Properties

Label 1216.2.i.r.961.3
Level $1216$
Weight $2$
Character 1216.961
Analytic conductor $9.710$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(577,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1216.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.i (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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 28 x^{9} - 121 x^{8} - 238 x^{7} + 392 x^{6} + 2534 x^{5} + 5589 x^{4} + 6426 x^{3} + 4802 x^{2} + 2548 x + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 608)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.3
Root \(-1.15100 - 1.56354i\) of defining polynomial
Character \(\chi\) \(=\) 1216.961
Dual form 1216.2.i.r.577.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 1.50000i) q^{3} +(2.09926 - 3.63603i) q^{5} +4.28918 q^{7} +O(q^{10})\) \(q+(-0.866025 + 1.50000i) q^{3} +(2.09926 - 3.63603i) q^{5} +4.28918 q^{7} +4.71493 q^{11} +(-0.615275 - 1.06569i) q^{13} +(3.63603 + 6.29778i) q^{15} +(-1.61528 + 2.79774i) q^{17} +(-0.625413 - 4.31380i) q^{19} +(-3.71454 + 6.43377i) q^{21} +(1.90398 + 3.29778i) q^{23} +(-6.31380 - 10.9358i) q^{25} -5.19615 q^{27} +(-2.09926 - 3.63603i) q^{29} +0.825076 q^{31} +(-4.08325 + 7.07239i) q^{33} +(9.00411 - 15.5956i) q^{35} -2.96797 q^{37} +2.13138 q^{39} +(2.69852 - 4.67398i) q^{41} +(-1.49144 + 2.58325i) q^{43} +(-0.653150 - 1.13129i) q^{47} +11.3970 q^{49} +(-2.79774 - 4.84583i) q^{51} +(-1.38472 - 2.39841i) q^{53} +(9.89787 - 17.1436i) q^{55} +(7.01232 + 2.79774i) q^{57} +(-7.31300 + 12.6665i) q^{59} +(4.09926 + 7.10013i) q^{61} -5.16650 q^{65} +(1.69110 + 2.92907i) q^{67} -6.59557 q^{69} +(-4.04857 + 7.01232i) q^{71} +(-4.69852 + 8.13808i) q^{73} +21.8716 q^{75} +20.2232 q^{77} +(2.31652 - 4.01232i) q^{79} +(4.50000 - 7.79423i) q^{81} -7.32753 q^{83} +(6.78177 + 11.7464i) q^{85} +7.27206 q^{87} +(-5.61528 - 9.72594i) q^{89} +(-2.63903 - 4.57093i) q^{91} +(-0.714537 + 1.23761i) q^{93} +(-16.9980 - 6.78177i) q^{95} +(-5.92907 + 10.2695i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{5} + 2 q^{13} - 10 q^{17} - 12 q^{21} - 20 q^{25} - 2 q^{29} - 12 q^{33} - 8 q^{37} - 14 q^{41} + 44 q^{49} - 26 q^{53} - 18 q^{57} + 26 q^{61} + 12 q^{65} + 60 q^{69} - 10 q^{73} - 8 q^{77} + 54 q^{81} - 2 q^{85} - 58 q^{89} + 24 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −0.866025 + 1.50000i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 2.09926 3.63603i 0.938818 1.62608i 0.171139 0.985247i \(-0.445255\pi\)
0.767679 0.640834i \(-0.221411\pi\)
\(6\) 0 0
\(7\) 4.28918 1.62116 0.810578 0.585630i \(-0.199153\pi\)
0.810578 + 0.585630i \(0.199153\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.71493 1.42160 0.710802 0.703392i \(-0.248332\pi\)
0.710802 + 0.703392i \(0.248332\pi\)
\(12\) 0 0
\(13\) −0.615275 1.06569i −0.170647 0.295569i 0.767999 0.640451i \(-0.221252\pi\)
−0.938646 + 0.344882i \(0.887919\pi\)
\(14\) 0 0
\(15\) 3.63603 + 6.29778i 0.938818 + 1.62608i
\(16\) 0 0
\(17\) −1.61528 + 2.79774i −0.391762 + 0.678551i −0.992682 0.120758i \(-0.961468\pi\)
0.600920 + 0.799309i \(0.294801\pi\)
\(18\) 0 0
\(19\) −0.625413 4.31380i −0.143480 0.989653i
\(20\) 0 0
\(21\) −3.71454 + 6.43377i −0.810578 + 1.40396i
\(22\) 0 0
\(23\) 1.90398 + 3.29778i 0.397007 + 0.687636i 0.993355 0.115090i \(-0.0367157\pi\)
−0.596348 + 0.802726i \(0.703382\pi\)
\(24\) 0 0
\(25\) −6.31380 10.9358i −1.26276 2.18716i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) −2.09926 3.63603i −0.389823 0.675193i 0.602602 0.798042i \(-0.294131\pi\)
−0.992425 + 0.122848i \(0.960797\pi\)
\(30\) 0 0
