Properties

Label 684.5.y.c.145.5
Level $684$
Weight $5$
Character 684.145
Analytic conductor $70.705$
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] = [684,5,Mod(145,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.y (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: \(70.7050547493\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.5
Root \(0.500000 - 9.58497i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.5.y.c.217.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(12.7505 + 22.0845i) q^{5} +27.7589 q^{7} +O(q^{10})\) \(q+(12.7505 + 22.0845i) q^{5} +27.7589 q^{7} -143.293 q^{11} +(123.911 + 71.5400i) q^{13} +(-43.9489 - 76.1217i) q^{17} +(340.586 + 119.675i) q^{19} +(-350.792 + 607.590i) q^{23} +(-12.6488 + 21.9084i) q^{25} +(37.5252 + 21.6652i) q^{29} -111.254i q^{31} +(353.939 + 613.040i) q^{35} +1079.87i q^{37} +(-80.1784 + 46.2910i) q^{41} +(945.995 + 1638.51i) q^{43} +(-1366.72 + 2367.22i) q^{47} -1630.45 q^{49} +(-2153.80 - 1243.50i) q^{53} +(-1827.05 - 3164.54i) q^{55} +(5032.93 - 2905.77i) q^{59} +(1263.85 - 2189.06i) q^{61} +3648.67i q^{65} +(4300.24 + 2482.74i) q^{67} +(-3720.00 + 2147.74i) q^{71} +(-1588.81 - 2751.89i) q^{73} -3977.65 q^{77} +(4934.57 - 2848.97i) q^{79} -10463.5 q^{83} +(1120.74 - 1941.17i) q^{85} +(-4822.97 - 2784.54i) q^{89} +(3439.63 + 1985.87i) q^{91} +(1699.68 + 9047.57i) q^{95} +(-15618.3 + 9017.22i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{5} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 9 q^{5} - 52 q^{7} - 6 q^{11} - 93 q^{13} + 483 q^{17} - 533 q^{19} - 531 q^{23} - 217 q^{25} - 2025 q^{29} + 1128 q^{35} + 1692 q^{41} - 63 q^{43} + 3471 q^{47} + 420 q^{49} + 3771 q^{53} - 2014 q^{55} + 9594 q^{59} + 1229 q^{61} + 7590 q^{67} - 963 q^{71} - 2838 q^{73} + 15408 q^{77} + 11073 q^{79} + 14202 q^{83} + 9455 q^{85} - 6525 q^{89} - 7686 q^{91} - 1521 q^{95} - 34110 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 12.7505 + 22.0845i 0.510019 + 0.883378i 0.999933 + 0.0116075i \(0.00369485\pi\)
−0.489914 + 0.871771i \(0.662972\pi\)
\(6\) 0 0
\(7\) 27.7589 0.566508 0.283254 0.959045i \(-0.408586\pi\)
0.283254 + 0.959045i \(0.408586\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −143.293 −1.18424 −0.592119 0.805850i \(-0.701709\pi\)
−0.592119 + 0.805850i \(0.701709\pi\)
\(12\) 0 0
\(13\) 123.911 + 71.5400i 0.733201 + 0.423314i 0.819592 0.572948i \(-0.194200\pi\)
−0.0863914 + 0.996261i \(0.527534\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −43.9489 76.1217i −0.152072 0.263397i 0.779917 0.625883i \(-0.215261\pi\)
−0.931989 + 0.362486i \(0.881928\pi\)
\(18\) 0 0
\(19\) 340.586 + 119.675i 0.943452 + 0.331509i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −350.792 + 607.590i −0.663123 + 1.14856i 0.316667 + 0.948537i \(0.397436\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(24\) 0 0
\(25\) −12.6488 + 21.9084i −0.0202381 + 0.0350534i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 37.5252 + 21.6652i 0.0446197 + 0.0257612i 0.522144 0.852857i \(-0.325132\pi\)
−0.477524 + 0.878619i \(0.658466\pi\)
\(30\) 0 0
\(31\) 111.254i 0.115769i −0.998323 0.0578843i \(-0.981565\pi\)
0.998323 0.0578843i \(-0.0184354\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 353.939 + 613.040i 0.288929 + 0.500440i
\(36\) 0 0
\(37\) 1079.87i 0.788802i 0.918938 + 0.394401i \(0.129048\pi\)
−0.918938 + 0.394401i \(0.870952\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −80.1784 + 46.2910i −0.0476969 + 0.0275378i −0.523659 0.851928i \(-0.675433\pi\)
0.475962 + 0.879466i \(0.342100\pi\)
\(42\) 0 0
\(43\) 945.995 + 1638.51i 0.511625 + 0.886161i 0.999909 + 0.0134763i \(0.00428978\pi\)
−0.488284 + 0.872685i \(0.662377\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1366.72 + 2367.22i −0.618703 + 1.07163i 0.371019 + 0.928625i \(0.379008\pi\)
−0.989723 + 0.143001i \(0.954325\pi\)
\(48\) 0 0
\(49\) −1630.45 −0.679069
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2153.80 1243.50i −0.766749 0.442683i 0.0649646 0.997888i \(-0.479307\pi\)
−0.831714 + 0.555205i \(0.812640\pi\)
\(54\) 0 0
\(55\) −1827.05 3164.54i −0.603984 1.04613i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5032.93 2905.77i 1.44583 0.834750i 0.447600 0.894234i \(-0.352279\pi\)
