Properties

Label 1520.2.q.p.961.1
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $10$
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] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), degree \(2\), 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 13x^{8} - 2x^{7} + 125x^{6} - 48x^{5} + 345x^{4} - 202x^{3} + 788x^{2} - 416x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(1.69277 - 2.93196i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.p.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} +O(q^{10})\) \(q+(-1.69277 + 2.93196i) q^{3} +(0.500000 - 0.866025i) q^{5} -2.29919 q^{7} +(-4.23092 - 7.32817i) q^{9} -5.21642 q^{11} +(-0.116253 - 0.201356i) q^{13} +(1.69277 + 2.93196i) q^{15} +(-2.19923 + 3.80917i) q^{17} +(2.13900 - 3.79798i) q^{19} +(3.89199 - 6.74113i) q^{21} +(0.643136 + 1.11395i) q^{23} +(-0.500000 - 0.866025i) q^{25} +18.4912 q^{27} +(3.31046 + 5.73388i) q^{29} -0.286273 q^{31} +(8.83018 - 15.2943i) q^{33} +(-1.14959 + 1.99116i) q^{35} +9.07026 q^{37} +0.787155 q^{39} +(-2.14959 + 3.72321i) q^{41} +(5.53994 - 9.59546i) q^{43} -8.46184 q^{45} +(2.57651 + 4.46265i) q^{47} -1.71373 q^{49} +(-7.44555 - 12.8961i) q^{51} +(-3.54942 - 6.14778i) q^{53} +(-2.60821 + 4.51755i) q^{55} +(7.51470 + 12.7006i) q^{57} +(3.24377 - 5.61838i) q^{59} +(-4.47153 - 7.74492i) q^{61} +(9.72769 + 16.8489i) q^{63} -0.232505 q^{65} +(5.09926 + 8.83218i) q^{67} -4.35472 q^{69} +(7.36352 - 12.7540i) q^{71} +(-0.282901 + 0.489999i) q^{73} +3.38553 q^{75} +11.9935 q^{77} +(-2.18328 + 3.78156i) q^{79} +(-18.6086 + 32.2311i) q^{81} +6.67867 q^{83} +(2.19923 + 3.80917i) q^{85} -22.4153 q^{87} +(-2.21463 - 3.83586i) q^{89} +(0.267287 + 0.462955i) q^{91} +(0.484593 - 0.839340i) q^{93} +(-2.21965 - 3.75142i) q^{95} +(1.75616 - 3.04175i) q^{97} +(22.0703 + 38.2268i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} + 5 q^{5} - 10 q^{7} - 10 q^{9} - 2 q^{11} - 5 q^{13} + q^{15} + 5 q^{17} + 4 q^{19} - 4 q^{21} + 11 q^{23} - 5 q^{25} + 20 q^{27} + 3 q^{29} - 12 q^{31} + 15 q^{33} - 5 q^{35} + 14 q^{37} + 2 q^{39} - 15 q^{41} + 11 q^{43} - 20 q^{45} + 6 q^{47} - 8 q^{49} - 20 q^{51} - 2 q^{53} - q^{55} + 25 q^{57} + 23 q^{59} - 4 q^{61} + 30 q^{63} - 10 q^{65} + 10 q^{67} - 2 q^{69} - 10 q^{71} + 2 q^{75} + 50 q^{77} + 5 q^{79} - 25 q^{81} - 22 q^{83} - 5 q^{85} + 44 q^{87} - 18 q^{91} - 7 q^{95} - 9 q^{97} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.69277 + 2.93196i −0.977319 + 1.69277i −0.305261 + 0.952269i \(0.598744\pi\)
−0.672058 + 0.740498i \(0.734589\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −2.29919 −0.869012 −0.434506 0.900669i \(-0.643077\pi\)
−0.434506 + 0.900669i \(0.643077\pi\)
\(8\) 0 0
\(9\) −4.23092 7.32817i −1.41031 2.44272i
\(10\) 0 0
\(11\) −5.21642 −1.57281 −0.786405 0.617712i \(-0.788060\pi\)
−0.786405 + 0.617712i \(0.788060\pi\)
\(12\) 0 0
\(13\) −0.116253 0.201356i −0.0322427 0.0558460i 0.849454 0.527663i \(-0.176932\pi\)
−0.881696 + 0.471817i \(0.843598\pi\)
\(14\) 0 0
\(15\) 1.69277 + 2.93196i 0.437071 + 0.757028i
\(16\) 0 0
\(17\) −2.19923 + 3.80917i −0.533390 + 0.923859i 0.465849 + 0.884864i \(0.345749\pi\)
−0.999239 + 0.0389952i \(0.987584\pi\)
\(18\) 0 0
\(19\) 2.13900 3.79798i 0.490720 0.871317i
\(20\) 0 0
\(21\) 3.89199 6.74113i 0.849302 1.47103i
\(22\) 0 0
\(23\) 0.643136 + 1.11395i 0.134103 + 0.232274i 0.925255 0.379347i \(-0.123851\pi\)
−0.791151 + 0.611620i \(0.790518\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 18.4912 3.55864
\(28\) 0 0
\(29\) 3.31046 + 5.73388i 0.614737 + 1.06476i 0.990431 + 0.138012i \(0.0440711\pi\)
−0.375694 + 0.926744i \(0.622596\pi\)
\(30\) 0 0
\(31\) −0.286273 −0.0514161 −0.0257081 0.999669i \(-0.508184\pi\)
−0.0257081 + 0.999669i \(0.508184\pi\)
\(32\) 0 0
\(33\) 8.83018 15.2943i 1.53714 2.66240i
\(34\) 0 0
\(35\) −1.14959 + 1.99116i −0.194317 + 0.336567i
\(36\) 0 0
\(37\) 9.07026 1.49114 0.745571 0.666427i \(-0.232177\pi\)
0.745571 + 0.666427i \(0.232177\pi\)
\(38\) 0 0
\(39\) 0.787155 0.126046
\(40\) 0 0
\(41\) −2.14959 + 3.72321i −0.335710 + 0.581467i −0.983621 0.180249i \(-0.942310\pi\)
0.647911 + 0.761716i \(0.275643\pi\)
\(42\) 0 0
\(43\) 5.53994 9.59546i 0.844833 1.46329i −0.0409333 0.999162i \(-0.513033\pi\)
0.885766 0.464132i \(-0.153634\pi\)
\(44\) 0 0
\(45\) −8.46184 −1.26142
\(46\) 0 0
\(47\) 2.57651 + 4.46265i 0.375823 + 0.650945i 0.990450 0.137873i \(-0.0440266\pi\)
−0.614627 + 0.788818i \(0.710693\pi\)
\(48\) 0 0
\(49\) −1.71373 −0.244818
\(50\) 0 0
\(51\) −7.44555 12.8961i −1.04259 1.80581i
\(52\) 0 0
\(53\) −3.54942 6.14778i −0.487551 0.844463i 0.512347 0.858779i \(-0.328776\pi\)
−0.999898 + 0.0143160i \(0.995443\pi\)
\(54\) 0 0
\(55\) −2.60821 + 4.51755i −0.351691 + 0.609146i
\(56\) 0 0
\(57\) 7.51470 + 12.7006i 0.995347 + 1.68223i
\(58\) 0 0
\(59\) 3.24377 5.61838i 0.422304 0.731451i −0.573861 0.818953i \(-0.694555\pi\)
0.996164 + 0.0875017i \(0.0278883\pi\)
\(60\) 0 0
\(61\) −4.47153 7.74492i −0.572521 0.991635i −0.996306 0.0858726i \(-0.972632\pi\)
0.423785 0.905763i \(-0.360701\pi\)
\(62\) 0 0
\(63\) 9.72769 + 16.8489i 1.22557 + 2.12276i
\(64\) 0 0
\(65\) −0.232505 −0.0288387
\(66\) 0 0