\(31\) 0.825076 0.148188 0.0740940 0.997251i \(-0.476394\pi\)
0.0740940 + 0.997251i \(0.476394\pi\)
\(32\) 0 0
\(33\) −4.08325 + 7.07239i −0.710802 + 1.23115i
\(34\) 0 0
\(35\) 9.00411 15.5956i 1.52197 2.63613i
\(36\) 0 0
\(37\) −2.96797 −0.487932 −0.243966 0.969784i \(-0.578448\pi\)
−0.243966 + 0.969784i \(0.578448\pi\)
\(38\) 0 0
\(39\) 2.13138 0.341293
\(40\) 0 0
\(41\) 2.69852 4.67398i 0.421439 0.729953i −0.574642 0.818405i \(-0.694859\pi\)
0.996080 + 0.0884520i \(0.0281920\pi\)
\(42\) 0 0
\(43\) −1.49144 + 2.58325i −0.227442 + 0.393942i −0.957049 0.289925i \(-0.906370\pi\)
0.729607 + 0.683867i \(0.239703\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.653150 1.13129i −0.0952717 0.165015i 0.814450 0.580233i \(-0.197039\pi\)
−0.909722 + 0.415218i \(0.863705\pi\)
\(48\) 0 0
\(49\) 11.3970 1.62815
\(50\) 0 0
\(51\) −2.79774 4.84583i −0.391762 0.678551i
\(52\) 0 0
\(53\) −1.38472 2.39841i −0.190207 0.329447i 0.755112 0.655596i \(-0.227582\pi\)
−0.945319 + 0.326148i \(0.894249\pi\)
\(54\) 0 0
\(55\) 9.89787 17.1436i 1.33463 2.31164i
\(56\) 0 0
\(57\) 7.01232 + 2.79774i 0.928805 + 0.370570i
\(58\) 0 0
\(59\) −7.31300 + 12.6665i −0.952072 + 1.64904i −0.211142 + 0.977455i \(0.567718\pi\)
−0.740930 + 0.671582i \(0.765615\pi\)
\(60\) 0 0
\(61\) 4.09926 + 7.10013i 0.524857 + 0.909078i 0.999581 + 0.0289439i \(0.00921441\pi\)
−0.474724 + 0.880135i \(0.657452\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.16650 −0.640825
\(66\) 0 0
\(67\) 1.69110 + 2.92907i 0.206601 + 0.357843i 0.950642 0.310291i \(-0.100426\pi\)
−0.744041 + 0.668134i \(0.767093\pi\)
\(68\) 0 0
\(69\) −6.59557 −0.794013
\(70\) 0 0
\(71\) −4.04857 + 7.01232i −0.480476 + 0.832209i −0.999749 0.0223991i \(-0.992870\pi\)
0.519273 + 0.854609i \(0.326203\pi\)
\(72\) 0 0
\(73\) −4.69852 + 8.13808i −0.549921 + 0.952490i 0.448359 + 0.893854i \(0.352009\pi\)
−0.998279 + 0.0586367i \(0.981325\pi\)
\(74\) 0 0
\(75\) 21.8716 2.52552
\(76\) 0 0
\(77\) 20.2232 2.30464
\(78\) 0 0
\(79\) 2.31652 4.01232i 0.260628 0.451421i −0.705781 0.708430i \(-0.749404\pi\)
0.966409 + 0.257009i \(0.0827370\pi\)
\(80\) 0 0
\(81\) 4.50000 7.79423i 0.500000 0.866025i
\(82\) 0 0
\(83\) −7.32753 −0.804301 −0.402150 0.915574i \(-0.631737\pi\)
−0.402150 + 0.915574i \(0.631737\pi\)
\(84\) 0 0
\(85\) 6.78177 + 11.7464i 0.735586 + 1.27407i
\(86\) 0 0
\(87\) 7.27206 0.779646
\(88\) 0 0
\(89\) −5.61528 9.72594i −0.595218 1.03095i −0.993516 0.113692i \(-0.963732\pi\)
0.398298 0.917256i \(-0.369601\pi\)
\(90\) 0 0
\(91\) −2.63903 4.57093i −0.276645 0.479163i
\(92\) 0 0
\(93\) −0.714537 + 1.23761i −0.0740940 + 0.128335i
\(94\) 0 0
\(95\) −16.9980 6.78177i −1.74396 0.695795i
\(96\) 0 0
\(97\) −5.92907 + 10.2695i −0.602006 + 1.04271i 0.390511 + 0.920598i \(0.372298\pi\)
−0.992517 + 0.122107i \(0.961035\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.868711 + 1.50465i 0.0864400 + 0.149718i 0.906004 0.423269i \(-0.139118\pi\)
−0.819564 + 0.572988i \(0.805784\pi\)
\(102\) 0 0
\(103\) 9.42986 0.929151 0.464576 0.885533i \(-0.346207\pi\)
0.464576 + 0.885533i \(0.346207\pi\)
\(104\) 0 0
\(105\) 15.5956 + 27.0123i 1.52197 + 2.63613i
\(106\) 0 0
\(107\) 15.5066 1.49908 0.749538 0.661962i \(-0.230276\pi\)
0.749538 + 0.661962i \(0.230276\pi\)
\(108\) 0 0
\(109\) 10.0123 17.3419i 0.959006 1.66105i 0.234085 0.972216i \(-0.424790\pi\)
0.724921 0.688832i \(-0.241876\pi\)
\(110\) 0 0
\(111\) 2.57034 4.45196i 0.243966 0.422561i
\(112\) 0 0
\(113\) 2.23055 0.209833 0.104916 0.994481i \(-0.466543\pi\)
0.104916 + 0.994481i \(0.466543\pi\)
\(114\) 0 0
\(115\) 15.9878 1.49087
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.92820 + 12.0000i −0.635107 + 1.10004i