0.998229 + 0.0594837i \(0.0189454\pi\)
\(60\) 0 0
\(61\) 1263.85 2189.06i 0.339654 0.588299i −0.644713 0.764425i \(-0.723023\pi\)
0.984368 + 0.176126i \(0.0563565\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3648.67i 0.863591i
\(66\) 0 0
\(67\) 4300.24 + 2482.74i 0.957950 + 0.553073i 0.895542 0.444978i \(-0.146788\pi\)
0.0624086 + 0.998051i \(0.480122\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3720.00 + 2147.74i −0.737949 + 0.426055i −0.821323 0.570463i \(-0.806764\pi\)
0.0833740 + 0.996518i \(0.473430\pi\)
\(72\) 0 0
\(73\) −1588.81 2751.89i −0.298143 0.516399i 0.677568 0.735460i \(-0.263034\pi\)
−0.975711 + 0.219061i \(0.929701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3977.65 −0.670880
\(78\) 0 0
\(79\) 4934.57 2848.97i 0.790669 0.456493i −0.0495288 0.998773i \(-0.515772\pi\)
0.840198 + 0.542280i \(0.182439\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10463.5 −1.51888 −0.759439 0.650579i \(-0.774526\pi\)
−0.759439 + 0.650579i \(0.774526\pi\)
\(84\) 0 0
\(85\) 1120.74 1941.17i 0.155119 0.268674i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4822.97 2784.54i −0.608884 0.351539i 0.163645 0.986519i \(-0.447675\pi\)
−0.772528 + 0.634980i \(0.781008\pi\)
\(90\) 0 0
\(91\) 3439.63 + 1985.87i 0.415364 + 0.239810i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1699.68 + 9047.57i 0.188330 + 1.00250i
\(96\) 0 0
\(97\) −15618.3 + 9017.22i −1.65993 + 0.958361i −0.687185 + 0.726482i \(0.741154\pi\)
−0.972745 + 0.231878i \(0.925513\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 48.5674 84.1212i 0.00476104 0.00824636i −0.863635 0.504118i \(-0.831818\pi\)
0.868396 + 0.495871i \(0.165151\pi\)
\(102\) 0 0
\(103\) 1061.32i 0.100040i −0.998748 0.0500198i \(-0.984072\pi\)
0.998748 0.0500198i \(-0.0159285\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 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 −0.514260 + 0.857634i \(0.671933\pi\)
−0.999863 + 0.0165449i \(0.994733\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 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
\(117\) 0 0
\(118\) 0 0
\(119\) −1219.97 2113.05i −0.0861500 0.149216i
\(120\) 0 0
\(121\) 5891.84 0.402421
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 15293.0 0.978750
\(126\) 0 0
\(127\) 16075.2 + 9281.03i 0.996665 + 0.575425i 0.907260 0.420571i \(-0.138170\pi\)
0.0894051 + 0.995995i \(0.471503\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7020.17 + 12159.3i 0.409077 + 0.708542i 0.994787 0.101979i \(-0.0325175\pi\)
−0.585710 + 0.810521i \(0.699184\pi\)
\(132\) 0 0
\(133\) 9454.29 + 3322.04i 0.534473 + 0.187803i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8900.54 + 15416.2i −0.474215 + 0.821364i −0.999564 0.0295224i \(-0.990601\pi\)
0.525349 + 0.850887i \(0.323935\pi\)
\(138\) 0 0
\(139\) −18227.9 + 31571.7i −0.943426 + 1.63406i −0.184554 + 0.982822i \(0.559084\pi\)
−0.758872 + 0.651240i \(0.774249\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17755.5 10251.2i −0.868284 0.501304i
\(144\) 0 0
\(145\) 1104.96i 0.0525548i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10085.2 + 17468.1i 0.454268 + 0.786816i 0.998646 0.0520247i \(-0.0165675\pi\)
−0.544378 + 0.838840i \(0.683234\pi\)
\(150\) 0 0
\(151\) 1839.49i 0.0806759i 0.999186 + 0.0403379i \(0.0128434\pi\)
−0.999186 + 0.0403379i \(0.987157\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2456.97 1418.54i 0.102267 0.0590441i
\(156\) 0 0
\(157\) −13219.3 22896.5i −0.536301 0.928901i −0.999099 0.0424369i \(-0.986488\pi\)
0.462798 0.886464i \(-0.346845\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 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
\(165\) 0 0
\(166\) 0 0
\(167\) 13289.5 + 7672.69i 0.476514 + 0.275115i 0.718962 0.695049i \(-0.244617\pi\)
−0.242449 + 0.970164i \(0.577951\pi\)
\(168\) 0 0
\(169\) −4044.56 7005.39i −0.141611 0.245278i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14804.8 + 8547.56i −0.494664 + 0.285594i −0.726507 0.687159i \(-0.758858\pi\)
0.231843 + 0.972753i \(0.425524\pi\)
\(174\) 0 0
\(175\) −351.116 + 608.152i −0.0114650 + 0.0198580i
\(176\) 0 0
\(177\) 0 0
