 0.913667 0.527506i
\(448\) 0 0
\(449\) 240682.i 1.19385i −0.802297 0.596926i \(-0.796389\pi\)
0.802297 0.596926i \(-0.203611\pi\)
\(450\) 0 0
\(451\) 11489.0 + 6633.17i 0.0564845 + 0.0326113i
\(452\) 0 0
\(453\) −9612.25 + 16648.9i −0.0468413 + 0.0811315i
\(454\) 0 0
\(455\) 101283.i 0.489231i
\(456\) 0 0
\(457\) 227190. 1.08782 0.543911 0.839143i \(-0.316943\pi\)
0.543911 + 0.839143i \(0.316943\pi\)
\(458\) 0 0
\(459\) 41986.4 + 24240.8i 0.199289 + 0.115059i
\(460\) 0 0
\(461\) −14069.9 + 24369.7i −0.0662046 + 0.114670i −0.897228 0.441568i \(-0.854422\pi\)
0.831023 + 0.556238i \(0.187756\pi\)
\(462\) 0 0
\(463\) 81970.7 0.382381 0.191191 0.981553i \(-0.438765\pi\)
0.191191 + 0.981553i \(0.438765\pi\)
\(464\) 0 0
\(465\) 14825.1 + 25677.8i 0.0685633 + 0.118755i
\(466\) 0 0
\(467\) −126102. −0.578213 −0.289106 0.957297i \(-0.593358\pi\)
−0.289106 + 0.957297i \(0.593358\pi\)
\(468\) 0 0
\(469\) 119370. 68918.2i 0.542686 0.313320i
\(470\) 0 0
\(471\) 239291. 138155.i 1.07866 0.622764i
\(472\) 0 0
\(473\) 135554. 234787.i 0.605886 1.04943i
\(474\) 0 0
\(475\) −6929.89 5947.94i −0.0307142 0.0263621i
\(476\) 0 0
\(477\) 60787.3 + 35095.5i 0.267163 + 0.154246i
\(478\) 0 0
\(479\) −57621.7 99803.7i −0.251139 0.434986i 0.712700 0.701469i \(-0.247472\pi\)
−0.963840 + 0.266482i \(0.914139\pi\)
\(480\) 0 0
\(481\) −77253.9 133808.i −0.333911 0.578350i
\(482\) 0 0
\(483\) 203535.i 0.872459i
\(484\) 0 0
\(485\) 398281. 229947.i 1.69319 0.977564i
\(486\) 0 0
\(487\) 135408.i 0.570935i 0.958388 + 0.285467i \(0.0921488\pi\)
−0.958388 + 0.285467i \(0.907851\pi\)
\(488\) 0 0
\(489\) 125619. + 72526.1i 0.525336 + 0.303303i
\(490\) 0 0
\(491\) −202101. + 350049.i −0.838312 + 1.45200i 0.0529937 + 0.998595i \(0.483124\pi\)
−0.891305 + 0.453404i \(0.850210\pi\)
\(492\) 0 0
\(493\) 3808.64i 0.0156702i
\(494\) 0 0
\(495\) −103131. −0.420899
\(496\) 0 0
\(497\) 103263. + 59619.0i 0.418054 + 0.241363i
\(498\) 0 0
\(499\) 220006. 381062.i 0.883557 1.53037i 0.0361977 0.999345i \(-0.488475\pi\)
0.847359 0.531020i \(-0.178191\pi\)
\(500\) 0 0
\(501\) 160375. 0.638940
\(502\) 0 0
\(503\) 140225. + 242877.i 0.554229 + 0.959953i 0.997963 + 0.0637944i \(0.0203202\pi\)
−0.443734 + 0.896159i \(0.646346\pi\)
\(504\) 0 0
\(505\) 2477.03 0.00971288
\(506\) 0 0
\(507\) 73213.2 42269.7i 0.284822 0.164442i
\(508\) 0 0
\(509\) −87780.5 + 50680.1i −0.338815 + 0.195615i −0.659748 0.751487i \(-0.729337\pi\)
0.320933 + 0.947102i \(0.396004\pi\)
\(510\) 0 0
\(511\) −44103.5 + 76389.4i −0.168900 + 0.292544i
\(512\) 0 0
\(513\) 66009.0 + 187857.i 0.250824 + 0.713826i
\(514\) 0 0
\(515\) −23438.7 13532.3i −0.0883729 0.0510221i
\(516\) 0 0