\(67\) 5.09926 + 8.83218i 0.622974 + 1.07902i 0.988929 + 0.148390i \(0.0474091\pi\)
−0.365955 + 0.930633i \(0.619258\pi\)
\(68\) 0 0
\(69\) −4.35472 −0.524247
\(70\) 0 0
\(71\) 7.36352 12.7540i 0.873889 1.51362i 0.0159484 0.999873i \(-0.494923\pi\)
0.857941 0.513748i \(-0.171743\pi\)
\(72\) 0 0
\(73\) −0.282901 + 0.489999i −0.0331111 + 0.0573501i −0.882106 0.471051i \(-0.843875\pi\)
0.848995 + 0.528401i \(0.177208\pi\)
\(74\) 0 0
\(75\) 3.38553 0.390928
\(76\) 0 0
\(77\) 11.9935 1.36679
\(78\) 0 0
\(79\) −2.18328 + 3.78156i −0.245639 + 0.425459i −0.962311 0.271951i \(-0.912331\pi\)
0.716672 + 0.697410i \(0.245664\pi\)
\(80\) 0 0
\(81\) −18.6086 + 32.2311i −2.06762 + 3.58123i
\(82\) 0 0
\(83\) 6.67867 0.733080 0.366540 0.930402i \(-0.380542\pi\)
0.366540 + 0.930402i \(0.380542\pi\)
\(84\) 0 0
\(85\) 2.19923 + 3.80917i 0.238539 + 0.413163i
\(86\) 0 0
\(87\) −22.4153 −2.40318
\(88\) 0 0
\(89\) −2.21463 3.83586i −0.234751 0.406600i 0.724450 0.689328i \(-0.242094\pi\)
−0.959200 + 0.282728i \(0.908761\pi\)
\(90\) 0 0
\(91\) 0.267287 + 0.462955i 0.0280193 + 0.0485308i
\(92\) 0 0
\(93\) 0.484593 0.839340i 0.0502500 0.0870355i
\(94\) 0 0
\(95\) −2.21965 3.75142i −0.227731 0.384888i
\(96\) 0 0
\(97\) 1.75616 3.04175i 0.178311 0.308843i −0.762991 0.646409i \(-0.776270\pi\)
0.941302 + 0.337565i \(0.109603\pi\)
\(98\) 0 0
\(99\) 22.0703 + 38.2268i 2.21814 + 3.84194i
\(100\) 0 0
\(101\) 7.15963 + 12.4008i 0.712409 + 1.23393i 0.963950 + 0.266083i \(0.0857294\pi\)
−0.251541 + 0.967847i \(0.580937\pi\)
\(102\) 0 0
\(103\) −19.0649 −1.87852 −0.939258 0.343211i \(-0.888485\pi\)
−0.939258 + 0.343211i \(0.888485\pi\)
\(104\) 0 0
\(105\) −3.89199 6.74113i −0.379820 0.657867i
\(106\) 0 0
\(107\) 0.712658 0.0688952 0.0344476 0.999407i \(-0.489033\pi\)
0.0344476 + 0.999407i \(0.489033\pi\)
\(108\) 0 0
\(109\) 3.47423 6.01754i 0.332771 0.576376i −0.650283 0.759692i \(-0.725350\pi\)
0.983054 + 0.183316i \(0.0586831\pi\)
\(110\) 0 0
\(111\) −15.3538 + 26.5936i −1.45732 + 2.52415i
\(112\) 0 0
\(113\) 17.1125 1.60980 0.804902 0.593407i \(-0.202218\pi\)
0.804902 + 0.593407i \(0.202218\pi\)
\(114\) 0 0
\(115\) 1.28627 0.119946
\(116\) 0 0
\(117\) −0.983712 + 1.70384i −0.0909442 + 0.157520i
\(118\) 0 0
\(119\) 5.05644 8.75800i 0.463523 0.802845i
\(120\) 0 0
\(121\) 16.2110 1.47373
\(122\) 0 0
\(123\) −7.27753 12.6050i −0.656192 1.13656i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.191120 + 0.331029i 0.0169591 + 0.0293741i 0.874380 0.485241i \(-0.161268\pi\)
−0.857421 + 0.514615i \(0.827935\pi\)
\(128\) 0 0
\(129\) 18.7557 + 32.4857i 1.65134 + 2.86021i
\(130\) 0 0
\(131\) 1.76406 3.05544i 0.154127 0.266955i −0.778614 0.627503i \(-0.784077\pi\)
0.932741 + 0.360548i \(0.117410\pi\)
\(132\) 0 0
\(133\) −4.91797 + 8.73229i −0.426442 + 0.757185i
\(134\) 0 0
\(135\) 9.24562 16.0139i 0.795737 1.37826i
\(136\) 0 0
\(137\) −0.975668 1.68991i −0.0833570 0.144378i 0.821333 0.570449i \(-0.193231\pi\)
−0.904690 + 0.426071i \(0.859898\pi\)
\(138\) 0 0
\(139\) −4.20548 7.28410i −0.356704 0.617829i 0.630704 0.776023i \(-0.282766\pi\)
−0.987408 + 0.158194i \(0.949433\pi\)
\(140\) 0 0
\(141\) −17.4458 −1.46920
\(142\) 0 0
\(143\) 0.606423 + 1.05035i 0.0507116 + 0.0878351i
\(144\) 0 0
\(145\) 6.62092 0.549837
\(146\) 0 0
\(147\) 2.90094 5.02458i 0.239266 0.414420i
\(148\) 0 0
\(149\) 3.94411 6.83140i 0.323114 0.559650i −0.658015 0.753005i \(-0.728603\pi\)
0.981129 + 0.193355i \(0.0619368\pi\)
\(150\) 0 0
\(151\) −12.3188 −1.00249 −0.501247 0.865304i \(-0.667125\pi\)
−0.501247 + 0.865304i \(0.667125\pi\)
\(152\) 0 0
\(153\) 37.2190 3.00898
\(154\) 0 0
\(155\) −0.143136 + 0.247920i −0.0114970 + 0.0199134i
\(156\) 0 0
\(157\) 5.85061 10.1335i 0.466929 0.808745i −0.532357 0.846520i \(-0.678694\pi\)
0.999286 + 0.0377747i \(0.0120269\pi\)
\(158\) 0 0
\(159\) 24.0334 1.90597
\(160\) 0 0
\(161\) −1.47869 2.56117i −0.116537 0.201849i
\(162\) 0 0
\(163\) 2.98968 0.234170 0.117085 0.993122i \(-0.462645\pi\)
0.117085 + 0.993122i \(0.462645\pi\)
\(164\) 0 0
\(165\) −8.83018 15.2943i −0.687429 1.19066i
\(166\) 0 0
\(167\) 1.68129 + 2.91208i 0.130102 + 0.225343i 0.923716 0.383079i \(-0.125136\pi\)
−0.793614 + 0.608422i \(0.791803\pi\)
\(168\) 0 0
\(169\) 6.47297 11.2115i 0.497921 0.862424i
\(170\) 0 0
\(171\) −36.8822 + 0.394018i −2.82045 + 0.0301313i
\(172\) 0 0
\(173\) 0.926712 1.60511i 0.0704566 0.122034i −0.828645 0.559775i \(-0.810888\pi\)
0.899101 + 0.437740i \(0.144221\pi\)
\(174\) 0 0
\(175\) 1.14959 + 1.99116i 0.0869012 + 0.150517i
\(176\) 0 0
\(177\) 10.9819 + 19.0212i 0.825451 + 1.42972i
\(178\) 0 0
\(179\) 18.9406 1.41569 0.707843 0.706370i \(-0.249668\pi\)
0.707843 + 0.706370i \(0.249668\pi\)
\(180\) 0 0
\(181\) −6.27285 10.8649i −0.466258 0.807582i 0.533000 0.846115i \(-0.321065\pi\)
−0.999257 + 0.0385335i \(0.987731\pi\)
\(182\) 0 0
\(183\) 30.2770 2.23814
\(184\) 0 0
\(185\) 4.53513 7.85507i 0.333429 0.577516i
\(186\) 0 0
\(187\) 11.4721 19.8702i 0.838922 1.45305i
\(188\) 0 0
\(189\) −42.5149 −3.09250
\(190\) 0 0
\(191\) −6.68472 −0.483690 −0.241845 0.970315i \(-0.577752\pi\)
−0.241845 + 0.970315i \(0.577752\pi\)
\(192\) 0 0
\(193\) 4.51808 7.82554i 0.325218 0.563294i −0.656338 0.754467i \(-0.727896\pi\)