\(178\) 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 0.999915 0.0130215i \(-0.00414497\pi\)
0.488681 + 0.872463i \(0.337478\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 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
\(189\) 0 0
\(190\) 0 0
\(191\) 20263.1 0.555443 0.277721 0.960662i \(-0.410421\pi\)
0.277721 + 0.960662i \(0.410421\pi\)
\(192\) 0 0
\(193\) 902.539 521.081i 0.0242299 0.0139891i −0.487836 0.872935i \(-0.662214\pi\)
0.512066 + 0.858946i \(0.328880\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −59986.7 −1.54569 −0.772845 0.634594i \(-0.781167\pi\)
−0.772845 + 0.634594i \(0.781167\pi\)
\(198\) 0 0
\(199\) −24312.1 + 42109.9i −0.613928 + 1.06335i 0.376644 + 0.926358i \(0.377078\pi\)
−0.990572 + 0.136996i \(0.956255\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 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
\(207\) 0 0
\(208\) 0 0
\(209\) −48803.6 17148.6i −1.11727 0.392586i
\(210\) 0 0
\(211\) 30887.6 17833.0i 0.693777 0.400552i −0.111248 0.993793i \(-0.535485\pi\)
0.805025 + 0.593240i \(0.202152\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −24123.8 + 41783.6i −0.521877 + 0.903918i
\(216\) 0 0
\(217\) 3088.27i 0.0655838i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12576.4i 0.257497i
\(222\) 0 0
\(223\) 31201.2 18014.0i 0.627425 0.362244i −0.152329 0.988330i \(-0.548677\pi\)
0.779754 + 0.626086i \(0.215344\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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
\(231\) 0 0
\(232\) 0 0
\(233\) 23210.6 + 40202.0i 0.427538 + 0.740518i 0.996654 0.0817394i \(-0.0260475\pi\)
−0.569115 + 0.822258i \(0.692714\pi\)
\(234\) 0 0
\(235\) −69705.1 −1.26220
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 35290.9 0.617828 0.308914 0.951090i \(-0.400035\pi\)
0.308914 + 0.951090i \(0.400035\pi\)
\(240\) 0 0
\(241\) −96639.8 55795.0i −1.66388 0.960642i −0.970835 0.239748i \(-0.922935\pi\)
−0.693045 0.720894i \(-0.743731\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 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
\(249\) 0 0
\(250\) 0 0
\(251\) −18656.9 + 32314.8i −0.296137 + 0.512924i −0.975249 0.221111i \(-0.929032\pi\)
0.679112 + 0.734035i \(0.262365\pi\)
\(252\) 0 0
\(253\) 50266.0 87063.3i 0.785296 1.36017i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 54521.2 + 31477.8i 0.825466 + 0.476583i 0.852298 0.523057i \(-0.175208\pi\)
−0.0268315 + 0.999640i \(0.508542\pi\)
\(258\) 0 0
\(259\) 29976.0i 0.446862i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −43980.5 76176.4i −0.635841 1.10131i −0.986336 0.164745i \(-0.947320\pi\)
0.350495 0.936565i \(-0.386013\pi\)
\(264\) 0 0
\(265\) 63420.6i 0.903106i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 51001.2 29445.6i 0.704816 0.406926i −0.104323 0.994544i \(-0.533267\pi\)
0.809139 + 0.587618i \(0.199934\pi\)
\(270\) 0 0
\(271\) 44150.5 + 76470.9i 0.601170 + 1.04126i 0.992644 + 0.121068i \(0.0386318\pi\)
−0.391475 + 0.920189i \(0.628035\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 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
\(279\) 0 0
\(280\) 0 0
\(281\) −51830.2 29924.2i −0.656403 0.378975i 0.134502 0.990913i \(-0.457057\pi\)
−0.790905 + 0.611939i \(0.790390\pi\)
\(282\) 0 0
\(283\) −70780.2 122595.i −0.883769 1.53073i −0.847118 0.531404i \(-0.821664\pi\)
−0.0366506 0.999328i \(-0.511669\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 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
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 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
\(303\) 0 0
\(304\) 0 0
\(305\) 64458.9 0.692921
\(306\) 0 0
\(307\) 23997.7 13855.1i 0.254620 0.147005i −0.367258 0.930119i \(-0.619703\pi\)
0.621878 + 0.783114i \(0.286370\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 189674. 1.96105 0.980524 0.196401i \(-0.0629254\pi\)
0.980524 + 0.196401i \(0.0629254\pi\)
\(312\) 0 0
\(313\) −79050.5 + 136919.i −0.806893 + 1.39758i 0.108113 + 0.994139i \(0.465519\pi\)
−0.915006 + 0.403441i \(0.867814\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −39512.7 22812.7i −0.393204 0.227017i 0.290343 0.956923i \(-0.406231\pi\)
−0.683548 + 0.729906i \(0.739564\pi\)
\(318\) 0 0