\(517\) 195841. + 339206.i 0.732692 + 1.26906i
\(518\) 0 0
\(519\) −89330.5 154725.i −0.331638 0.574415i
\(520\) 0 0
\(521\) 3847.69i 0.0141751i −0.999975 0.00708753i \(-0.997744\pi\)
0.999975 0.00708753i \(-0.00225605\pi\)
\(522\) 0 0
\(523\) 126432. 72995.3i 0.462223 0.266865i −0.250755 0.968051i \(-0.580679\pi\)
0.712979 + 0.701186i \(0.247346\pi\)
\(524\) 0 0
\(525\) 7339.03i 0.0266269i
\(526\) 0 0
\(527\) 8468.81 + 4889.47i 0.0304931 + 0.0176052i
\(528\) 0 0
\(529\) −106190. + 183926.i −0.379464 + 0.657252i
\(530\) 0 0
\(531\) 164021.i 0.581713i
\(532\) 0 0
\(533\) 13246.6 0.0466285
\(534\) 0 0
\(535\) −1914.22 1105.17i −0.00668781 0.00386121i
\(536\) 0 0
\(537\) −159688. + 276587.i −0.553762 + 0.959144i
\(538\) 0 0
\(539\) −233631. −0.804180
\(540\) 0 0
\(541\) 210834. + 365176.i 0.720355 + 1.24769i 0.960857 + 0.277043i \(0.0893546\pi\)
−0.240502 + 0.970649i \(0.577312\pi\)
\(542\) 0 0
\(543\) −588520. −1.99600
\(544\) 0 0
\(545\) 458744. 264856.i 1.54446 0.891696i
\(546\) 0 0
\(547\) 86529.9 49958.1i 0.289196 0.166967i −0.348383 0.937352i \(-0.613269\pi\)
0.637579 + 0.770385i \(0.279936\pi\)
\(548\) 0 0
\(549\) −35670.1 + 61782.5i −0.118348 + 0.204984i
\(550\) 0 0
\(551\) −10187.8 + 11869.7i −0.0335565 + 0.0390963i
\(552\) 0 0
\(553\) 136978. + 79084.3i 0.447920 + 0.258607i
\(554\) 0 0
\(555\) −143898. 249239.i −0.467164 0.809152i
\(556\) 0 0
\(557\) −128227. 222096.i −0.413304 0.715864i 0.581945 0.813228i \(-0.302292\pi\)
−0.995249 + 0.0973645i \(0.968959\pi\)
\(558\) 0 0
\(559\) 270706.i 0.866312i
\(560\) 0 0
\(561\) −113996. + 65815.7i −0.362213 + 0.209124i
\(562\) 0 0
\(563\) 5702.78i 0.0179916i 0.999960 + 0.00899580i \(0.00286349\pi\)
−0.999960 + 0.00899580i \(0.997137\pi\)
\(564\) 0 0
\(565\) −408639. 235928.i −1.28010 0.739064i
\(566\) 0 0
\(567\) 111737. 193534.i 0.347560 0.601992i
\(568\) 0 0
\(569\) 264278.i 0.816275i 0.912920 + 0.408138i \(0.133822\pi\)
−0.912920 + 0.408138i \(0.866178\pi\)
\(570\) 0 0
\(571\) −140891. −0.432126 −0.216063 0.976379i \(-0.569322\pi\)
−0.216063 + 0.976379i \(0.569322\pi\)
\(572\) 0 0
\(573\) 183398. + 105885.i 0.558579 + 0.322496i
\(574\) 0 0
\(575\) 8874.20 15370.6i 0.0268407 0.0464894i
\(576\) 0 0
\(577\) −33455.7 −0.100489 −0.0502445 0.998737i \(-0.516000\pi\)
−0.0502445 + 0.998737i \(0.516000\pi\)
\(578\) 0 0
\(579\) −5445.82 9432.43i −0.0162445 0.0281363i
\(580\) 0 0
\(581\) 290456. 0.860456
\(582\) 0 0
\(583\) 308624. 178184.i 0.908014 0.524242i
\(584\) 0 0
\(585\) −89181.2 + 51488.8i −0.260592 + 0.150453i
\(586\) 0 0
\(587\) −151917. + 263128.i −0.440890 + 0.763644i −0.997756 0.0669588i \(-0.978670\pi\)
0.556866 + 0.830602i \(0.312004\pi\)
\(588\) 0 0