\(120\) 0 0
\(121\) 11.2306 1.02096
\(122\) 0 0
\(123\) 4.67398 + 8.09557i 0.421439 + 0.729953i
\(124\) 0 0
\(125\) −32.0246 −2.86437
\(126\) 0 0
\(127\) 2.39841 + 4.15417i 0.212825 + 0.368623i 0.952597 0.304234i \(-0.0984003\pi\)
−0.739773 + 0.672857i \(0.765067\pi\)
\(128\) 0 0
\(129\) −2.58325 4.47432i −0.227442 0.393942i
\(130\) 0 0
\(131\) −3.50505 + 6.07093i −0.306238 + 0.530419i −0.977536 0.210768i \(-0.932404\pi\)
0.671298 + 0.741187i \(0.265737\pi\)
\(132\) 0 0
\(133\) −2.68251 18.5026i −0.232603 1.60438i
\(134\) 0 0
\(135\) −10.9081 + 18.8934i −0.938818 + 1.62608i
\(136\) 0 0
\(137\) −8.89705 15.4101i −0.760126 1.31658i −0.942785 0.333401i \(-0.891804\pi\)
0.182659 0.983176i \(-0.441529\pi\)
\(138\) 0 0
\(139\) 7.71233 + 13.3581i 0.654151 + 1.13302i 0.982106 + 0.188330i \(0.0603074\pi\)
−0.327955 + 0.944694i \(0.606359\pi\)
\(140\) 0 0
\(141\) 2.26258 0.190543
\(142\) 0 0
\(143\) −2.90098 5.02464i −0.242592 0.420182i
\(144\) 0 0
\(145\) −17.6276 −1.46389
\(146\) 0 0
\(147\) −9.87013 + 17.0956i −0.814075 + 1.41002i
\(148\) 0 0
\(149\) 1.09926 1.90398i 0.0900550 0.155980i −0.817479 0.575958i \(-0.804629\pi\)
0.907534 + 0.419978i \(0.137962\pi\)
\(150\) 0 0
\(151\) 13.7190 1.11644 0.558220 0.829693i \(-0.311485\pi\)
0.558220 + 0.829693i \(0.311485\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.73205 3.00000i 0.139122 0.240966i
\(156\) 0 0
\(157\) 4.78177 8.28227i 0.381627 0.660997i −0.609668 0.792657i \(-0.708697\pi\)
0.991295 + 0.131660i \(0.0420306\pi\)
\(158\) 0 0
\(159\) 4.79683 0.380413
\(160\) 0 0
\(161\) 8.16650 + 14.1448i 0.643610 + 1.11477i
\(162\) 0 0
\(163\) 0.563122 0.0441071 0.0220536 0.999757i \(-0.492980\pi\)
0.0220536 + 0.999757i \(0.492980\pi\)
\(164\) 0 0
\(165\) 17.1436 + 29.6936i 1.33463 + 2.31164i
\(166\) 0 0
\(167\) −3.64924 6.32067i −0.282387 0.489108i 0.689585 0.724204i \(-0.257793\pi\)
−0.971972 + 0.235096i \(0.924459\pi\)
\(168\) 0 0
\(169\) 5.74287 9.94695i 0.441759 0.765150i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.78177 + 16.9425i −0.743694 + 1.28812i 0.207108 + 0.978318i \(0.433595\pi\)
−0.950802 + 0.309798i \(0.899739\pi\)
\(174\) 0 0
\(175\) −27.0810 46.9057i −2.04713 3.54574i
\(176\) 0 0
\(177\) −12.6665 21.9390i −0.952072 1.64904i
\(178\) 0 0
\(179\) −17.7198 −1.32444 −0.662221 0.749308i \(-0.730386\pi\)
−0.662221 + 0.749308i \(0.730386\pi\)
\(180\) 0 0
\(181\) 9.29778 + 16.1042i 0.691099 + 1.19702i 0.971478 + 0.237129i \(0.0762064\pi\)
−0.280379 + 0.959889i \(0.590460\pi\)
\(182\) 0 0
\(183\) −14.2003 −1.04971
\(184\) 0 0
\(185\) −6.23055 + 10.7916i −0.458079 + 0.793416i
\(186\) 0 0
\(187\) −7.61591 + 13.1911i −0.556930 + 0.964632i
\(188\) 0 0
\(189\) −22.2872 −1.62116
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 2.01232 3.48544i 0.144850 0.250888i −0.784467 0.620171i \(-0.787063\pi\)
0.929317 + 0.369283i \(0.120397\pi\)
\(194\) 0 0
\(195\) 4.47432 7.74974i 0.320412 0.554971i
\(196\) 0 0
\(197\) −2.96797 −0.211459 −0.105730 0.994395i \(-0.533718\pi\)
−0.105730 + 0.994395i \(0.533718\pi\)
\(198\) 0 0
\(199\) −7.51267 13.0123i −0.532559 0.922419i −0.999277 0.0380130i \(-0.987897\pi\)
0.466718 0.884406i \(-0.345436\pi\)
\(200\) 0 0
\(201\) −5.85815 −0.413202
\(202\) 0 0
\(203\) −9.00411 15.5956i −0.631964 1.09459i
\(204\) 0 0
\(205\) −11.3298 19.6238i −0.791308 1.37059i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.94878 20.3393i −0.203971 1.40690i
\(210\) 0 0
\(211\) −1.54691 + 2.67933i −0.106494 + 0.184453i −0.914348 0.404930i \(-0.867296\pi\)
0.807854 + 0.589383i \(0.200629\pi\)
\(212\) 0 0