\(319\) −5377.09 3104.46i −0.0528404 0.0305074i
\(320\) 0 0
\(321\) 0 0
\(322\) 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
\(327\) 0 0
\(328\) 0 0
\(329\) −37938.5 + 65711.4i −0.350500 + 0.607084i
\(330\) 0 0
\(331\) 51241.9i 0.467702i 0.972272 + 0.233851i \(0.0751328\pi\)
−0.972272 + 0.233851i \(0.924867\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 126625.i 1.12831i
\(336\) 0 0
\(337\) 7928.49 4577.52i 0.0698121 0.0403060i −0.464688 0.885475i \(-0.653833\pi\)
0.534500 + 0.845169i \(0.320500\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15941.8i 0.137098i
\(342\) 0 0
\(343\) −111908. −0.951205
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −76816.1 133049.i −0.637960 1.10498i −0.985880 0.167455i \(-0.946445\pi\)
0.347920 0.937524i \(-0.386888\pi\)
\(348\) 0 0
\(349\) 168335. 1.38205 0.691023 0.722832i \(-0.257160\pi\)
0.691023 + 0.722832i \(0.257160\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 130340. 1.04599 0.522994 0.852336i \(-0.324815\pi\)
0.522994 + 0.852336i \(0.324815\pi\)
\(354\) 0 0
\(355\) −94863.5 54769.5i −0.752736 0.434592i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18849.9 + 32649.0i 0.146258 + 0.253327i 0.929842 0.367960i \(-0.119944\pi\)
−0.783583 + 0.621287i \(0.786610\pi\)
\(360\) 0 0
\(361\) 101677. + 81519.2i 0.780203 + 0.625526i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40516.0 70175.8i 0.304117 0.526747i
\(366\) 0 0
\(367\) 48448.8 83915.8i 0.359709 0.623034i −0.628203 0.778049i \(-0.716209\pi\)
0.987912 + 0.155015i \(0.0495427\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −59787.0 34518.0i −0.434369 0.250783i
\(372\) 0 0
\(373\) 125723.i 0.903644i 0.892108 + 0.451822i \(0.149226\pi\)
−0.892108 + 0.451822i \(0.850774\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3099.85 + 5369.10i 0.0218101 + 0.0377762i
\(378\) 0 0
\(379\) 7410.05i 0.0515873i 0.999667 + 0.0257936i \(0.00821128\pi\)
−0.999667 + 0.0257936i \(0.991789\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 105519. 60921.6i 0.719340 0.415311i −0.0951699 0.995461i \(-0.530339\pi\)
0.814510 + 0.580150i \(0.197006\pi\)
\(384\) 0 0
\(385\) −50716.9 87844.2i −0.342161 0.592641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 129529. 224351.i 0.855990 1.48262i −0.0197344 0.999805i \(-0.506282\pi\)
0.875724 0.482812i \(-0.160385\pi\)
\(390\) 0 0
\(391\) 61667.6 0.403370
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 125836. + 72651.5i 0.806512 + 0.465640i
\(396\) 0 0
\(397\) −109875. 190310.i −0.697139 1.20748i −0.969454 0.245273i \(-0.921122\pi\)
0.272315 0.962208i \(-0.412211\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 200338. 115665.i 1.24588 0.719307i 0.275593 0.961275i \(-0.411126\pi\)
0.970284 + 0.241967i \(0.0777926\pi\)
\(402\) 0 0
\(403\) 7959.08 13785.5i 0.0490064 0.0848816i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 154738.i 0.934130i
\(408\) 0 0
\(409\) 36379.7 + 21003.8i 0.217477 + 0.125560i 0.604781 0.796392i \(-0.293261\pi\)
−0.387305 + 0.921952i \(0.626594\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 139709. 80660.8i 0.819073 0.472892i
\(414\) 0 0
\(415\) −133415. 231082.i −0.774656 1.34174i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17406.8 0.0991494 0.0495747 0.998770i \(-0.484213\pi\)
0.0495747 + 0.998770i \(0.484213\pi\)
\(420\) 0 0
\(421\) −161802. + 93416.2i −0.912891 + 0.527058i −0.881360 0.472445i \(-0.843371\pi\)
−0.0315305 + 0.999503i \(0.510038\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2223.60 0.0123106
\(426\) 0 0
\(427\) 35083.2 60765.8i 0.192417 0.333276i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9163.54 5290.57i −0.0493297 0.0284805i 0.475132 0.879914i \(-0.342400\pi\)
−0.524462 + 0.851434i \(0.675734\pi\)
\(432\) 0 0
\(433\) −188173. 108642.i −1.00365 0.579458i −0.0943244 0.995542i \(-0.530069\pi\)
−0.909326 + 0.416083i \(0.863402\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −192188. + 164956.i −1.00638 + 0.863782i
\(438\) 0 0
\(439\) −27000.5 + 15588.7i −0.140101 + 0.0808875i −0.568412 0.822744i \(-0.692442\pi\)
0.428311 + 0.903631i \(0.359109\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 84484.4 146331.i 0.430496 0.745641i −0.566420 0.824117i \(-0.691672\pi\)