\(589\) 13314.3 + 37891.4i 0.0383784 + 0.109222i
\(590\) 0 0
\(591\) −542929. 313460.i −1.55442 0.897445i
\(592\) 0 0
\(593\) −130746. 226459.i −0.371809 0.643992i 0.618035 0.786151i \(-0.287929\pi\)
−0.989844 + 0.142158i \(0.954596\pi\)
\(594\) 0 0
\(595\) 31110.4 + 53884.8i 0.0878763 + 0.152206i
\(596\) 0 0
\(597\) 508172.i 1.42581i
\(598\) 0 0
\(599\) −422627. + 244004.i −1.17789 + 0.680053i −0.955524 0.294912i \(-0.904710\pi\)
−0.222361 + 0.974964i \(0.571376\pi\)
\(600\) 0 0
\(601\) 362708.i 1.00417i −0.864818 0.502086i \(-0.832566\pi\)
0.864818 0.502086i \(-0.167434\pi\)
\(602\) 0 0
\(603\) 121367. + 70071.2i 0.333784 + 0.192710i
\(604\) 0 0
\(605\) −75123.7 + 130118.i −0.205242 + 0.355490i
\(606\) 0 0
\(607\) 560324.i 1.52077i −0.649475 0.760383i \(-0.725011\pi\)
0.649475 0.760383i \(-0.274989\pi\)
\(608\) 0 0
\(609\) 12570.5 0.0338935
\(610\) 0 0
\(611\) 338702. + 195550.i 0.907267 + 0.523811i
\(612\) 0 0
\(613\) −197665. + 342365.i −0.526027 + 0.911106i 0.473513 + 0.880787i \(0.342986\pi\)
−0.999540 + 0.0303191i \(0.990348\pi\)
\(614\) 0 0
\(615\) 24674.1 0.0652365
\(616\) 0 0
\(617\) −47374.0 82054.2i −0.124443 0.215541i 0.797072 0.603884i \(-0.206381\pi\)
−0.921515 + 0.388343i \(0.873048\pi\)
\(618\) 0 0
\(619\) 255301. 0.666303 0.333151 0.942873i \(-0.391888\pi\)
0.333151 + 0.942873i \(0.391888\pi\)
\(620\) 0 0
\(621\) −335128. + 193486.i −0.869015 + 0.501726i
\(622\) 0 0
\(623\) 133880. 77295.7i 0.344937 0.199150i
\(624\) 0 0
\(625\) 202898. 351430.i 0.519419 0.899660i
\(626\) 0 0
\(627\) −531322. 99814.2i −1.35152 0.253897i
\(628\) 0 0
\(629\) −82201.5 47459.1i −0.207768 0.119955i
\(630\) 0 0
\(631\) 392479. + 679794.i 0.985730 + 1.70733i 0.638646 + 0.769501i \(0.279495\pi\)
0.347084 + 0.937834i \(0.387172\pi\)
\(632\) 0 0
\(633\) −186372. 322807.i −0.465130 0.805629i
\(634\) 0 0
\(635\) 473350.i 1.17391i
\(636\) 0 0
\(637\) −202030. + 116642.i −0.497894 + 0.287459i
\(638\) 0 0
\(639\) 121233.i 0.296906i
\(640\) 0 0
\(641\) −55185.8 31861.5i −0.134311 0.0775444i 0.431339 0.902190i \(-0.358041\pi\)
−0.565650 + 0.824646i \(0.691375\pi\)
\(642\) 0 0
\(643\) −114558. + 198420.i −0.277079 + 0.479915i −0.970657 0.240466i \(-0.922700\pi\)
0.693579 + 0.720381i \(0.256033\pi\)
\(644\) 0 0
\(645\) 504235.i 1.21203i
\(646\) 0 0
\(647\) 439341. 1.04953 0.524763 0.851248i \(-0.324154\pi\)
0.524763 + 0.851248i \(0.324154\pi\)
\(648\) 0 0
\(649\) −721183. 416375.i −1.71221 0.988543i
\(650\) 0 0
\(651\) 16137.8 27951.4i 0.0380786 0.0659541i
\(652\) 0 0
\(653\) 296297. 0.694866 0.347433 0.937705i \(-0.387053\pi\)
0.347433 + 0.937705i \(0.387053\pi\)
\(654\) 0 0
\(655\) −179021. 310073.i −0.417274 0.722739i