\(213\) −7.01232 12.1457i −0.480476 0.832209i
\(214\) 0 0
\(215\) 6.26184 + 10.8458i 0.427054 + 0.739679i
\(216\) 0 0
\(217\) 3.53890 0.240236
\(218\) 0 0
\(219\) −8.13808 14.0956i −0.549921 0.952490i
\(220\) 0 0
\(221\) 3.97536 0.267411
\(222\) 0 0
\(223\) −3.21028 + 5.56036i −0.214976 + 0.372349i −0.953265 0.302135i \(-0.902301\pi\)
0.738289 + 0.674484i \(0.235634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.8716 −1.45167 −0.725836 0.687868i \(-0.758547\pi\)
−0.725836 + 0.687868i \(0.758547\pi\)
\(228\) 0 0
\(229\) −16.9222 −1.11825 −0.559125 0.829083i \(-0.688863\pi\)
−0.559125 + 0.829083i \(0.688863\pi\)
\(230\) 0 0
\(231\) −17.5138 + 30.3348i −1.15232 + 1.99588i
\(232\) 0 0
\(233\) 2.07093 3.58695i 0.135671 0.234989i −0.790183 0.612871i \(-0.790014\pi\)
0.925854 + 0.377883i \(0.123348\pi\)
\(234\) 0 0
\(235\) −5.48453 −0.357771
\(236\) 0 0
\(237\) 4.01232 + 6.94955i 0.260628 + 0.451421i
\(238\) 0 0
\(239\) −4.26275 −0.275735 −0.137867 0.990451i \(-0.544025\pi\)
−0.137867 + 0.990451i \(0.544025\pi\)
\(240\) 0 0
\(241\) 6.12760 + 10.6133i 0.394713 + 0.683663i 0.993065 0.117571i \(-0.0375106\pi\)
−0.598351 + 0.801234i \(0.704177\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 23.9254 41.4400i 1.52854 2.64750i
\(246\) 0 0
\(247\) −4.21236 + 3.32067i −0.268026 + 0.211289i
\(248\) 0 0
\(249\) 6.34583 10.9913i 0.402150 0.696545i
\(250\) 0 0
\(251\) −2.51618 4.35815i −0.158820 0.275084i 0.775624 0.631196i \(-0.217436\pi\)
−0.934443 + 0.356112i \(0.884102\pi\)
\(252\) 0 0
\(253\) 8.97712 + 15.5488i 0.564386 + 0.977546i
\(254\) 0 0
\(255\) −23.4927 −1.47117
\(256\) 0 0
\(257\) 3.30148 + 5.71833i 0.205940 + 0.356699i 0.950432 0.310933i \(-0.100641\pi\)
−0.744492 + 0.667632i \(0.767308\pi\)
\(258\) 0 0
\(259\) −12.7302 −0.791014
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.58135 13.1313i 0.467486 0.809710i −0.531824 0.846855i \(-0.678493\pi\)
0.999310 + 0.0371452i \(0.0118264\pi\)
\(264\) 0 0
\(265\) −11.6276 −0.714278
\(266\) 0 0
\(267\) 19.4519 1.19044
\(268\) 0 0
\(269\) 6.01232 10.4136i 0.366578 0.634931i −0.622450 0.782659i \(-0.713863\pi\)
0.989028 + 0.147728i \(0.0471960\pi\)
\(270\) 0 0
\(271\) −10.5642 + 18.2978i −0.641731 + 1.11151i 0.343315 + 0.939220i \(0.388450\pi\)
−0.985046 + 0.172291i \(0.944883\pi\)
\(272\) 0 0
\(273\) 9.14185 0.553290
\(274\) 0 0
\(275\) −29.7691 51.5616i −1.79514 3.10928i
\(276\) 0 0
\(277\) −16.5709 −0.995650 −0.497825 0.867277i \(-0.665868\pi\)
−0.497825 + 0.867277i \(0.665868\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.69852 + 9.87013i 0.339945 + 0.588803i 0.984422 0.175821i \(-0.0562581\pi\)
−0.644477 + 0.764624i \(0.722925\pi\)
\(282\) 0 0
\(283\) −0.866025 + 1.50000i −0.0514799 + 0.0891657i −0.890617 0.454754i \(-0.849727\pi\)
0.839137 + 0.543920i \(0.183060\pi\)
\(284\) 0 0
\(285\) 24.8934 19.6238i 1.47456 1.16241i
\(286\) 0 0
\(287\) 11.5744 20.0475i 0.683218 1.18337i
\(288\) 0 0
\(289\) 3.28177 + 5.68419i 0.193045 + 0.334364i
\(290\) 0 0
\(291\) −10.2695 17.7872i −0.602006 1.04271i
\(292\) 0 0
\(293\) 4.44282 0.259552 0.129776 0.991543i \(-0.458574\pi\)
0.129776 + 0.991543i \(0.458574\pi\)
\(294\) 0 0
\(295\) 30.7038 + 53.1806i 1.78765 + 3.09629i
\(296\) 0 0
\(297\) −24.4995 −1.42160
\(298\) 0 0
\(299\) 2.34294 4.05809i 0.135496 0.234685i
\(300\) 0 0
\(301\) −6.39705 + 11.0800i −0.368720 + 0.638641i
\(302\) 0 0
\(303\) −3.00930 −0.172880
\(304\) 0 0
\(305\) 34.4217 1.97098
\(306\) 0 0
\(307\) −15.4656 + 26.7872i −0.882669 + 1.52883i −0.0343067 + 0.999411i \(0.510922\pi\)
−0.848362 + 0.529416i \(0.822411\pi\)
\(308\) 0 0
\(309\) −8.16650 + 14.1448i −0.464576 + 0.804669i