0.996916 + 0.0784760i \(0.0250054\pi\)
\(444\) 0 0
\(445\) 142017.i 0.717166i
\(446\) 0 0
\(447\) 0 0
\(448\) 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
\(453\) 0 0
\(454\) 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
\(459\) 0 0
\(460\) 0 0
\(461\) 14069.9 + 24369.7i 0.0662046 + 0.114670i 0.897228 0.441568i \(-0.145578\pi\)
−0.831023 + 0.556238i \(0.812244\pi\)
\(462\) 0 0
\(463\) 81970.7 0.382381 0.191191 0.981553i \(-0.438765\pi\)
0.191191 + 0.981553i \(0.438765\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 126102. 0.578213 0.289106 0.957297i \(-0.406642\pi\)
0.289106 + 0.957297i \(0.406642\pi\)
\(468\) 0 0
\(469\) 119370. + 68918.2i 0.542686 + 0.313320i
\(470\) 0 0
\(471\) 0 0
\(472\) 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
\(477\) 0 0
\(478\) 0 0
\(479\) 57621.7 99803.7i 0.251139 0.434986i −0.712700 0.701469i \(-0.752528\pi\)
0.963840 + 0.266482i \(0.0858614\pi\)
\(480\) 0 0
\(481\) −77253.9 + 133808.i −0.333911 + 0.578350i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −398281. 229947.i −1.69319 0.977564i
\(486\) 0 0
\(487\) 135408.i 0.570935i −0.958388 0.285467i \(-0.907851\pi\)
0.958388 0.285467i \(-0.0921488\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 202101. + 350049.i 0.838312 + 1.45200i 0.891305 + 0.453404i \(0.149790\pi\)
−0.0529937 + 0.998595i \(0.516876\pi\)
\(492\) 0 0
\(493\) 3808.64i 0.0156702i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −103263. + 59619.0i −0.418054 + 0.241363i
\(498\) 0 0
\(499\) 220006. + 381062.i 0.883557 + 1.53037i 0.847359 + 0.531020i \(0.178191\pi\)
0.0361977 + 0.999345i \(0.488475\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −140225. + 242877.i −0.554229 + 0.959953i 0.443734 + 0.896159i \(0.353654\pi\)
−0.997963 + 0.0637944i \(0.979680\pi\)
\(504\) 0 0
\(505\) 2477.03 0.00971288
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 87780.5 + 50680.1i 0.338815 + 0.195615i 0.659748 0.751487i \(-0.270663\pi\)
−0.320933 + 0.947102i \(0.603996\pi\)
\(510\) 0 0
\(511\) −44103.5 76389.4i −0.168900 0.292544i
\(512\) 0 0
\(513\) 0 0
\(514\) 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
\(519\) 0 0
\(520\) 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 0.712979 0.701186i \(-0.247346\pi\)
−0.250755 + 0.968051i \(0.580679\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(531\) 0 0
\(532\) 0 0
\(533\) −13246.6 −0.0466285
\(534\) 0 0
\(535\) −1914.22 + 1105.17i −0.00668781 + 0.00386121i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 233631. 0.804180
\(540\) 0 0
\(541\) 210834. 365176.i 0.720355 1.24769i −0.240502 0.970649i \(-0.577312\pi\)
0.960857 0.277043i \(-0.0893546\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −458744. 264856.i −1.54446 0.891696i
\(546\) 0 0
\(547\) 86529.9 + 49958.1i 0.289196 + 0.166967i 0.637579 0.770385i \(-0.279936\pi\)
−0.348383 + 0.937352i \(0.613269\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 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
\(555\) 0 0
\(556\) 0 0
\(557\) 128227. 222096.i 0.413304 0.715864i −0.581945 0.813228i \(-0.697708\pi\)
0.995249 + 0.0973645i \(0.0310413\pi\)
\(558\) 0 0
\(559\) 270706.i 0.866312i
\(560\) 0 0
\(561\) 0 0
\(562\) 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
\(567\) 0 0
\(568\) 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
\(573\) 0 0
\(574\) 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
\(579\) 0 0
\(580\) 0 0
\(581\) −290456. −0.860456
\(582\) 0 0
\(583\) 308624. + 178184.i 0.908014 + 0.524242i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 151917. + 263128.i 0.440890 + 0.763644i 0.997756 0.0669588i \(-0.0213296\pi\)
−0.556866 + 0.830602i \(0.687996\pi\)
\(588\) 0 0
\(589\) 13314.3 37891.4i 0.0383784 0.109222i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 130746. 226459.i 0.371809 0.643992i −0.618035 0.786151i \(-0.712071\pi\)
0.989844 + 0.142158i \(0.0454043\pi\)
\(594\) 0 0
\(595\) 31110.4 53884.8i 0.0878763 0.152206i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 422627. + 244004.i 1.17789 + 0.680053i 0.955524 0.294912i \(-0.0952903\pi\)
0.222361 + 0.974964i \(0.428624\pi\)
\(600\) 0 0