\(656\) 0 0
\(657\) −89682.7 −0.207768
\(658\) 0 0
\(659\) −642280. + 370820.i −1.47895 + 0.853872i −0.999716 0.0238155i \(-0.992419\pi\)
−0.479233 + 0.877688i \(0.659085\pi\)
\(660\) 0 0
\(661\) −287667. + 166084.i −0.658395 + 0.380125i −0.791665 0.610955i \(-0.790786\pi\)
0.133270 + 0.991080i \(0.457452\pi\)
\(662\) 0 0
\(663\) −65718.0 + 113827.i −0.149505 + 0.258951i
\(664\) 0 0
\(665\) −47181.1 + 251150.i −0.106690 + 0.567924i
\(666\) 0 0
\(667\) −26327.1 15199.9i −0.0591767 0.0341657i
\(668\) 0 0
\(669\) −188265. 326084.i −0.420646 0.728580i
\(670\) 0 0
\(671\) 181101. + 313677.i 0.402232 + 0.696686i
\(672\) 0 0
\(673\) 40533.1i 0.0894912i 0.998998 + 0.0447456i \(0.0142477\pi\)
−0.998998 + 0.0447456i \(0.985752\pi\)
\(674\) 0 0
\(675\) 12084.0 6976.69i 0.0265218 0.0153123i
\(676\) 0 0
\(677\) 392617.i 0.856627i 0.903630 + 0.428314i \(0.140892\pi\)
−0.903630 + 0.428314i \(0.859108\pi\)
\(678\) 0 0
\(679\) −433546. 250308.i −0.940363 0.542919i
\(680\) 0 0
\(681\) 485255. 840487.i 1.04635 1.81233i
\(682\) 0 0
\(683\) 509603.i 1.09242i 0.837648 + 0.546211i \(0.183930\pi\)
−0.837648 + 0.546211i \(0.816070\pi\)
\(684\) 0 0
\(685\) −453944. −0.967434
\(686\) 0 0
\(687\) 510485. + 294729.i 1.08161 + 0.624466i
\(688\) 0 0
\(689\) 177919. 308165.i 0.374787 0.649150i
\(690\) 0 0
\(691\) 753165. 1.57737 0.788686 0.614796i \(-0.210762\pi\)
0.788686 + 0.614796i \(0.210762\pi\)
\(692\) 0 0
\(693\) 56131.2 + 97222.0i 0.116879 + 0.202441i
\(694\) 0 0
\(695\) 929659. 1.92466
\(696\) 0 0
\(697\) 7047.50 4068.88i 0.0145067 0.00837547i
\(698\) 0 0
\(699\) 420151. 242574.i 0.859906 0.496467i
\(700\) 0 0
\(701\) 286342. 495960.i 0.582706 1.00928i −0.412451 0.910980i \(-0.635327\pi\)
0.995157 0.0982968i \(-0.0313395\pi\)
\(702\) 0 0
\(703\) −129233. 367789.i −0.261495 0.744197i
\(704\) 0 0
\(705\) 630888. + 364244.i 1.26933 + 0.732848i
\(706\) 0 0
\(707\) −1348.18 2335.11i −0.00269717 0.00467163i
\(708\) 0 0
\(709\) −84487.5 146337.i −0.168074 0.291112i 0.769669 0.638443i \(-0.220421\pi\)
−0.937743 + 0.347331i \(0.887088\pi\)
\(710\) 0 0
\(711\) 160815.i 0.318117i
\(712\) 0 0
\(713\) −67596.5 + 39026.9i −0.132967 + 0.0767688i
\(714\) 0 0
\(715\) 522829.i 1.02270i
\(716\) 0 0
\(717\) 319412. + 184413.i 0.621317 + 0.358717i
\(718\) 0 0
\(719\) −6416.59 + 11113.9i −0.0124121 + 0.0214985i −0.872165 0.489212i \(-0.837284\pi\)
0.859753 + 0.510711i \(0.170618\pi\)
\(720\) 0 0
\(721\) 29461.1i 0.0566732i
\(722\) 0 0
\(723\) 1.16623e6 2.23104
\(724\) 0 0
\(725\) 949.297 + 548.077i 0.00180603 + 0.00104271i
\(726\) 0 0
\(727\) 385862. 668332.i 0.730067 1.26451i −0.226787 0.973944i \(-0.572822\pi\)
0.956854 0.290569i \(-0.0938447\pi\)