0.981556 + 0.191172i \(0.0612289\pi\)
\(194\) 0 0
\(195\) 0.393577 0.681696i 0.0281847 0.0488173i
\(196\) 0 0
\(197\) −16.7458 −1.19309 −0.596543 0.802581i \(-0.703460\pi\)
−0.596543 + 0.802581i \(0.703460\pi\)
\(198\) 0 0
\(199\) −2.01555 3.49103i −0.142878 0.247472i 0.785701 0.618606i \(-0.212302\pi\)
−0.928579 + 0.371134i \(0.878969\pi\)
\(200\) 0 0
\(201\) −34.5274 −2.43538
\(202\) 0 0
\(203\) −7.61137 13.1833i −0.534214 0.925285i
\(204\) 0 0
\(205\) 2.14959 + 3.72321i 0.150134 + 0.260040i
\(206\) 0 0
\(207\) 5.44212 9.42603i 0.378253 0.655154i
\(208\) 0 0
\(209\) −11.1579 + 19.8119i −0.771809 + 1.37042i
\(210\) 0 0
\(211\) −3.19344 + 5.53120i −0.219846 + 0.380784i −0.954761 0.297376i \(-0.903889\pi\)
0.734915 + 0.678159i \(0.237222\pi\)
\(212\) 0 0
\(213\) 24.9295 + 43.1791i 1.70814 + 2.95858i
\(214\) 0 0
\(215\) −5.53994 9.59546i −0.377821 0.654405i
\(216\) 0 0
\(217\) 0.658196 0.0446812
\(218\) 0 0
\(219\) −0.957772 1.65891i −0.0647202 0.112099i
\(220\) 0 0
\(221\) 1.02266 0.0687918
\(222\) 0 0
\(223\) −6.44142 + 11.1569i −0.431349 + 0.747118i −0.996990 0.0775333i \(-0.975296\pi\)
0.565641 + 0.824652i \(0.308629\pi\)
\(224\) 0 0
\(225\) −4.23092 + 7.32817i −0.282061 + 0.488545i
\(226\) 0 0
\(227\) −19.2498 −1.27766 −0.638828 0.769350i \(-0.720580\pi\)
−0.638828 + 0.769350i \(0.720580\pi\)
\(228\) 0 0
\(229\) −19.3443 −1.27831 −0.639153 0.769080i \(-0.720715\pi\)
−0.639153 + 0.769080i \(0.720715\pi\)
\(230\) 0 0
\(231\) −20.3023 + 35.1645i −1.33579 + 2.31366i
\(232\) 0 0
\(233\) −4.63365 + 8.02572i −0.303561 + 0.525783i −0.976940 0.213515i \(-0.931509\pi\)
0.673379 + 0.739297i \(0.264842\pi\)
\(234\) 0 0
\(235\) 5.15303 0.336147
\(236\) 0 0
\(237\) −7.39158 12.8026i −0.480135 0.831618i
\(238\) 0 0
\(239\) −3.20995 −0.207635 −0.103817 0.994596i \(-0.533106\pi\)
−0.103817 + 0.994596i \(0.533106\pi\)
\(240\) 0 0
\(241\) 1.19291 + 2.06617i 0.0768419 + 0.133094i 0.901886 0.431975i \(-0.142183\pi\)
−0.825044 + 0.565069i \(0.808850\pi\)
\(242\) 0 0
\(243\) −35.2632 61.0777i −2.26214 3.91814i
\(244\) 0 0
\(245\) −0.856864 + 1.48413i −0.0547430 + 0.0948177i
\(246\) 0 0
\(247\) −1.01341 + 0.0108264i −0.0644817 + 0.000688868i
\(248\) 0 0
\(249\) −11.3054 + 19.5816i −0.716453 + 1.24093i
\(250\) 0 0
\(251\) −0.257596 0.446170i −0.0162593 0.0281620i 0.857781 0.514015i \(-0.171842\pi\)
−0.874041 + 0.485853i \(0.838509\pi\)
\(252\) 0 0
\(253\) −3.35487 5.81080i −0.210919 0.365322i
\(254\) 0 0
\(255\) −14.8911 −0.932517
\(256\) 0 0
\(257\) −3.87373 6.70950i −0.241637 0.418527i 0.719544 0.694447i \(-0.244351\pi\)
−0.961181 + 0.275920i \(0.911018\pi\)
\(258\) 0 0
\(259\) −20.8542 −1.29582
\(260\) 0 0
\(261\) 28.0126 48.5192i 1.73394 3.00326i
\(262\) 0 0
\(263\) 15.7196 27.2272i 0.969315 1.67890i 0.271769 0.962363i \(-0.412391\pi\)
0.697546 0.716540i \(-0.254275\pi\)
\(264\) 0 0
\(265\) −7.09885 −0.436079
\(266\) 0 0
\(267\) 14.9954 0.917705
\(268\) 0 0
\(269\) 6.14663 10.6463i 0.374767 0.649115i −0.615525 0.788117i \(-0.711056\pi\)
0.990292 + 0.139002i \(0.0443894\pi\)
\(270\) 0 0
\(271\) 6.15091 10.6537i 0.373641 0.647165i −0.616482 0.787369i \(-0.711442\pi\)
0.990123 + 0.140204i \(0.0447758\pi\)
\(272\) 0 0
\(273\) −1.80982 −0.109535
\(274\) 0 0
\(275\) 2.60821 + 4.51755i 0.157281 + 0.272419i
\(276\) 0 0
\(277\) 16.3814 0.984264 0.492132 0.870521i \(-0.336218\pi\)
0.492132 + 0.870521i \(0.336218\pi\)
\(278\) 0 0
\(279\) 1.21120 + 2.09786i 0.0725125 + 0.125595i
\(280\) 0 0
\(281\) 2.81293 + 4.87213i 0.167805 + 0.290647i 0.937648 0.347587i \(-0.112999\pi\)
−0.769843 + 0.638234i \(0.779665\pi\)
\(282\) 0 0
\(283\) 12.2775 21.2653i 0.729823 1.26409i −0.227135 0.973863i \(-0.572936\pi\)
0.956958 0.290227i \(-0.0937310\pi\)
\(284\) 0 0
\(285\) 14.7564 0.157644i 0.874091 0.00933805i
\(286\) 0 0
\(287\) 4.94233 8.56036i 0.291736 0.505302i
\(288\) 0 0
\(289\) −1.17318 2.03201i −0.0690108 0.119530i
\(290\) 0 0
\(291\) 5.94553 + 10.2980i 0.348533 + 0.603677i
\(292\) 0 0
\(293\) 0.437953 0.0255855 0.0127928 0.999918i \(-0.495928\pi\)
0.0127928 + 0.999918i \(0.495928\pi\)
\(294\) 0 0
\(295\) −3.24377 5.61838i −0.188860 0.327115i
\(296\) 0 0
\(297\) −96.4581 −5.59707
\(298\) 0 0
\(299\) 0.149533 0.258998i 0.00864770 0.0149783i
\(300\) 0 0
\(301\) −12.7374 + 22.0618i −0.734170 + 1.27162i
\(302\) 0 0
\(303\) −48.4783 −2.78501
\(304\) 0 0
\(305\) −8.94306 −0.512078
\(306\) 0 0
\(307\) −8.56269 + 14.8310i −0.488699 + 0.846451i −0.999915 0.0130008i \(-0.995862\pi\)
0.511217 + 0.859452i \(0.329195\pi\)
\(308\) 0 0
\(309\) 32.2724 55.8974i 1.83591 3.17989i
\(310\) 0 0
\(311\) 31.6381 1.79403 0.897016 0.441999i \(-0.145730\pi\)
0.897016 + 0.441999i \(0.145730\pi\)
\(312\) 0 0
\(313\) 7.71574 + 13.3641i 0.436119 + 0.755381i 0.997386 0.0722540i \(-0.0230192\pi\)
−0.561267 + 0.827635i \(0.689686\pi\)
\(314\) 0 0
\(315\) 19.4554 1.09619
\(316\) 0 0
\(317\) −4.24022 7.34427i −0.238154 0.412495i 0.722030 0.691861i \(-0.243209\pi\)
−0.960185 + 0.279366i \(0.909876\pi\)
\(318\) 0 0
\(319\) −17.2687 29.9103i −0.966864 1.67466i
\(320\) 0 0
\(321\) −1.20636 + 2.08948i −0.0673327 + 0.116624i
\(322\) 0 0
\(323\) 9.76302 + 16.5004i 0.543229 + 0.918109i