\(601\) 362708.i 1.00417i 0.864818 + 0.502086i \(0.167434\pi\)
−0.864818 + 0.502086i \(0.832566\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 75123.7 + 130118.i 0.205242 + 0.355490i
\(606\) 0 0
\(607\) 560324.i 1.52077i 0.649475 + 0.760383i \(0.274989\pi\)
−0.649475 + 0.760383i \(0.725011\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −338702. + 195550.i −0.907267 + 0.523811i
\(612\) 0 0
\(613\) −197665. 342365.i −0.526027 0.911106i −0.999540 0.0303191i \(-0.990348\pi\)
0.473513 0.880787i \(-0.342986\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47374.0 82054.2i 0.124443 0.215541i −0.797072 0.603884i \(-0.793619\pi\)
0.921515 + 0.388343i \(0.126952\pi\)
\(618\) 0 0
\(619\) 255301. 0.666303 0.333151 0.942873i \(-0.391888\pi\)
0.333151 + 0.942873i \(0.391888\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −133880. 77295.7i −0.344937 0.199150i
\(624\) 0 0
\(625\) 202898. + 351430.i 0.519419 + 0.899660i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 82201.5 47459.1i 0.207768 0.119955i
\(630\) 0 0
\(631\) 392479. 679794.i 0.985730 1.70733i 0.347084 0.937834i \(-0.387172\pi\)
0.638646 0.769501i \(-0.279495\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 473350.i 1.17391i
\(636\) 0 0
\(637\) −202030. 116642.i −0.497894 0.287459i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 55185.8 31861.5i 0.134311 0.0775444i −0.431339 0.902190i \(-0.641959\pi\)
0.565650 + 0.824646i \(0.308625\pi\)
\(642\) 0 0
\(643\) −114558. 198420.i −0.277079 0.479915i 0.693579 0.720381i \(-0.256033\pi\)
−0.970657 + 0.240466i \(0.922700\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −439341. −1.04953 −0.524763 0.851248i \(-0.675846\pi\)
−0.524763 + 0.851248i \(0.675846\pi\)
\(648\) 0 0
\(649\) −721183. + 416375.i −1.71221 + 0.988543i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −296297. −0.694866 −0.347433 0.937705i \(-0.612947\pi\)
−0.347433 + 0.937705i \(0.612947\pi\)
\(654\) 0 0
\(655\) −179021. + 310073.i −0.417274 + 0.722739i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 642280. + 370820.i 1.47895 + 0.853872i 0.999716 0.0238155i \(-0.00758143\pi\)
0.479233 + 0.877688i \(0.340915\pi\)
\(660\) 0 0
\(661\) −287667. 166084.i −0.658395 0.380125i 0.133270 0.991080i \(-0.457452\pi\)
−0.791665 + 0.610955i \(0.790786\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 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
\(669\) 0 0
\(670\) 0 0
\(671\) −181101. + 313677.i −0.402232 + 0.696686i
\(672\) 0 0
\(673\) 40533.1i 0.0894912i −0.998998 0.0447456i \(-0.985752\pi\)
0.998998 0.0447456i \(-0.0142477\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(681\) 0 0
\(682\) 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
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 0 0
\(695\) −929659. −1.92466
\(696\) 0 0
\(697\) 7047.50 + 4068.88i 0.0145067 + 0.00837547i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −286342. 495960.i −0.582706 1.00928i −0.995157 0.0982968i \(-0.968661\pi\)
0.412451 0.910980i \(-0.364673\pi\)
\(702\) 0 0
\(703\) −129233. + 367789.i −0.261495 + 0.744197i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1348.18 2335.11i 0.00269717 0.00467163i
\(708\) 0 0
\(709\) −84487.5 + 146337.i −0.168074 + 0.291112i −0.937743 0.347331i \(-0.887088\pi\)
0.769669 + 0.638443i \(0.220421\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 67596.5 + 39026.9i 0.132967 + 0.0767688i
\(714\) 0 0
\(715\) 522829.i 1.02270i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6416.59 + 11113.9i 0.0124121 + 0.0214985i 0.872165 0.489212i \(-0.162716\pi\)
−0.859753 + 0.510711i \(0.829382\pi\)
\(720\) 0 0
\(721\) 29461.1i 0.0566732i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −949.297 + 548.077i −0.00180603 + 0.00104271i
\(726\) 0 0
\(727\) 385862. + 668332.i 0.730067 + 1.26451i 0.956854 + 0.290569i \(0.0938447\pi\)
−0.226787 + 0.973944i \(0.572822\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 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
\(735\) 0 0
\(736\) 0 0
\(737\) −616193. 355759.i −1.13444 0.654970i
\(738\) 0 0
\(739\) −56363.8 97624.9i −0.103207 0.178761i 0.809797 0.586710i \(-0.199577\pi\)