\(728\) 0 0
\(729\) −239708. −0.451052
\(730\) 0 0
\(731\) −83150.8 144021.i −0.155608 0.269521i
\(732\) 0 0
\(733\) 598661. 1.11423 0.557113 0.830437i \(-0.311909\pi\)
0.557113 + 0.830437i \(0.311909\pi\)
\(734\) 0 0
\(735\) −376314. + 217265.i −0.696588 + 0.402175i
\(736\) 0 0
\(737\) 616193. 355759.i 1.13444 0.654970i
\(738\) 0 0
\(739\) −56363.8 + 97624.9i −0.103207 + 0.178761i −0.913004 0.407950i \(-0.866244\pi\)
0.809797 + 0.586710i \(0.199577\pi\)
\(740\) 0 0
\(741\) −509288. + 178953.i −0.927528 + 0.325914i
\(742\) 0 0
\(743\) 444706. + 256751.i 0.805555 + 0.465088i 0.845410 0.534118i \(-0.179356\pi\)
−0.0398546 + 0.999205i \(0.512689\pi\)
\(744\) 0 0
\(745\) −257182. 445453.i −0.463370 0.802581i
\(746\) 0 0
\(747\) 147658. + 255751.i 0.264616 + 0.458328i
\(748\) 0 0
\(749\) 2406.06i 0.00428887i
\(750\) 0 0
\(751\) −740064. + 427276.i −1.31217 + 0.757580i −0.982455 0.186501i \(-0.940285\pi\)
−0.329713 + 0.944081i \(0.606952\pi\)
\(752\) 0 0
\(753\) 389967.i 0.687761i
\(754\) 0 0
\(755\) 40624.2 + 23454.4i 0.0712673 + 0.0411462i
\(756\) 0 0
\(757\) −74063.5 + 128282.i −0.129245 + 0.223858i −0.923384 0.383877i \(-0.874589\pi\)
0.794139 + 0.607736i \(0.207922\pi\)
\(758\) 0 0
\(759\) 1.05066e6i 1.82380i
\(760\) 0 0
\(761\) 262528. 0.453322 0.226661 0.973974i \(-0.427219\pi\)
0.226661 + 0.973974i \(0.427219\pi\)
\(762\) 0 0
\(763\) −499363. 288307.i −0.857762 0.495229i
\(764\) 0 0
\(765\) −31630.9 + 54786.3i −0.0540491 + 0.0936158i
\(766\) 0 0
\(767\) −831514. −1.41344
\(768\) 0 0
\(769\) −23000.9 39838.8i −0.0388949 0.0673680i 0.845923 0.533306i \(-0.179050\pi\)
−0.884818 + 0.465938i \(0.845717\pi\)
\(770\) 0 0
\(771\) 657950. 1.10684
\(772\) 0 0
\(773\) −590492. + 340921.i −0.988224 + 0.570551i −0.904743 0.425958i \(-0.859937\pi\)
−0.0834809 + 0.996509i \(0.526604\pi\)
\(774\) 0 0
\(775\) 2437.38 1407.22i 0.00405808 0.00234293i
\(776\) 0 0
\(777\) −156639. + 271307.i −0.259453 + 0.449386i
\(778\) 0 0
\(779\) 32847.5 + 6170.74i 0.0541287 + 0.0101686i
\(780\) 0 0
\(781\) 533050. + 307756.i 0.873908 + 0.504551i
\(782\) 0 0
\(783\) −11949.8 20697.7i −0.0194912 0.0337597i
\(784\) 0 0
\(785\) −337104. 583881.i −0.547047 0.947513i
\(786\) 0 0
\(787\) 356499.i 0.575584i 0.957693 + 0.287792i \(0.0929212\pi\)
−0.957693 + 0.287792i \(0.907079\pi\)
\(788\) 0 0
\(789\) −796120. + 459640.i −1.27886 + 0.738352i
\(790\) 0 0
\(791\) 513635.i 0.820921i
\(792\) 0 0
\(793\) 313211. + 180832.i 0.498070 + 0.287561i
\(794\) 0 0
\(795\) 331404. 574009.i 0.524353 0.908206i
\(796\) 0 0
\(797\) 356520.i 0.561265i −0.959815 0.280632i \(-0.909456\pi\)
0.959815 0.280632i \(-0.0905442\pi\)
\(798\) 0 0