\(324\) 0 0
\(325\) −0.116253 + 0.201356i −0.00644854 + 0.0111692i
\(326\) 0 0
\(327\) 11.7621 + 20.3726i 0.650447 + 1.12661i
\(328\) 0 0
\(329\) −5.92390 10.2605i −0.326595 0.565679i
\(330\) 0 0
\(331\) 7.39022 0.406203 0.203102 0.979158i \(-0.434898\pi\)
0.203102 + 0.979158i \(0.434898\pi\)
\(332\) 0 0
\(333\) −38.3755 66.4684i −2.10297 3.64244i
\(334\) 0 0
\(335\) 10.1985 0.557205
\(336\) 0 0
\(337\) 18.2139 31.5475i 0.992177 1.71850i 0.387972 0.921671i \(-0.373176\pi\)
0.604204 0.796829i \(-0.293491\pi\)
\(338\) 0 0
\(339\) −28.9674 + 50.1730i −1.57329 + 2.72502i
\(340\) 0 0
\(341\) 1.49332 0.0808678
\(342\) 0 0
\(343\) 20.0345 1.08176
\(344\) 0 0
\(345\) −2.17736 + 3.77130i −0.117225 + 0.203040i
\(346\) 0 0
\(347\) −2.04868 + 3.54842i −0.109979 + 0.190489i −0.915761 0.401722i \(-0.868412\pi\)
0.805783 + 0.592212i \(0.201745\pi\)
\(348\) 0 0
\(349\) −27.4105 −1.46725 −0.733626 0.679554i \(-0.762173\pi\)
−0.733626 + 0.679554i \(0.762173\pi\)
\(350\) 0 0
\(351\) −2.14966 3.72332i −0.114740 0.198736i
\(352\) 0 0
\(353\) −18.0481 −0.960605 −0.480302 0.877103i \(-0.659473\pi\)
−0.480302 + 0.877103i \(0.659473\pi\)
\(354\) 0 0
\(355\) −7.36352 12.7540i −0.390815 0.676912i
\(356\) 0 0
\(357\) 17.1187 + 29.6505i 0.906020 + 1.56927i
\(358\) 0 0
\(359\) 6.76679 11.7204i 0.357138 0.618580i −0.630344 0.776316i \(-0.717086\pi\)
0.987481 + 0.157736i \(0.0504194\pi\)
\(360\) 0 0
\(361\) −9.84936 16.2478i −0.518388 0.855146i
\(362\) 0 0
\(363\) −27.4415 + 47.5300i −1.44030 + 2.49468i
\(364\) 0 0
\(365\) 0.282901 + 0.489999i 0.0148077 + 0.0256477i
\(366\) 0 0
\(367\) 15.2090 + 26.3428i 0.793906 + 1.37509i 0.923532 + 0.383522i \(0.125289\pi\)
−0.129626 + 0.991563i \(0.541378\pi\)
\(368\) 0 0
\(369\) 36.3791 1.89382
\(370\) 0 0
\(371\) 8.16080 + 14.1349i 0.423687 + 0.733848i
\(372\) 0 0
\(373\) 20.6468 1.06905 0.534525 0.845153i \(-0.320491\pi\)
0.534525 + 0.845153i \(0.320491\pi\)
\(374\) 0 0
\(375\) 1.69277 2.93196i 0.0874141 0.151406i
\(376\) 0 0
\(377\) 0.769699 1.33316i 0.0396415 0.0686612i
\(378\) 0 0
\(379\) 27.5790 1.41664 0.708319 0.705893i \(-0.249454\pi\)
0.708319 + 0.705893i \(0.249454\pi\)
\(380\) 0 0
\(381\) −1.29409 −0.0662980
\(382\) 0 0
\(383\) −4.19855 + 7.27210i −0.214536 + 0.371587i −0.953129 0.302564i \(-0.902157\pi\)
0.738593 + 0.674152i \(0.235491\pi\)
\(384\) 0 0
\(385\) 5.99677 10.3867i 0.305624 0.529356i
\(386\) 0 0
\(387\) −93.7562 −4.76589
\(388\) 0 0
\(389\) 5.54246 + 9.59983i 0.281014 + 0.486731i 0.971635 0.236487i \(-0.0759959\pi\)
−0.690621 + 0.723217i \(0.742663\pi\)
\(390\) 0 0
\(391\) −5.65761 −0.286118
\(392\) 0 0
\(393\) 5.97229 + 10.3443i 0.301262 + 0.521801i
\(394\) 0 0
\(395\) 2.18328 + 3.78156i 0.109853 + 0.190271i
\(396\) 0 0
\(397\) 18.0199 31.2114i 0.904394 1.56646i 0.0826655 0.996577i \(-0.473657\pi\)
0.821729 0.569879i \(-0.193010\pi\)
\(398\) 0 0
\(399\) −17.2777 29.2010i −0.864968 1.46188i
\(400\) 0 0
\(401\) 3.09515 5.36095i 0.154564 0.267713i −0.778336 0.627848i \(-0.783936\pi\)
0.932900 + 0.360135i \(0.117269\pi\)
\(402\) 0 0
\(403\) 0.0332800 + 0.0576426i 0.00165779 + 0.00287138i
\(404\) 0 0
\(405\) 18.6086 + 32.2311i 0.924669 + 1.60157i
\(406\) 0 0
\(407\) −47.3143 −2.34528
\(408\) 0 0
\(409\) −14.1432 24.4967i −0.699335 1.21128i −0.968697 0.248244i \(-0.920146\pi\)
0.269363 0.963039i \(-0.413187\pi\)
\(410\) 0 0
\(411\) 6.60632 0.325866
\(412\) 0 0
\(413\) −7.45805 + 12.9177i −0.366987 + 0.635640i
\(414\) 0 0
\(415\) 3.33934 5.78390i 0.163922 0.283921i
\(416\) 0 0
\(417\) 28.4756 1.39445
\(418\) 0 0
\(419\) 23.9433 1.16971 0.584854 0.811139i \(-0.301152\pi\)
0.584854 + 0.811139i \(0.301152\pi\)
\(420\) 0 0
\(421\) −3.33251 + 5.77207i −0.162417 + 0.281314i −0.935735 0.352704i \(-0.885262\pi\)
0.773318 + 0.634018i \(0.218595\pi\)
\(422\) 0 0
\(423\) 21.8021 37.7623i 1.06005 1.83606i
\(424\) 0 0
\(425\) 4.39845 0.213356
\(426\) 0 0
\(427\) 10.2809 + 17.8070i 0.497528 + 0.861743i
\(428\) 0 0
\(429\) −4.10613 −0.198246
\(430\) 0 0
\(431\) 12.0568 + 20.8829i 0.580754 + 1.00590i 0.995390 + 0.0959080i \(0.0305755\pi\)
−0.414636 + 0.909987i \(0.636091\pi\)
\(432\) 0 0
\(433\) 2.02923 + 3.51474i 0.0975188 + 0.168907i 0.910657 0.413163i \(-0.135576\pi\)
−0.813138 + 0.582071i \(0.802243\pi\)
\(434\) 0 0
\(435\) −11.2077 + 19.4123i −0.537367 + 0.930747i
\(436\) 0 0
\(437\) 5.60641 0.0598942i 0.268191 0.00286513i
\(438\) 0 0
\(439\) 9.21029 15.9527i 0.439583 0.761381i −0.558074 0.829791i \(-0.688459\pi\)
0.997657 + 0.0684106i \(0.0217928\pi\)
\(440\) 0 0
\(441\) 7.25064 + 12.5585i 0.345269 + 0.598023i
\(442\) 0 0
\(443\) 1.71375 + 2.96830i 0.0814226 + 0.141028i 0.903861 0.427826i \(-0.140720\pi\)
−0.822439 + 0.568854i \(0.807387\pi\)
\(444\) 0 0
\(445\) −4.42926 −0.209967
\(446\) 0 0
\(447\) 13.3529 + 23.1279i 0.631572 + 1.09391i
\(448\) 0 0
\(449\) −5.30746 −0.250475 −0.125237 0.992127i \(-0.539969\pi\)
−0.125237 + 0.992127i \(0.539969\pi\)
\(450\) 0 0
\(451\) 11.2132 19.4218i 0.528008 0.914537i
\(452\) 0 0
\(453\) 20.8529 36.1184i 0.979757 1.69699i
\(454\) 0 0
\(455\) 0.534574 0.0250612
\(456\) 0 0
\(457\) −3.75416 −0.175612 −0.0878062 0.996138i \(-0.527986\pi\)
−0.0878062 + 0.996138i \(0.527986\pi\)