−0.913004 + 0.407950i \(0.866244\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −444706. + 256751.i −0.805555 + 0.465088i −0.845410 0.534118i \(-0.820644\pi\)
0.0398546 + 0.999205i \(0.487311\pi\)
\(744\) 0 0
\(745\) −257182. + 445453.i −0.463370 + 0.802581i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2406.06i 0.00428887i
\(750\) 0 0
\(751\) −740064. 427276.i −1.31217 0.757580i −0.329713 0.944081i \(-0.606952\pi\)
−0.982455 + 0.186501i \(0.940285\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −40624.2 + 23454.4i −0.0712673 + 0.0411462i
\(756\) 0 0
\(757\) −74063.5 128282.i −0.129245 0.223858i 0.794139 0.607736i \(-0.207922\pi\)
−0.923384 + 0.383877i \(0.874589\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −262528. −0.453322 −0.226661 0.973974i \(-0.572781\pi\)
−0.226661 + 0.973974i \(0.572781\pi\)
\(762\) 0 0
\(763\) −499363. + 288307.i −0.857762 + 0.495229i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 831514. 1.41344
\(768\) 0 0
\(769\) −23000.9 + 39838.8i −0.0388949 + 0.0673680i −0.884818 0.465938i \(-0.845717\pi\)
0.845923 + 0.533306i \(0.179050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 590492. + 340921.i 0.988224 + 0.570551i 0.904743 0.425958i \(-0.140063\pi\)
0.0834809 + 0.996509i \(0.473396\pi\)
\(774\) 0 0
\(775\) 2437.38 + 1407.22i 0.00405808 + 0.00234293i
\(776\) 0 0
\(777\) 0 0
\(778\) 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
\(783\) 0 0
\(784\) 0 0
\(785\) 337104. 583881.i 0.547047 0.947513i
\(786\) 0 0
\(787\) 356499.i 0.575584i −0.957693 0.287792i \(-0.907079\pi\)
0.957693 0.287792i \(-0.0929212\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 513635.i 0.820921i
\(792\) 0 0
\(793\) 313211. 180832.i 0.498070 0.287561i
\(794\) 0 0
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) 0 0
\(803\) 227665. + 394327.i 0.353073 + 0.611540i
\(804\) 0 0
\(805\) −496635. −0.766383
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −146248. −0.223456 −0.111728 0.993739i \(-0.535639\pi\)
−0.111728 + 0.993739i \(0.535639\pi\)
\(810\) 0 0
\(811\) −1.09333e6 631237.i −1.66231 0.959733i −0.971610 0.236588i \(-0.923971\pi\)
−0.690696 0.723145i \(-0.742696\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −176967. 306516.i −0.266427 0.461465i
\(816\) 0 0
\(817\) 126104. + 671266.i 0.188923 + 1.00566i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 86526.3 149868.i 0.128370 0.222343i −0.794676 0.607034i \(-0.792359\pi\)
0.923045 + 0.384692i \(0.125692\pi\)
\(822\) 0 0
\(823\) −164144. + 284306.i −0.242340 + 0.419745i −0.961380 0.275223i \(-0.911248\pi\)
0.719040 + 0.694968i \(0.244582\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 894676. + 516541.i 1.30814 + 0.755256i 0.981786 0.189991i \(-0.0608459\pi\)
0.326356 + 0.945247i \(0.394179\pi\)
\(828\) 0 0
\(829\) 876630.i 1.27558i −0.770211 0.637789i \(-0.779849\pi\)
0.770211 0.637789i \(-0.220151\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 71656.2 + 124112.i 0.103268 + 0.178865i
\(834\) 0 0
\(835\) 391322.i 0.561256i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −228597. + 131981.i −0.324748 + 0.187493i −0.653507 0.756921i \(-0.726703\pi\)
0.328759 + 0.944414i \(0.393370\pi\)
\(840\) 0 0
\(841\) −352702. 610897.i −0.498673 0.863726i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 103140. 178644.i 0.144449 0.250193i
\(846\) 0 0
\(847\) 163551. 0.227974
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −656118. 378810.i −0.905989 0.523073i
\(852\) 0 0
\(853\) 537972. + 931795.i 0.739370 + 1.28063i 0.952779 + 0.303664i \(0.0982099\pi\)
−0.213409 + 0.976963i \(0.568457\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −346531. + 200070.i −0.471824 + 0.272408i −0.717003 0.697070i \(-0.754487\pi\)
0.245179 + 0.969478i \(0.421153\pi\)
\(858\) 0 0
\(859\) −274871. + 476090.i −0.372514 + 0.645213i −0.989952 0.141407i \(-0.954837\pi\)
0.617438 + 0.786620i \(0.288171\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 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
\(867\) 0 0
\(868\) 0 0
\(869\) −707088. + 408238.i −0.936341 + 0.540597i
\(870\) 0 0
\(871\) 355231. + 615278.i 0.468246 + 0.811027i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 424516. 0.554469
\(876\) 0 0