\(799\) 240262. 0.376350
\(800\) 0 0
\(801\) 136120. + 78588.9i 0.212157 + 0.122489i
\(802\) 0 0
\(803\) −227665. + 394327.i −0.353073 + 0.611540i
\(804\) 0 0
\(805\) −496635. −0.766383
\(806\) 0 0
\(807\) 307735. + 533013.i 0.472531 + 0.818448i
\(808\) 0 0
\(809\) 146248. 0.223456 0.111728 0.993739i \(-0.464361\pi\)
0.111728 + 0.993739i \(0.464361\pi\)
\(810\) 0 0
\(811\) −1.09333e6 + 631237.i −1.66231 + 0.959733i −0.690696 + 0.723145i \(0.742696\pi\)
−0.971610 + 0.236588i \(0.923971\pi\)
\(812\) 0 0
\(813\) −799197. + 461417.i −1.20913 + 0.698091i
\(814\) 0 0
\(815\) 176967. 306516.i 0.266427 0.461465i
\(816\) 0 0
\(817\) 126104. 671266.i 0.188923 1.00566i
\(818\) 0 0
\(819\) 97077.6 + 56047.8i 0.144728 + 0.0835585i
\(820\) 0 0
\(821\) −86526.3 149868.i −0.128370 0.222343i 0.794676 0.607034i \(-0.207641\pi\)
−0.923045 + 0.384692i \(0.874308\pi\)
\(822\) 0 0
\(823\) −164144. 284306.i −0.242340 0.419745i 0.719040 0.694968i \(-0.244582\pi\)
−0.961380 + 0.275223i \(0.911248\pi\)
\(824\) 0 0
\(825\) 37884.5i 0.0556613i
\(826\) 0 0
\(827\) −894676. + 516541.i −1.30814 + 0.755256i −0.981786 0.189991i \(-0.939154\pi\)
−0.326356 + 0.945247i \(0.605821\pi\)
\(828\) 0 0
\(829\) 876630.i 1.27558i 0.770211 + 0.637789i \(0.220151\pi\)
−0.770211 + 0.637789i \(0.779849\pi\)
\(830\) 0 0
\(831\) −667604. 385441.i −0.966756 0.558157i
\(832\) 0 0
\(833\) −71656.2 + 124112.i −0.103268 + 0.178865i
\(834\) 0 0
\(835\) 391322.i 0.561256i
\(836\) 0 0
\(837\) −61364.0 −0.0875917
\(838\) 0 0
\(839\) 228597. + 131981.i 0.324748 + 0.187493i 0.653507 0.756921i \(-0.273297\pi\)
−0.328759 + 0.944414i \(0.606630\pi\)
\(840\) 0 0
\(841\) −352702. + 610897.i −0.498673 + 0.863726i
\(842\) 0 0
\(843\) −625475. −0.880147
\(844\) 0 0
\(845\) −103140. 178644.i −0.144449 0.250193i
\(846\) 0 0
\(847\) 163551. 0.227974
\(848\) 0 0
\(849\) 1.28124e6 739723.i 1.77752 1.02625i
\(850\) 0 0
\(851\) 656118. 378810.i 0.905989 0.523073i
\(852\) 0 0
\(853\) 537972. 931795.i 0.739370 1.28063i −0.213409 0.976963i \(-0.568457\pi\)
0.952779 0.303664i \(-0.0982099\pi\)
\(854\) 0 0
\(855\) −245127. + 86132.5i −0.335319 + 0.117824i
\(856\) 0 0
\(857\) 346531. + 200070.i 0.471824 + 0.272408i 0.717003 0.697070i \(-0.245513\pi\)
−0.245179 + 0.969478i \(0.578847\pi\)
\(858\) 0 0
\(859\) −274871. 476090.i −0.372514 0.645213i 0.617438 0.786620i \(-0.288171\pi\)
−0.989952 + 0.141407i \(0.954837\pi\)
\(860\) 0 0
\(861\) −13429.4 23260.4i −0.0181155 0.0313769i
\(862\) 0 0
\(863\) 443814.i 0.595908i 0.954580 + 0.297954i \(0.0963042\pi\)
−0.954580 + 0.297954i \(0.903696\pi\)
\(864\) 0 0
\(865\) −377536. + 217971.i −0.504576 + 0.291317i
\(866\) 0 0
\(867\) 792133.i 1.05380i