\(458\) 0 0
\(459\) −40.6664 + 70.4363i −1.89815 + 3.28769i
\(460\) 0 0
\(461\) −17.8776 + 30.9649i −0.832643 + 1.44218i 0.0632916 + 0.997995i \(0.479840\pi\)
−0.895935 + 0.444185i \(0.853493\pi\)
\(462\) 0 0
\(463\) −32.9059 −1.52927 −0.764634 0.644465i \(-0.777080\pi\)
−0.764634 + 0.644465i \(0.777080\pi\)
\(464\) 0 0
\(465\) −0.484593 0.839340i −0.0224725 0.0389235i
\(466\) 0 0
\(467\) 17.9580 0.830999 0.415499 0.909593i \(-0.363607\pi\)
0.415499 + 0.909593i \(0.363607\pi\)
\(468\) 0 0
\(469\) −11.7242 20.3069i −0.541372 0.937683i
\(470\) 0 0
\(471\) 19.8074 + 34.3075i 0.912678 + 1.58080i
\(472\) 0 0
\(473\) −28.8986 + 50.0539i −1.32876 + 2.30148i
\(474\) 0 0
\(475\) −4.35865 + 0.0465641i −0.199989 + 0.00213651i
\(476\) 0 0
\(477\) −30.0347 + 52.0215i −1.37519 + 2.38190i
\(478\) 0 0
\(479\) −12.5142 21.6752i −0.571787 0.990364i −0.996383 0.0849806i \(-0.972917\pi\)
0.424596 0.905383i \(-0.360416\pi\)
\(480\) 0 0
\(481\) −1.05444 1.82635i −0.0480784 0.0832742i
\(482\) 0 0
\(483\) 10.0123 0.455577
\(484\) 0 0
\(485\) −1.75616 3.04175i −0.0797430 0.138119i
\(486\) 0 0
\(487\) −15.4665 −0.700855 −0.350428 0.936590i \(-0.613964\pi\)
−0.350428 + 0.936590i \(0.613964\pi\)
\(488\) 0 0
\(489\) −5.06084 + 8.76563i −0.228859 + 0.396395i
\(490\) 0 0
\(491\) 2.88154 4.99097i 0.130042 0.225239i −0.793651 0.608374i \(-0.791822\pi\)
0.923693 + 0.383135i \(0.125155\pi\)
\(492\) 0 0
\(493\) −29.1218 −1.31158
\(494\) 0 0
\(495\) 44.1405 1.98397
\(496\) 0 0
\(497\) −16.9301 + 29.3239i −0.759420 + 1.31535i
\(498\) 0 0
\(499\) 12.4539 21.5708i 0.557512 0.965640i −0.440191 0.897904i \(-0.645089\pi\)
0.997703 0.0677354i \(-0.0215774\pi\)
\(500\) 0 0
\(501\) −11.3841 −0.508605
\(502\) 0 0
\(503\) 16.7754 + 29.0559i 0.747978 + 1.29554i 0.948790 + 0.315907i \(0.102309\pi\)
−0.200812 + 0.979630i \(0.564358\pi\)
\(504\) 0 0
\(505\) 14.3193 0.637198
\(506\) 0 0
\(507\) 21.9145 + 37.9570i 0.973255 + 1.68573i
\(508\) 0 0
\(509\) 9.48949 + 16.4363i 0.420614 + 0.728525i 0.996000 0.0893573i \(-0.0284813\pi\)
−0.575386 + 0.817882i \(0.695148\pi\)
\(510\) 0 0
\(511\) 0.650444 1.12660i 0.0287739 0.0498379i
\(512\) 0 0
\(513\) 39.5528 70.2295i 1.74630 3.10071i
\(514\) 0 0
\(515\) −9.53243 + 16.5107i −0.420049 + 0.727546i
\(516\) 0 0
\(517\) −13.4402 23.2791i −0.591098 1.02381i
\(518\) 0 0
\(519\) 3.13741 + 5.43416i 0.137717 + 0.238533i
\(520\) 0 0
\(521\) 34.4750 1.51038 0.755189 0.655507i \(-0.227545\pi\)
0.755189 + 0.655507i \(0.227545\pi\)
\(522\) 0 0
\(523\) −5.64257 9.77322i −0.246732 0.427353i 0.715885 0.698218i \(-0.246024\pi\)
−0.962617 + 0.270865i \(0.912690\pi\)
\(524\) 0 0
\(525\) −7.78398 −0.339721
\(526\) 0 0
\(527\) 0.629579 1.09046i 0.0274249 0.0475013i
\(528\) 0 0
\(529\) 10.6728 18.4857i 0.464033 0.803728i
\(530\) 0 0
\(531\) −54.8966 −2.38231
\(532\) 0 0
\(533\) 0.999585 0.0432968
\(534\) 0 0
\(535\) 0.356329 0.617180i 0.0154054 0.0266830i
\(536\) 0 0
\(537\) −32.0620 + 55.5330i −1.38358 + 2.39643i
\(538\) 0 0
\(539\) 8.93952 0.385052
\(540\) 0 0
\(541\) 3.86609 + 6.69627i 0.166216 + 0.287895i 0.937087 0.349097i \(-0.113512\pi\)
−0.770870 + 0.636992i \(0.780178\pi\)
\(542\) 0 0
\(543\) 42.4739 1.82273
\(544\) 0 0
\(545\) −3.47423 6.01754i −0.148820 0.257763i
\(546\) 0 0
\(547\) 8.48302 + 14.6930i 0.362708 + 0.628228i 0.988406 0.151837i \(-0.0485190\pi\)
−0.625698 + 0.780066i \(0.715186\pi\)
\(548\) 0 0
\(549\) −37.8374 + 65.5363i −1.61486 + 2.79702i
\(550\) 0 0
\(551\) 28.8583 0.308297i 1.22940 0.0131339i
\(552\) 0 0
\(553\) 5.01978 8.69452i 0.213463 0.369729i
\(554\) 0 0
\(555\) 15.3538 + 26.5936i 0.651734 + 1.12884i
\(556\) 0 0
\(557\) 2.88450 + 4.99611i 0.122220 + 0.211692i 0.920643 0.390406i \(-0.127665\pi\)
−0.798423 + 0.602097i \(0.794332\pi\)
\(558\) 0 0
\(559\) −2.57613 −0.108959
\(560\) 0 0
\(561\) 38.8391 + 67.2713i 1.63979 + 2.84020i
\(562\) 0 0
\(563\) 36.3722 1.53291 0.766453 0.642301i \(-0.222020\pi\)
0.766453 + 0.642301i \(0.222020\pi\)
\(564\) 0 0
\(565\) 8.55623 14.8198i 0.359963 0.623475i
\(566\) 0 0
\(567\) 42.7847 74.1053i 1.79679 3.11213i
\(568\) 0 0
\(569\) −45.9450 −1.92611 −0.963057 0.269298i \(-0.913208\pi\)
−0.963057 + 0.269298i \(0.913208\pi\)
\(570\) 0 0
\(571\) −28.7409 −1.20277 −0.601385 0.798960i \(-0.705384\pi\)
−0.601385 + 0.798960i \(0.705384\pi\)
\(572\) 0 0
\(573\) 11.3157 19.5993i 0.472719 0.818774i
\(574\) 0 0
\(575\) 0.643136 1.11395i 0.0268206 0.0464547i
\(576\) 0 0
\(577\) −3.39938 −0.141518 −0.0707591 0.997493i \(-0.522542\pi\)
−0.0707591 + 0.997493i \(0.522542\pi\)
\(578\) 0 0
\(579\) 15.2961 + 26.4936i 0.635684 + 1.10104i
\(580\) 0 0
\(581\) −15.3555 −0.637055
\(582\) 0 0
\(583\) 18.5153 + 32.0694i 0.766824 + 1.32818i
\(584\) 0 0
\(585\) 0.983712 + 1.70384i 0.0406715 + 0.0704451i
\(586\) 0 0
\(587\) −8.77194 + 15.1935i −0.362057 + 0.627101i −0.988299 0.152528i \(-0.951259\pi\)
0.626242 + 0.779628i \(0.284592\pi\)
\(588\) 0 0
\(589\) −0.612338 + 1.08726i −0.0252309 + 0.0447998i
\(590\) 0 0
\(591\) 28.3467 49.0979i 1.16603 2.01962i
\(592\) 0 0
\(593\) 14.8445 + 25.7114i 0.609590 + 1.05584i 0.991308 + 0.131562i \(0.0419994\pi\)
−0.381718 + 0.924279i \(0.624667\pi\)
\(594\) 0 0