\(310\) 0 0
\(311\) −29.9133 −1.69623 −0.848114 0.529814i \(-0.822262\pi\)
−0.848114 + 0.529814i \(0.822262\pi\)
\(312\) 0 0
\(313\) 3.15962 + 5.47263i 0.178593 + 0.309331i 0.941399 0.337296i \(-0.109512\pi\)
−0.762806 + 0.646627i \(0.776179\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.38472 + 12.7907i 0.414767 + 0.718398i 0.995404 0.0957646i \(-0.0305296\pi\)
−0.580637 + 0.814163i \(0.697196\pi\)
\(318\) 0 0
\(319\) −9.89787 17.1436i −0.554174 0.959858i
\(320\) 0 0
\(321\) −13.4291 + 23.2598i −0.749538 + 1.29824i
\(322\) 0 0
\(323\) 13.0791 + 5.21823i 0.727740 + 0.290350i
\(324\) 0 0
\(325\) −7.76945 + 13.4571i −0.430972 + 0.746465i
\(326\) 0 0
\(327\) 17.3419 + 30.0370i 0.959006 + 1.66105i
\(328\) 0 0
\(329\) −2.80148 4.85230i −0.154450 0.267516i
\(330\) 0 0
\(331\) −0.563122 −0.0309520 −0.0154760 0.999880i \(-0.504926\pi\)
−0.0154760 + 0.999880i \(0.504926\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14.2003 0.775843
\(336\) 0 0
\(337\) −1.30148 + 2.25422i −0.0708960 + 0.122795i −0.899294 0.437344i \(-0.855919\pi\)
0.828398 + 0.560140i \(0.189252\pi\)
\(338\) 0 0
\(339\) −1.93171 + 3.34583i −0.104916 + 0.181720i
\(340\) 0 0
\(341\) 3.89018 0.210665
\(342\) 0 0
\(343\) 18.8597 1.01833
\(344\) 0 0
\(345\) −13.8458 + 23.9817i −0.745434 + 1.29113i
\(346\) 0 0
\(347\) −11.6022 + 20.0956i −0.622838 + 1.07879i 0.366117 + 0.930569i \(0.380687\pi\)
−0.988955 + 0.148218i \(0.952646\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 3.19706 + 5.53748i 0.170647 + 0.295569i
\(352\) 0 0
\(353\) 4.29461 0.228579 0.114289 0.993447i \(-0.463541\pi\)
0.114289 + 0.993447i \(0.463541\pi\)
\(354\) 0 0
\(355\) 16.9980 + 29.4414i 0.902160 + 1.56259i
\(356\) 0 0
\(357\) −12.0000 20.7846i −0.635107 1.10004i
\(358\) 0 0
\(359\) −0.571252 + 0.989437i −0.0301495 + 0.0522205i −0.880706 0.473662i \(-0.842932\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(360\) 0 0
\(361\) −18.2177 + 5.39582i −0.958827 + 0.283990i
\(362\) 0 0
\(363\) −9.72594 + 16.8458i −0.510480 + 0.884177i
\(364\) 0 0
\(365\) 19.7269 + 34.1679i 1.03255 + 1.78843i
\(366\) 0 0
\(367\) 4.06178 + 7.03521i 0.212023 + 0.367235i 0.952348 0.305015i \(-0.0986615\pi\)
−0.740324 + 0.672250i \(0.765328\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.93933 10.2872i −0.308355 0.534086i
\(372\) 0 0
\(373\) 18.4611 0.955880 0.477940 0.878393i \(-0.341384\pi\)
0.477940 + 0.878393i \(0.341384\pi\)
\(374\) 0 0
\(375\) 27.7342 48.0370i 1.43219 2.48062i
\(376\) 0 0
\(377\) −2.58325 + 4.47432i −0.133044 + 0.230439i
\(378\) 0 0
\(379\) −19.6584 −1.00978 −0.504891 0.863183i \(-0.668467\pi\)
−0.504891 + 0.863183i \(0.668467\pi\)
\(380\) 0 0
\(381\) −8.30835 −0.425650
\(382\) 0 0
\(383\) −11.0455 + 19.1313i −0.564396 + 0.977563i 0.432709 + 0.901534i \(0.357558\pi\)
−0.997106 + 0.0760296i \(0.975776\pi\)
\(384\) 0 0
\(385\) 42.4537 73.5320i 2.16364 3.74754i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.55122 16.5432i −0.484266 0.838773i 0.515571 0.856847i \(-0.327580\pi\)
−0.999837 + 0.0180736i \(0.994247\pi\)
\(390\) 0 0
\(391\) −12.3018 −0.622128
\(392\) 0 0
\(393\) −6.07093 10.5152i −0.306238 0.530419i
\(394\) 0 0
\(395\) −9.72594 16.8458i −0.489365 0.847605i
\(396\) 0 0
\(397\) −5.49631 + 9.51988i −0.275852 + 0.477789i −0.970350 0.241705i \(-0.922293\pi\)
0.694498 + 0.719495i \(0.255627\pi\)
\(398\) 0 0
\(399\) 30.0771 + 12.0000i 1.50574 + 0.600751i
\(400\) 0 0
\(401\) −18.8330 + 32.6197i −0.940475 + 1.62895i −0.175907 + 0.984407i \(0.556286\pi\)
−0.764568 + 0.644543i \(0.777048\pi\)
\(402\) 0 0
\(403\) −0.507649 0.879274i −0.0252878 0.0437998i
\(404\) 0 0