\(868\) 0 0
\(869\) 707088. + 408238.i 0.936341 + 0.540597i
\(870\) 0 0
\(871\) 355231. 615278.i 0.468246 0.811027i
\(872\) 0 0
\(873\) 508991.i 0.667854i
\(874\) 0 0
\(875\) −424516. −0.554469
\(876\) 0 0
\(877\) −919656. 530964.i −1.19571 0.690344i −0.236115 0.971725i \(-0.575874\pi\)
−0.959596 + 0.281381i \(0.909207\pi\)
\(878\) 0 0
\(879\) −895020. + 1.55022e6i −1.15839 + 2.00639i
\(880\) 0 0
\(881\) −337096. −0.434312 −0.217156 0.976137i \(-0.569678\pi\)
−0.217156 + 0.976137i \(0.569678\pi\)
\(882\) 0 0
\(883\) −23184.8 40157.3i −0.0297360 0.0515043i 0.850774 0.525531i \(-0.176133\pi\)
−0.880510 + 0.474027i \(0.842800\pi\)
\(884\) 0 0
\(885\) −1.54883e6 −1.97751
\(886\) 0 0
\(887\) −94128.5 + 54345.1i −0.119639 + 0.0690738i −0.558625 0.829420i \(-0.688671\pi\)
0.438986 + 0.898494i \(0.355338\pi\)
\(888\) 0 0
\(889\) 446230. 257631.i 0.564618 0.325982i
\(890\) 0 0
\(891\) 576792. 999033.i 0.726547 1.25842i
\(892\) 0 0
\(893\) 748781. + 642681.i 0.938971 + 0.805921i
\(894\) 0 0
\(895\) 674887. + 389646.i 0.842529 + 0.486434i
\(896\) 0 0
\(897\) −524549. 908546.i −0.651931 1.12918i
\(898\) 0 0
\(899\) −2410.33 4174.81i −0.00298234 0.00516556i
\(900\) 0 0
\(901\) 218601.i 0.269279i
\(902\) 0 0
\(903\) −475345. + 274441.i −0.582953 + 0.336568i
\(904\) 0 0
\(905\) 1.43602e6i 1.75332i
\(906\) 0 0
\(907\) −930792. 537393.i −1.13146 0.653247i −0.187156 0.982330i \(-0.559927\pi\)
−0.944301 + 0.329083i \(0.893260\pi\)
\(908\) 0 0
\(909\) 1370.73 2374.18i 0.00165892 0.00287333i
\(910\) 0 0
\(911\) 586315.i 0.706471i −0.935534 0.353235i \(-0.885081\pi\)
0.935534 0.353235i \(-0.114919\pi\)
\(912\) 0 0
\(913\) 1.49935e6 1.79871
\(914\) 0 0
\(915\) 583407. + 336830.i 0.696834 + 0.402317i
\(916\) 0 0
\(917\) −194872. + 337528.i −0.231745 + 0.401394i
\(918\) 0 0
\(919\) 678021. 0.802809 0.401405 0.915901i \(-0.368522\pi\)
0.401405 + 0.915901i \(0.368522\pi\)
\(920\) 0 0
\(921\) −144799. 250800.i −0.170705 0.295671i
\(922\) 0 0
\(923\) 614598. 0.721420
\(924\) 0 0
\(925\) −23658.2 + 13659.1i −0.0276502 + 0.0159638i
\(926\) 0 0
\(927\) −25940.9 + 14977.0i −0.0301874 + 0.0174287i
\(928\) 0 0
\(929\) −364161. + 630746.i −0.421951 + 0.730841i −0.996130 0.0878888i \(-0.971988\pi\)
0.574179 + 0.818730i \(0.305321\pi\)
\(930\) 0 0
\(931\) −555307. + 195123.i −0.640669 + 0.225118i
\(932\) 0 0
\(933\) 1.71671e6 + 991143.i 1.97212 + 1.13861i
\(934\) 0 0
\(935\) 160594. + 278156.i 0.183698 + 0.318175i
\(936\) 0 0
\(937\) −338424. 586168.i −0.385463 0.667641i 0.606370 0.795182i \(-0.292625\pi\)
−0.991833 + 0.127541i \(0.959292\pi\)
\(938\) 0 0
\(939\) 1.65231e6i 1.87396i
\(940\) 0 0