\(595\) −5.05644 8.75800i −0.207294 0.359043i
\(596\) 0 0
\(597\) 13.6474 0.558551
\(598\) 0 0
\(599\) −18.2305 31.5762i −0.744879 1.29017i −0.950251 0.311484i \(-0.899174\pi\)
0.205373 0.978684i \(-0.434159\pi\)
\(600\) 0 0
\(601\) −1.25119 −0.0510372 −0.0255186 0.999674i \(-0.508124\pi\)
−0.0255186 + 0.999674i \(0.508124\pi\)
\(602\) 0 0
\(603\) 43.1491 74.7365i 1.75717 3.04351i
\(604\) 0 0
\(605\) 8.10551 14.0392i 0.329536 0.570773i
\(606\) 0 0
\(607\) −21.1279 −0.857554 −0.428777 0.903410i \(-0.641055\pi\)
−0.428777 + 0.903410i \(0.641055\pi\)
\(608\) 0 0
\(609\) 51.5371 2.08839
\(610\) 0 0
\(611\) 0.599053 1.03759i 0.0242351 0.0419764i
\(612\) 0 0
\(613\) 18.0094 31.1932i 0.727393 1.25988i −0.230588 0.973052i \(-0.574065\pi\)
0.957981 0.286831i \(-0.0926017\pi\)
\(614\) 0 0
\(615\) −14.5551 −0.586916
\(616\) 0 0
\(617\) 10.1693 + 17.6138i 0.409401 + 0.709103i 0.994823 0.101626i \(-0.0324044\pi\)
−0.585422 + 0.810729i \(0.699071\pi\)
\(618\) 0 0
\(619\) 8.04168 0.323222 0.161611 0.986855i \(-0.448331\pi\)
0.161611 + 0.986855i \(0.448331\pi\)
\(620\) 0 0
\(621\) 11.8924 + 20.5982i 0.477225 + 0.826579i
\(622\) 0 0
\(623\) 5.09186 + 8.81936i 0.204001 + 0.353340i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −39.1998 66.2514i −1.56549 2.64583i
\(628\) 0 0
\(629\) −19.9475 + 34.5502i −0.795361 + 1.37760i
\(630\) 0 0
\(631\) −17.3979 30.1340i −0.692599 1.19962i −0.970983 0.239147i \(-0.923132\pi\)
0.278385 0.960470i \(-0.410201\pi\)
\(632\) 0 0
\(633\) −10.8115 18.7261i −0.429719 0.744295i
\(634\) 0 0
\(635\) 0.382240 0.0151687
\(636\) 0 0
\(637\) 0.199225 + 0.345068i 0.00789360 + 0.0136721i
\(638\) 0 0
\(639\) −124.618 −4.92981
\(640\) 0 0
\(641\) 13.8100 23.9196i 0.545461 0.944766i −0.453117 0.891451i \(-0.649688\pi\)
0.998578 0.0533147i \(-0.0169786\pi\)
\(642\) 0 0
\(643\) −5.96205 + 10.3266i −0.235120 + 0.407240i −0.959308 0.282363i \(-0.908882\pi\)
0.724187 + 0.689603i \(0.242215\pi\)
\(644\) 0 0
\(645\) 37.5113 1.47701
\(646\) 0 0
\(647\) −25.5124 −1.00300 −0.501498 0.865159i \(-0.667218\pi\)
−0.501498 + 0.865159i \(0.667218\pi\)
\(648\) 0 0
\(649\) −16.9209 + 29.3078i −0.664203 + 1.15043i
\(650\) 0 0
\(651\) −1.11417 + 1.92980i −0.0436678 + 0.0756349i
\(652\) 0 0
\(653\) 18.5410 0.725565 0.362782 0.931874i \(-0.381827\pi\)
0.362782 + 0.931874i \(0.381827\pi\)
\(654\) 0 0
\(655\) −1.76406 3.05544i −0.0689276 0.119386i
\(656\) 0 0
\(657\) 4.78773 0.186787
\(658\) 0 0
\(659\) −4.42424 7.66301i −0.172344 0.298509i 0.766895 0.641773i \(-0.221801\pi\)
−0.939239 + 0.343264i \(0.888467\pi\)
\(660\) 0 0
\(661\) 14.3789 + 24.9050i 0.559275 + 0.968692i 0.997557 + 0.0698549i \(0.0222536\pi\)
−0.438283 + 0.898837i \(0.644413\pi\)
\(662\) 0 0
\(663\) −1.73113 + 2.99841i −0.0672315 + 0.116448i
\(664\) 0 0
\(665\) 5.10340 + 8.62523i 0.197901 + 0.334472i
\(666\) 0 0
\(667\) −4.25815 + 7.37534i −0.164876 + 0.285574i
\(668\) 0 0
\(669\) −21.8076 37.7719i −0.843132 1.46035i
\(670\) 0 0
\(671\) 23.3254 + 40.4007i 0.900466 + 1.55965i
\(672\) 0 0
\(673\) −47.2200 −1.82020 −0.910098 0.414393i \(-0.863994\pi\)
−0.910098 + 0.414393i \(0.863994\pi\)
\(674\) 0 0
\(675\) −9.24562 16.0139i −0.355864 0.616375i
\(676\) 0 0
\(677\) −21.2638 −0.817236 −0.408618 0.912706i \(-0.633989\pi\)
−0.408618 + 0.912706i \(0.633989\pi\)
\(678\) 0 0
\(679\) −4.03774 + 6.99357i −0.154954 + 0.268388i
\(680\) 0 0
\(681\) 32.5855 56.4397i 1.24868 2.16277i
\(682\) 0 0
\(683\) 14.9535 0.572180 0.286090 0.958203i \(-0.407644\pi\)
0.286090 + 0.958203i \(0.407644\pi\)
\(684\) 0 0
\(685\) −1.95134 −0.0745567
\(686\) 0 0
\(687\) 32.7453 56.7166i 1.24931 2.16387i
\(688\) 0 0
\(689\) −0.825260 + 1.42939i −0.0314399 + 0.0544555i
\(690\) 0 0
\(691\) 14.1119 0.536842 0.268421 0.963302i \(-0.413498\pi\)
0.268421 + 0.963302i \(0.413498\pi\)
\(692\) 0 0
\(693\) −50.7437 87.8907i −1.92759 3.33869i
\(694\) 0 0
\(695\) −8.41095 −0.319046
\(696\) 0 0
\(697\) −9.45489 16.3763i −0.358129 0.620298i
\(698\) 0 0
\(699\) −15.6874 27.1714i −0.593352 1.02772i
\(700\) 0 0
\(701\) 10.1936 17.6559i 0.385008 0.666854i −0.606762 0.794884i \(-0.707532\pi\)
0.991770 + 0.128030i \(0.0408652\pi\)
\(702\) 0 0
\(703\) 19.4013 34.4487i 0.731733 1.29926i
\(704\) 0 0
\(705\) −8.72288 + 15.1085i −0.328523 + 0.569018i
\(706\) 0 0
\(707\) −16.4613 28.5119i −0.619092 1.07230i
\(708\) 0 0
\(709\) −13.1360 22.7522i −0.493332 0.854477i 0.506638 0.862159i \(-0.330888\pi\)
−0.999970 + 0.00768201i \(0.997555\pi\)
\(710\) 0 0
\(711\) 36.9492 1.38570
\(712\) 0 0
\(713\) −0.184113 0.318892i −0.00689507 0.0119426i
\(714\) 0 0
\(715\) 1.21285 0.0453578
\(716\) 0 0
\(717\) 5.43370 9.41145i 0.202925 0.351477i
\(718\) 0 0
\(719\) 15.5443 26.9235i 0.579703 1.00408i −0.415810 0.909452i \(-0.636502\pi\)
0.995513 0.0946239i \(-0.0301649\pi\)
\(720\) 0 0
\(721\) 43.8337 1.63245
\(722\) 0 0
\(723\) −8.07725 −0.300396
\(724\) 0 0
\(725\) 3.31046 5.73388i 0.122947 0.212951i
\(726\) 0 0
\(727\) −9.26593 + 16.0491i −0.343654 + 0.595227i −0.985108 0.171935i \(-0.944998\pi\)
0.641454 + 0.767161i \(0.278332\pi\)
\(728\) 0 0
\(729\) 127.118 4.70807
\(730\) 0 0
\(731\) 24.3672 + 42.2051i 0.901252 + 1.56101i
\(732\) 0 0