\(405\) −18.8934 32.7242i −0.938818 1.62608i
\(406\) 0 0
\(407\) −13.9938 −0.693646
\(408\) 0 0
\(409\) 4.50000 + 7.79423i 0.222511 + 0.385400i 0.955570 0.294765i \(-0.0952414\pi\)
−0.733059 + 0.680165i \(0.761908\pi\)
\(410\) 0 0
\(411\) 30.8203 1.52025
\(412\) 0 0
\(413\) −31.3668 + 54.3289i −1.54346 + 2.67335i
\(414\) 0 0
\(415\) −15.3824 + 26.6431i −0.755092 + 1.30786i
\(416\) 0 0
\(417\) −26.7163 −1.30830
\(418\) 0 0
\(419\) 8.63121 0.421662 0.210831 0.977523i \(-0.432383\pi\)
0.210831 + 0.977523i \(0.432383\pi\)
\(420\) 0 0
\(421\) 14.2658 24.7090i 0.695270 1.20424i −0.274819 0.961496i \(-0.588618\pi\)
0.970089 0.242748i \(-0.0780487\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 40.7941 1.97880
\(426\) 0 0
\(427\) 17.5825 + 30.4537i 0.850875 + 1.47376i
\(428\) 0 0
\(429\) 10.0493 0.485184
\(430\) 0 0
\(431\) −7.03144 12.1788i −0.338693 0.586633i 0.645494 0.763765i \(-0.276651\pi\)
−0.984187 + 0.177132i \(0.943318\pi\)
\(432\) 0 0
\(433\) −8.27490 14.3325i −0.397666 0.688778i 0.595771 0.803154i \(-0.296846\pi\)
−0.993438 + 0.114376i \(0.963513\pi\)
\(434\) 0 0
\(435\) 15.2659 26.4414i 0.731946 1.26777i
\(436\) 0 0
\(437\) 13.0352 10.2759i 0.623558 0.491561i
\(438\) 0 0
\(439\) 10.8262 18.7515i 0.516706 0.894960i −0.483106 0.875562i \(-0.660492\pi\)
0.999812 0.0193986i \(-0.00617516\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.9460 20.6911i −0.567573 0.983066i −0.996805 0.0798714i \(-0.974549\pi\)
0.429232 0.903194i \(-0.358784\pi\)
\(444\) 0 0
\(445\) −47.1517 −2.23521
\(446\) 0 0
\(447\) 1.90398 + 3.29778i 0.0900550 + 0.155980i
\(448\) 0 0
\(449\) 12.9606 0.611648 0.305824 0.952088i \(-0.401068\pi\)
0.305824 + 0.952088i \(0.401068\pi\)
\(450\) 0 0
\(451\) 12.7233 22.0375i 0.599119 1.03770i
\(452\) 0 0
\(453\) −11.8810 + 20.5786i −0.558220 + 0.966865i
\(454\) 0 0
\(455\) −22.1600 −1.03888
\(456\) 0 0
\(457\) 5.70539 0.266887 0.133444 0.991056i \(-0.457397\pi\)
0.133444 + 0.991056i \(0.457397\pi\)
\(458\) 0 0
\(459\) 8.39322 14.5375i 0.391762 0.678551i
\(460\) 0 0
\(461\) −2.61528 + 4.52979i −0.121806 + 0.210973i −0.920480 0.390790i \(-0.872202\pi\)
0.798674 + 0.601764i \(0.205535\pi\)
\(462\) 0 0
\(463\) 8.46741 0.393514 0.196757 0.980452i \(-0.436959\pi\)
0.196757 + 0.980452i \(0.436959\pi\)
\(464\) 0 0
\(465\) 3.00000 + 5.19615i 0.139122 + 0.240966i
\(466\) 0 0
\(467\) 29.5985 1.36965 0.684827 0.728705i \(-0.259878\pi\)
0.684827 + 0.728705i \(0.259878\pi\)
\(468\) 0 0
\(469\) 7.25344 + 12.5633i 0.334933 + 0.580120i
\(470\) 0 0
\(471\) 8.28227 + 14.3453i 0.381627 + 0.660997i
\(472\) 0 0
\(473\) −7.03203 + 12.1798i −0.323333 + 0.560029i
\(474\) 0 0
\(475\) −43.2262 + 34.0759i −1.98335 + 1.56351i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −11.3761 19.7040i −0.519787 0.900298i −0.999735 0.0230008i \(-0.992678\pi\)
0.479948 0.877297i \(-0.340655\pi\)
\(480\) 0 0
\(481\) 1.82612 + 3.16293i 0.0832639 + 0.144217i
\(482\) 0 0
\(483\) −28.2896 −1.28722
\(484\) 0 0
\(485\) 24.8934 + 43.1166i 1.13035 + 1.95782i
\(486\) 0 0
\(487\) 5.14068 0.232946 0.116473 0.993194i \(-0.462841\pi\)
0.116473 + 0.993194i \(0.462841\pi\)
\(488\) 0 0
\(489\) −0.487678 + 0.844683i −0.0220536 + 0.0381979i
\(490\) 0 0
\(491\) 4.95554 8.58325i 0.223640 0.387357i −0.732270 0.681014i \(-0.761539\pi\)
0.955911 + 0.293658i \(0.0948725\pi\)
\(492\) 0 0
\(493\) 13.5635 0.610871
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −17.3650 + 30.0771i −0.778928 + 1.34914i
\(498\) 0 0
\(499\) 1.69110 2.92907i 0.0757041 0.131123i −0.825688 0.564127i \(-0.809213\pi\)
0.901392 + 0.433003i \(0.142546\pi\)
\(500\) 0 0
\(501\) 12.6413 0.564773