\(941\) −211702. + 122226.i −0.239081 + 0.138034i −0.614754 0.788719i \(-0.710745\pi\)
0.375673 + 0.926752i \(0.377412\pi\)
\(942\) 0 0
\(943\) 64954.1i 0.0730438i
\(944\) 0 0
\(945\) −338134. 195222.i −0.378639 0.218607i
\(946\) 0 0
\(947\) 392652. 680093.i 0.437832 0.758348i −0.559690 0.828702i \(-0.689080\pi\)
0.997522 + 0.0703543i \(0.0224130\pi\)
\(948\) 0 0
\(949\) 454653.i 0.504832i
\(950\) 0 0
\(951\) −476830. −0.527233
\(952\) 0 0
\(953\) −504490. 291268.i −0.555478 0.320705i 0.195850 0.980634i \(-0.437253\pi\)
−0.751329 + 0.659928i \(0.770587\pi\)
\(954\) 0 0
\(955\) 258364. 447500.i 0.283286 0.490666i
\(956\) 0 0
\(957\) 64889.5 0.0708517
\(958\) 0 0
\(959\) 247069. + 427936.i 0.268646 + 0.465309i
\(960\) 0 0
\(961\) 911144. 0.986598
\(962\) 0 0
\(963\) −2118.57 + 1223.16i −0.00228450 + 0.00131895i
\(964\) 0 0
\(965\) −23015.6 + 13288.1i −0.0247154 + 0.0142694i
\(966\) 0 0
\(967\) 378832. 656157.i 0.405130 0.701705i −0.589207 0.807982i \(-0.700560\pi\)
0.994337 + 0.106277i \(0.0338930\pi\)
\(968\) 0 0
\(969\) −215984. + 251641.i −0.230025 + 0.268000i
\(970\) 0 0
\(971\) −412390. 238093.i −0.437390 0.252527i 0.265100 0.964221i \(-0.414595\pi\)
−0.702490 + 0.711694i \(0.747928\pi\)
\(972\) 0 0
\(973\) −505987. 876395.i −0.534458 0.925709i
\(974\) 0 0
\(975\) 18914.1 + 32760.2i 0.0198965 + 0.0344617i
\(976\) 0 0
\(977\) 439217.i 0.460140i −0.973174 0.230070i \(-0.926105\pi\)
0.973174 0.230070i \(-0.0738955\pi\)
\(978\) 0 0
\(979\) 691097. 399005.i 0.721063 0.416306i
\(980\) 0 0
\(981\) 586261.i 0.609191i
\(982\) 0 0
\(983\) −469744. 271207.i −0.486132 0.280668i 0.236837 0.971549i \(-0.423889\pi\)
−0.722968 + 0.690881i \(0.757223\pi\)
\(984\) 0 0
\(985\) −764859. + 1.32477e6i −0.788331 + 1.36543i
\(986\) 0 0
\(987\) 792990.i 0.814017i
\(988\) 0 0
\(989\) 1.32739e6 1.35708
\(990\) 0 0
\(991\) 560235. + 323452.i 0.570457 + 0.329354i 0.757332 0.653030i \(-0.226502\pi\)
−0.186875 + 0.982384i \(0.559836\pi\)
\(992\) 0 0
\(993\) −267764. + 463781.i −0.271553 + 0.470343i
\(994\) 0 0
\(995\) 1.23996e6 1.25246
\(996\) 0 0
\(997\) 581238. + 1.00673e6i 0.584741 + 1.01280i 0.994908 + 0.100791i \(0.0321373\pi\)
−0.410166 + 0.912011i \(0.634529\pi\)
\(998\) 0 0
\(999\) 595623. 0.596816
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 76.5.h.a.65.2 12
3.2 odd 2 684.5.y.c.217.5 12
4.3 odd 2 304.5.r.b.65.5 12
19.12 odd 6 inner 76.5.h.a.69.2 yes 12
57.50 even 6 684.5.y.c.145.5 12
76.31 even 6 304.5.r.b.145.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.h.a.65.2 12 1.1 even 1 trivial
76.5.h.a.69.2 yes 12 19.12 odd 6 inner
304.5.r.b.65.5 12 4.3 odd 2
304.5.r.b.145.5 12 76.31 even 6
684.5.y.c.145.5 12 57.50 even 6
684.5.y.c.217.5 12 3.2 odd 2