\(733\) −5.60306 −0.206954 −0.103477 0.994632i \(-0.532997\pi\)
−0.103477 + 0.994632i \(0.532997\pi\)
\(734\) 0 0
\(735\) −2.90094 5.02458i −0.107003 0.185334i
\(736\) 0 0
\(737\) −26.5999 46.0723i −0.979819 1.69710i
\(738\) 0 0
\(739\) −3.45647 + 5.98678i −0.127148 + 0.220227i −0.922571 0.385828i \(-0.873916\pi\)
0.795422 + 0.606056i \(0.207249\pi\)
\(740\) 0 0
\(741\) 1.68372 2.98960i 0.0618531 0.109826i
\(742\) 0 0
\(743\) −13.7812 + 23.8698i −0.505585 + 0.875698i 0.494394 + 0.869238i \(0.335390\pi\)
−0.999979 + 0.00646075i \(0.997943\pi\)
\(744\) 0 0
\(745\) −3.94411 6.83140i −0.144501 0.250283i
\(746\) 0 0
\(747\) −28.2569 48.9425i −1.03387 1.79071i
\(748\) 0 0
\(749\) −1.63854 −0.0598708
\(750\) 0 0
\(751\) −13.5924 23.5428i −0.495995 0.859089i 0.503994 0.863707i \(-0.331863\pi\)
−0.999989 + 0.00461793i \(0.998530\pi\)
\(752\) 0 0
\(753\) 1.74420 0.0635623
\(754\) 0 0
\(755\) −6.15942 + 10.6684i −0.224164 + 0.388264i
\(756\) 0 0
\(757\) −17.8013 + 30.8327i −0.646998 + 1.12063i 0.336838 + 0.941562i \(0.390642\pi\)
−0.983836 + 0.179071i \(0.942691\pi\)
\(758\) 0 0
\(759\) 22.7160 0.824540
\(760\) 0 0
\(761\) −10.3170 −0.373992 −0.186996 0.982361i \(-0.559875\pi\)
−0.186996 + 0.982361i \(0.559875\pi\)
\(762\) 0 0
\(763\) −7.98791 + 13.8355i −0.289182 + 0.500878i
\(764\) 0 0
\(765\) 18.6095 32.2326i 0.672828 1.16537i
\(766\) 0 0
\(767\) −1.50839 −0.0544648
\(768\) 0 0
\(769\) −4.57335 7.92127i −0.164919 0.285648i 0.771707 0.635978i \(-0.219403\pi\)
−0.936627 + 0.350329i \(0.886070\pi\)
\(770\) 0 0
\(771\) 26.2293 0.944625
\(772\) 0 0
\(773\) 18.3662 + 31.8112i 0.660587 + 1.14417i 0.980462 + 0.196710i \(0.0630258\pi\)
−0.319875 + 0.947460i \(0.603641\pi\)
\(774\) 0 0
\(775\) 0.143136 + 0.247920i 0.00514161 + 0.00890554i
\(776\) 0 0
\(777\) 35.3014 61.1438i 1.26643 2.19352i
\(778\) 0 0
\(779\) 9.54270 + 16.1281i 0.341903 + 0.577848i
\(780\) 0 0
\(781\) −38.4112 + 66.5302i −1.37446 + 2.38064i
\(782\) 0 0
\(783\) 61.2145 + 106.027i 2.18763 + 3.78908i
\(784\) 0 0
\(785\) −5.85061 10.1335i −0.208817 0.361682i
\(786\) 0 0
\(787\) 44.3784 1.58192 0.790959 0.611870i \(-0.209582\pi\)
0.790959 + 0.611870i \(0.209582\pi\)
\(788\) 0 0
\(789\) 53.2194 + 92.1787i 1.89466 + 3.28165i
\(790\) 0 0
\(791\) −39.3448 −1.39894
\(792\) 0 0
\(793\) −1.03965 + 1.80074i −0.0369192 + 0.0639460i
\(794\) 0 0
\(795\) 12.0167 20.8135i 0.426188 0.738180i
\(796\) 0 0
\(797\) −2.98322 −0.105671 −0.0528355 0.998603i \(-0.516826\pi\)
−0.0528355 + 0.998603i \(0.516826\pi\)
\(798\) 0 0
\(799\) −22.6653 −0.801842
\(800\) 0 0
\(801\) −18.7399 + 32.4584i −0.662140 + 1.14686i
\(802\) 0 0
\(803\) 1.47573 2.55604i 0.0520774 0.0902007i
\(804\) 0 0
\(805\) −2.95739 −0.104234
\(806\) 0 0
\(807\) 20.8096 + 36.0433i 0.732534 + 1.26879i
\(808\) 0 0
\(809\) 51.5319 1.81176 0.905882 0.423530i \(-0.139209\pi\)
0.905882 + 0.423530i \(0.139209\pi\)
\(810\) 0 0
\(811\) −9.15679 15.8600i −0.321538 0.556921i 0.659267 0.751909i \(-0.270867\pi\)
−0.980806 + 0.194988i \(0.937533\pi\)
\(812\) 0 0
\(813\) 20.8241 + 36.0684i 0.730333 + 1.26497i
\(814\) 0 0
\(815\) 1.49484 2.58914i 0.0523620 0.0906937i
\(816\) 0 0
\(817\) −24.5935 41.5653i −0.860416 1.45419i
\(818\) 0 0
\(819\) 2.26174 3.91745i 0.0790316 0.136887i
\(820\) 0 0
\(821\) −6.02937 10.4432i −0.210426 0.364469i 0.741422 0.671039i \(-0.234152\pi\)
−0.951848 + 0.306570i \(0.900819\pi\)
\(822\) 0 0
\(823\) 20.8606 + 36.1316i 0.727154 + 1.25947i 0.958081 + 0.286497i \(0.0924907\pi\)
−0.230927 + 0.972971i \(0.574176\pi\)
\(824\) 0 0
\(825\) −17.6604 −0.614855
\(826\) 0 0
\(827\) −24.3614 42.1952i −0.847129 1.46727i −0.883760 0.467941i \(-0.844996\pi\)
0.0366312 0.999329i \(-0.488337\pi\)
\(828\) 0 0
\(829\) 7.83226 0.272026 0.136013 0.990707i \(-0.456571\pi\)
0.136013 + 0.990707i \(0.456571\pi\)
\(830\) 0 0
\(831\) −27.7299 + 48.0296i −0.961940 + 1.66613i
\(832\) 0 0
\(833\) 3.76887 6.52788i 0.130584 0.226178i
\(834\) 0 0
\(835\) 3.36258 0.116367
\(836\) 0 0
\(837\) −5.29354 −0.182972
\(838\) 0 0
\(839\) −21.7628 + 37.6943i −0.751336 + 1.30135i 0.195839 + 0.980636i \(0.437257\pi\)
−0.947175 + 0.320717i \(0.896076\pi\)
\(840\) 0 0
\(841\) −7.41828 + 12.8488i −0.255803 + 0.443063i
\(842\) 0 0
\(843\) −19.0465 −0.655997
\(844\) 0 0
\(845\) −6.47297 11.2115i −0.222677 0.385688i
\(846\) 0 0
\(847\) −37.2722 −1.28069
\(848\) 0 0
\(849\) 41.5660 + 71.9944i 1.42654 + 2.47084i
\(850\) 0 0
\(851\) 5.83341 + 10.1038i 0.199967 + 0.346353i
\(852\) 0 0
\(853\) 1.44597 2.50449i 0.0495090 0.0857521i −0.840209 0.542263i \(-0.817568\pi\)
0.889718 + 0.456511i \(0.150901\pi\)
\(854\) 0 0
\(855\) −18.0999 + 32.1379i −0.619003 + 1.09909i
\(856\) 0 0
\(857\) −5.32599 + 9.22488i −0.181932 + 0.315116i −0.942539 0.334098i \(-0.891569\pi\)
0.760606 + 0.649214i \(0.224902\pi\)
\(858\) 0 0
\(859\) −1.55176 2.68773i −0.0529455 0.0917043i 0.838338 0.545151i \(-0.183528\pi\)
−0.891283 + 0.453447i \(0.850194\pi\)
\(860\) 0 0
\(861\) 16.7324 + 28.9814i 0.570239 + 0.987683i
\(862\) 0 0
\(863\) −16.4219 −0.559007 −0.279503 0.960145i \(-0.590170\pi\)
−0.279503 + 0.960145i \(0.590170\pi\)
\(864\) 0 0
\(865\) −0.926712 1.60511i −0.0315091 0.0545754i
\(866\) 0 0
\(867\) 7.94371 0.269783
\(868\) 0 0