\(502\) 0 0
\(503\) −4.03535 6.98944i −0.179928 0.311644i 0.761928 0.647662i \(-0.224253\pi\)
−0.941856 + 0.336018i \(0.890920\pi\)
\(504\) 0 0
\(505\) 7.29461 0.324606
\(506\) 0 0
\(507\) 9.94695 + 17.2286i 0.441759 + 0.765150i
\(508\) 0 0
\(509\) −16.7818 29.0669i −0.743839 1.28837i −0.950735 0.310004i \(-0.899670\pi\)
0.206896 0.978363i \(-0.433664\pi\)
\(510\) 0 0
\(511\) −20.1528 + 34.9057i −0.891507 + 1.54414i
\(512\) 0 0
\(513\) 3.24974 + 22.4152i 0.143480 + 0.989653i
\(514\) 0 0
\(515\) 19.7957 34.2872i 0.872304 1.51088i
\(516\) 0 0
\(517\) −3.07956 5.33395i −0.135439 0.234587i
\(518\) 0 0
\(519\) −16.9425 29.3453i −0.743694 1.28812i
\(520\) 0 0
\(521\) 22.1665 0.971132 0.485566 0.874200i \(-0.338614\pi\)
0.485566 + 0.874200i \(0.338614\pi\)
\(522\) 0 0
\(523\) 17.3683 + 30.0827i 0.759462 + 1.31543i 0.943125 + 0.332437i \(0.107871\pi\)
−0.183664 + 0.982989i \(0.558796\pi\)
\(524\) 0 0
\(525\) 93.8114 4.09426
\(526\) 0 0
\(527\) −1.33273 + 2.30835i −0.0580544 + 0.100553i
\(528\) 0 0
\(529\) 4.24974 7.36077i 0.184771 0.320034i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.64134 −0.287668
\(534\) 0 0
\(535\) 32.5523 56.3823i 1.40736 2.43762i
\(536\) 0 0
\(537\) 15.3458 26.5798i 0.662221 1.14700i
\(538\) 0 0
\(539\) 53.7363 2.31458
\(540\) 0 0
\(541\) −17.2429 29.8655i −0.741329 1.28402i −0.951890 0.306439i \(-0.900863\pi\)
0.210561 0.977581i \(-0.432471\pi\)
\(542\) 0 0
\(543\) −32.2085 −1.38220
\(544\) 0 0
\(545\) −42.0370 72.8102i −1.80067 3.11884i
\(546\) 0 0
\(547\) −9.75237 16.8916i −0.416981 0.722232i 0.578653 0.815574i \(-0.303579\pi\)
−0.995634 + 0.0933413i \(0.970245\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.3722 + 11.3298i −0.612276 + 0.482666i
\(552\) 0 0
\(553\) 9.93594 17.2096i 0.422519 0.731825i
\(554\) 0 0
\(555\) −10.7916 18.6917i −0.458079 0.793416i
\(556\) 0 0
\(557\) −14.4734 25.0687i −0.613259 1.06220i −0.990687 0.136156i \(-0.956525\pi\)
0.377429 0.926039i \(-0.376808\pi\)
\(558\) 0 0
\(559\) 3.67058 0.155249
\(560\) 0 0
\(561\) −13.1911 22.8477i −0.556930 0.964632i
\(562\) 0 0
\(563\) −18.5713 −0.782688 −0.391344 0.920244i \(-0.627990\pi\)
−0.391344 + 0.920244i \(0.627990\pi\)
\(564\) 0 0
\(565\) 4.68251 8.11034i 0.196995 0.341205i
\(566\) 0 0
\(567\) 19.3013 33.4308i 0.810578 1.40396i
\(568\) 0 0
\(569\) 12.9222 0.541727 0.270863 0.962618i \(-0.412691\pi\)
0.270863 + 0.962618i \(0.412691\pi\)
\(570\) 0 0
\(571\) 21.5969 0.903802 0.451901 0.892068i \(-0.350746\pi\)
0.451901 + 0.892068i \(0.350746\pi\)
\(572\) 0 0
\(573\) −12.0000 + 20.7846i −0.501307 + 0.868290i
\(574\) 0 0
\(575\) 24.0427 41.6431i 1.00265 1.73664i
\(576\) 0 0
\(577\) 5.43646 0.226323 0.113161 0.993577i \(-0.463902\pi\)
0.113161 + 0.993577i \(0.463902\pi\)
\(578\) 0 0
\(579\) 3.48544 + 6.03697i 0.144850 + 0.250888i
\(580\) 0 0
\(581\) −31.4291 −1.30390
\(582\) 0 0
\(583\) −6.52888 11.3083i −0.270398 0.468344i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.7028 + 25.4660i −0.606851 + 1.05110i 0.384906 + 0.922956i \(0.374234\pi\)
−0.991756 + 0.128140i \(0.959099\pi\)
\(588\) 0 0
\(589\) −0.516014 3.55921i −0.0212620 0.146655i
\(590\) 0 0
\(591\) 2.57034 4.45196i 0.105730 0.183129i
\(592\) 0 0
\(593\) 13.3902 + 23.1925i 0.549869 + 0.952400i 0.998283 + 0.0585746i \(0.0186555\pi\)
−0.448414 + 0.893826i \(0.648011\pi\)
\(594\) 0 0
\(595\) 29.0882 + 50.3823i 1.19250 + 2.06547i
\(596\) 0 0
\(597\) 26.0246 1.06512
\(598\) 0 0
\(599\) −19.5419 33.8476i −0.798461 1.38297i −0.920618 0.390464i \(-0.872315\pi\)
0.122157 0.992511i \(-0.461019\pi\)
\(600\) 0 0
\(601\) −36.7409 −1.49869 −0.749347 0.662177i \(-0.769633\pi\)