\(877\) −919656. + 530964.i −1.19571 + 0.690344i −0.959596 0.281381i \(-0.909207\pi\)
−0.236115 + 0.971725i \(0.575874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 337096. 0.434312 0.217156 0.976137i \(-0.430322\pi\)
0.217156 + 0.976137i \(0.430322\pi\)
\(882\) 0 0
\(883\) −23184.8 + 40157.3i −0.0297360 + 0.0515043i −0.880510 0.474027i \(-0.842800\pi\)
0.850774 + 0.525531i \(0.176133\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 94128.5 + 54345.1i 0.119639 + 0.0690738i 0.558625 0.829420i \(-0.311329\pi\)
−0.438986 + 0.898494i \(0.644662\pi\)
\(888\) 0 0
\(889\) 446230. + 257631.i 0.564618 + 0.325982i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −748781. + 642681.i −0.938971 + 0.805921i
\(894\) 0 0
\(895\) 674887. 389646.i 0.842529 0.486434i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2410.33 4174.81i 0.00298234 0.00516556i
\(900\) 0 0
\(901\) 218601.i 0.269279i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.43602e6i 1.75332i
\(906\) 0 0
\(907\) −930792. + 537393.i −1.13146 + 0.653247i −0.944301 0.329083i \(-0.893260\pi\)
−0.187156 + 0.982330i \(0.559927\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(915\) 0 0
\(916\) 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
\(921\) 0 0
\(922\) 0 0
\(923\) −614598. −0.721420
\(924\) 0 0
\(925\) −23658.2 13659.1i −0.0276502 0.0159638i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 364161. + 630746.i 0.421951 + 0.730841i 0.996130 0.0878888i \(-0.0280120\pi\)
−0.574179 + 0.818730i \(0.694679\pi\)
\(930\) 0 0
\(931\) −555307. 195123.i −0.640669 0.225118i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −160594. + 278156.i −0.183698 + 0.318175i
\(936\) 0 0
\(937\) −338424. + 586168.i −0.385463 + 0.667641i −0.991833 0.127541i \(-0.959292\pi\)
0.606370 + 0.795182i \(0.292625\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 211702. + 122226.i 0.239081 + 0.138034i 0.614754 0.788719i \(-0.289255\pi\)
−0.375673 + 0.926752i \(0.622588\pi\)
\(942\) 0 0
\(943\) 64954.1i 0.0730438i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −392652. 680093.i −0.437832 0.758348i 0.559690 0.828702i \(-0.310920\pi\)
−0.997522 + 0.0703543i \(0.977587\pi\)
\(948\) 0 0
\(949\) 454653.i 0.504832i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 504490. 291268.i 0.555478 0.320705i −0.195850 0.980634i \(-0.562747\pi\)
0.751329 + 0.659928i \(0.229413\pi\)
\(954\) 0 0
\(955\) 258364. + 447500.i 0.283286 + 0.490666i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −247069. + 427936.i −0.268646 + 0.465309i
\(960\) 0 0
\(961\) 911144. 0.986598
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 23015.6 + 13288.1i 0.0247154 + 0.0142694i
\(966\) 0 0
\(967\) 378832. + 656157.i 0.405130 + 0.701705i 0.994337 0.106277i \(-0.0338930\pi\)
−0.589207 + 0.807982i \(0.700560\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 412390. 238093.i 0.437390 0.252527i −0.265100 0.964221i \(-0.585405\pi\)
0.702490 + 0.711694i \(0.252072\pi\)
\(972\) 0 0
\(973\) −505987. + 876395.i −0.534458 + 0.925709i
\(974\) 0 0
\(975\) 0 0
\(976\) 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
\(981\) 0 0
\(982\) 0 0
\(983\) 469744. 271207.i 0.486132 0.280668i −0.236837 0.971549i \(-0.576111\pi\)
0.722968 + 0.690881i \(0.242777\pi\)
\(984\) 0 0
\(985\) −764859. 1.32477e6i −0.788331 1.36543i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.32739e6 −1.35708
\(990\) 0 0
\(991\) 560235. 323452.i 0.570457 0.329354i −0.186875 0.982384i \(-0.559836\pi\)
0.757332 + 0.653030i \(0.226502\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.23996e6 −1.25246
\(996\) 0 0
\(997\) 581238. 1.00673e6i 0.584741 1.01280i −0.410166 0.912011i \(-0.634529\pi\)
0.994908 0.100791i \(-0.0321373\pi\)
\(998\) 0 0
\(999\) 0 0
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 684.5.y.c.145.5 12
3.2 odd 2 76.5.h.a.69.2 yes 12
12.11 even 2 304.5.r.b.145.5 12
19.8 odd 6 inner 684.5.y.c.217.5 12
57.8 even 6 76.5.h.a.65.2 12
228.179 odd 6 304.5.r.b.65.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.h.a.65.2 12 57.8 even 6
76.5.h.a.69.2 yes 12 3.2 odd 2
304.5.r.b.65.5 12 228.179 odd 6
304.5.r.b.145.5 12 12.11 even 2
684.5.y.c.145.5 12 1.1 even 1 trivial
684.5.y.c.217.5 12 19.8 odd 6 inner