\(869\) 11.3889 19.7262i 0.386343 0.669165i
\(870\) 0 0
\(871\) 1.18561 2.05353i 0.0401727 0.0695812i
\(872\) 0 0
\(873\) −29.7206 −1.00589
\(874\) 0 0
\(875\) 2.29919 0.0777268
\(876\) 0 0
\(877\) −4.29300 + 7.43570i −0.144964 + 0.251086i −0.929360 0.369176i \(-0.879640\pi\)
0.784395 + 0.620261i \(0.212973\pi\)
\(878\) 0 0
\(879\) −0.741353 + 1.28406i −0.0250052 + 0.0433103i
\(880\) 0 0
\(881\) −24.3052 −0.818862 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(882\) 0 0
\(883\) −9.59054 16.6113i −0.322747 0.559015i 0.658307 0.752750i \(-0.271273\pi\)
−0.981054 + 0.193735i \(0.937940\pi\)
\(884\) 0 0
\(885\) 21.9638 0.738306
\(886\) 0 0
\(887\) −0.691245 1.19727i −0.0232097 0.0402004i 0.854187 0.519966i \(-0.174055\pi\)
−0.877397 + 0.479765i \(0.840722\pi\)
\(888\) 0 0
\(889\) −0.439421 0.761099i −0.0147377 0.0255264i
\(890\) 0 0
\(891\) 97.0703 168.131i 3.25198 5.63259i
\(892\) 0 0
\(893\) 22.4602 0.239946i 0.751604 0.00802949i
\(894\) 0 0
\(895\) 9.47029 16.4030i 0.316557 0.548293i
\(896\) 0 0
\(897\) 0.506248 + 0.876847i 0.0169031 + 0.0292771i
\(898\) 0 0
\(899\) −0.947695 1.64146i −0.0316074 0.0547456i
\(900\) 0 0
\(901\) 31.2239 1.04022
\(902\) 0 0
\(903\) −43.1228 74.6909i −1.43504 2.48556i
\(904\) 0 0
\(905\) −12.5457 −0.417033
\(906\) 0 0
\(907\) 5.44382 9.42897i 0.180759 0.313084i −0.761380 0.648306i \(-0.775478\pi\)
0.942139 + 0.335222i \(0.108811\pi\)
\(908\) 0 0
\(909\) 60.5836 104.934i 2.00943 3.48044i
\(910\) 0 0
\(911\) −43.9872 −1.45736 −0.728681 0.684854i \(-0.759866\pi\)
−0.728681 + 0.684854i \(0.759866\pi\)
\(912\) 0 0
\(913\) −34.8388 −1.15299
\(914\) 0 0
\(915\) 15.1385 26.2207i 0.500464 0.866829i
\(916\) 0 0
\(917\) −4.05591 + 7.02504i −0.133938 + 0.231987i
\(918\) 0 0
\(919\) 18.1912 0.600074 0.300037 0.953928i \(-0.403001\pi\)
0.300037 + 0.953928i \(0.403001\pi\)
\(920\) 0 0
\(921\) −28.9893 50.2109i −0.955230 1.65451i
\(922\) 0 0
\(923\) −3.42412 −0.112706
\(924\) 0 0
\(925\) −4.53513 7.85507i −0.149114 0.258273i
\(926\) 0 0
\(927\) 80.6619 + 139.711i 2.64928 + 4.58870i
\(928\) 0 0
\(929\) 2.23962 3.87913i 0.0734795 0.127270i −0.826945 0.562283i \(-0.809923\pi\)
0.900424 + 0.435013i \(0.143256\pi\)
\(930\) 0 0
\(931\) −3.66566 + 6.50871i −0.120137 + 0.213314i
\(932\) 0 0
\(933\) −53.5559 + 92.7616i −1.75334 + 3.03688i
\(934\) 0 0
\(935\) −11.4721 19.8702i −0.375177 0.649826i
\(936\) 0 0
\(937\) 17.6774 + 30.6181i 0.577495 + 1.00025i 0.995766 + 0.0919277i \(0.0293029\pi\)
−0.418271 + 0.908322i \(0.637364\pi\)
\(938\) 0 0
\(939\) −52.2438 −1.70491
\(940\) 0 0
\(941\) 7.57022 + 13.1120i 0.246782 + 0.427439i 0.962631 0.270816i \(-0.0872934\pi\)
−0.715849 + 0.698255i \(0.753960\pi\)
\(942\) 0 0
\(943\) −5.52993 −0.180079
\(944\) 0 0
\(945\) −21.2574 + 36.8190i −0.691505 + 1.19772i
\(946\) 0 0
\(947\) −5.86541 + 10.1592i −0.190600 + 0.330129i −0.945449 0.325769i \(-0.894377\pi\)
0.754849 + 0.655898i \(0.227710\pi\)
\(948\) 0 0
\(949\) 0.131552 0.00427036
\(950\) 0 0
\(951\) 28.7108 0.931011
\(952\) 0 0
\(953\) −11.6229 + 20.1315i −0.376503 + 0.652122i −0.990551 0.137146i \(-0.956207\pi\)
0.614048 + 0.789269i \(0.289540\pi\)
\(954\) 0 0
\(955\) −3.34236 + 5.78914i −0.108156 + 0.187332i
\(956\) 0 0
\(957\) 116.928 3.77974
\(958\) 0 0
\(959\) 2.24325 + 3.88542i 0.0724382 + 0.125467i
\(960\) 0 0
\(961\) −30.9180 −0.997356
\(962\) 0 0
\(963\) −3.01520 5.22248i −0.0971634 0.168292i
\(964\) 0 0
\(965\) −4.51808 7.82554i −0.145442 0.251913i
\(966\) 0 0
\(967\) 2.93397 5.08178i 0.0943501 0.163419i −0.814987 0.579479i \(-0.803256\pi\)
0.909337 + 0.416060i \(0.136589\pi\)
\(968\) 0 0
\(969\) −64.9051 + 0.693391i −2.08505 + 0.0222749i
\(970\) 0 0
\(971\) 9.93432 17.2067i 0.318807 0.552191i −0.661432 0.750005i \(-0.730051\pi\)
0.980239 + 0.197814i \(0.0633844\pi\)
\(972\) 0 0
\(973\) 9.66919 + 16.7475i 0.309980 + 0.536901i
\(974\) 0 0
\(975\) −0.393577 0.681696i −0.0126046 0.0218317i
\(976\) 0 0
\(977\) 11.5198 0.368550 0.184275 0.982875i \(-0.441006\pi\)
0.184275 + 0.982875i \(0.441006\pi\)
\(978\) 0 0
\(979\) 11.5524 + 20.0094i 0.369218 + 0.639504i
\(980\) 0 0
\(981\) −58.7967 −1.87724
\(982\) 0 0
\(983\) −17.8709 + 30.9533i −0.569993 + 0.987256i 0.426573 + 0.904453i \(0.359721\pi\)
−0.996566 + 0.0828031i \(0.973613\pi\)
\(984\) 0 0
\(985\) −8.37288 + 14.5023i −0.266782 + 0.462080i
\(986\) 0 0
\(987\) 40.1111 1.27675
\(988\) 0 0
\(989\) 14.2518 0.453179
\(990\) 0 0
\(991\) 23.6565 40.9743i 0.751475 1.30159i −0.195633 0.980677i \(-0.562676\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(992\) 0 0
\(993\) −12.5099 + 21.6678i −0.396990 + 0.687608i
\(994\) 0 0
\(995\) −4.03109 −0.127794
\(996\) 0 0
\(997\) 0.148004 + 0.256350i 0.00468733 + 0.00811869i 0.868360 0.495935i \(-0.165175\pi\)
−0.863672 + 0.504054i \(0.831841\pi\)
\(998\) 0 0
\(999\) 167.720 5.30644
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 1520.2.q.p.961.1 10
4.3 odd 2 760.2.q.g.201.5 yes 10
19.7 even 3 inner 1520.2.q.p.881.1 10
76.7 odd 6 760.2.q.g.121.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.q.g.121.5 10 76.7 odd 6
760.2.q.g.201.5 yes 10 4.3 odd 2
1520.2.q.p.881.1 10 19.7 even 3 inner
1520.2.q.p.961.1 10 1.1 even 1 trivial