−0.749347 + 0.662177i \(0.769633\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.5759 40.8346i 0.958495 1.66016i
\(606\) 0 0
\(607\) 31.7272 1.28777 0.643885 0.765123i \(-0.277322\pi\)
0.643885 + 0.765123i \(0.277322\pi\)
\(608\) 0 0
\(609\) 31.1911 1.26393
\(610\) 0 0
\(611\) −0.803734 + 1.39211i −0.0325156 + 0.0563187i
\(612\) 0 0
\(613\) −0.551220 + 0.954742i −0.0222636 + 0.0385617i −0.876943 0.480595i \(-0.840421\pi\)
0.854679 + 0.519157i \(0.173754\pi\)
\(614\) 0 0
\(615\) 39.2476 1.58262
\(616\) 0 0
\(617\) 9.09557 + 15.7540i 0.366174 + 0.634232i 0.988964 0.148157i \(-0.0473342\pi\)
−0.622790 + 0.782389i \(0.714001\pi\)
\(618\) 0 0
\(619\) −37.6666 −1.51395 −0.756974 0.653445i \(-0.773323\pi\)
−0.756974 + 0.653445i \(0.773323\pi\)
\(620\) 0 0
\(621\) −9.89335 17.1358i −0.397007 0.687636i
\(622\) 0 0
\(623\) −24.0849 41.7163i −0.964942 1.67133i
\(624\) 0 0
\(625\) −35.6591 + 61.7634i −1.42636 + 2.47054i
\(626\) 0 0
\(627\) 33.0626 + 13.1911i 1.32039 + 0.526803i
\(628\) 0 0
\(629\) 4.79409 8.30361i 0.191153 0.331087i
\(630\) 0 0
\(631\) 14.8534 + 25.7269i 0.591305 + 1.02417i 0.994057 + 0.108861i \(0.0347203\pi\)
−0.402752 + 0.915309i \(0.631946\pi\)
\(632\) 0 0
\(633\) −2.67933 4.64074i −0.106494 0.184453i
\(634\) 0 0
\(635\) 20.1396 0.799215
\(636\) 0 0
\(637\) −7.01232 12.1457i −0.277838 0.481230i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.86502 10.1585i 0.231654 0.401237i −0.726641 0.687018i \(-0.758920\pi\)
0.958295 + 0.285781i \(0.0922528\pi\)
\(642\) 0 0
\(643\) 9.84371 17.0498i 0.388198 0.672378i −0.604009 0.796977i \(-0.706431\pi\)
0.992207 + 0.124599i \(0.0397644\pi\)
\(644\) 0 0
\(645\) −21.6917 −0.854108
\(646\) 0 0
\(647\) −29.8869 −1.17497 −0.587487 0.809234i \(-0.699883\pi\)
−0.587487 + 0.809234i \(0.699883\pi\)
\(648\) 0 0
\(649\) −34.4803 + 59.7216i −1.35347 + 2.34428i
\(650\) 0 0
\(651\) −3.06478 + 5.30835i −0.120118 + 0.208051i
\(652\) 0 0
\(653\) −41.5882 −1.62747 −0.813736 0.581235i \(-0.802570\pi\)
−0.813736 + 0.581235i \(0.802570\pi\)
\(654\) 0 0
\(655\) 14.7160 + 25.4889i 0.575003 + 0.995935i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.502566 0.870470i −0.0195772 0.0339087i 0.856071 0.516859i \(-0.172899\pi\)
−0.875648 + 0.482950i \(0.839565\pi\)
\(660\) 0 0
\(661\) 2.69483 + 4.66758i 0.104817 + 0.181548i 0.913663 0.406472i \(-0.133241\pi\)
−0.808847 + 0.588020i \(0.799908\pi\)
\(662\) 0 0
\(663\) −3.44276 + 5.96303i −0.133706 + 0.231585i
\(664\) 0 0
\(665\) −72.9074 29.0882i −2.82723 1.12799i
\(666\) 0 0
\(667\) 7.99389 13.8458i 0.309525 0.536113i
\(668\) 0 0
\(669\) −5.56036 9.63083i −0.214976 0.372349i
\(670\) 0 0
\(671\) 19.3277 + 33.4766i 0.746139 + 1.29235i
\(672\) 0 0
\(673\) −14.7300 −0.567801 −0.283901 0.958854i \(-0.591629\pi\)
−0.283901 + 0.958854i \(0.591629\pi\)
\(674\) 0 0
\(675\) 32.8075 + 56.8242i 1.26276 + 2.18716i
\(676\) 0 0
\(677\) −14.0493 −0.539958 −0.269979 0.962866i \(-0.587017\pi\)
−0.269979 + 0.962866i \(0.587017\pi\)
\(678\) 0 0
\(679\) −25.4309 + 44.0475i −0.975947 + 1.69039i
\(680\) 0 0
\(681\) 18.9414 32.8075i 0.725836 1.25718i
\(682\) 0 0
\(683\) 40.2792 1.54124 0.770620 0.637295i \(-0.219947\pi\)
0.770620 + 0.637295i \(0.219947\pi\)
\(684\) 0 0
\(685\) −74.7089 −2.85448
\(686\) 0 0
\(687\) 14.6551 25.3833i 0.559125 0.968433i
\(688\) 0 0
\(689\) −1.70397 + 2.95137i −0.0649162 + 0.112438i
\(690\) 0 0
\(691\) 29.0882 1.10657 0.553284 0.832993i \(-0.313374\pi\)
0.553284 + 0.832993i \(0.313374\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 64.7608 2.45652
\(696\) 0 0
\(697\) 8.71772 + 15.0995i 0.330207 + 0.571935i
\(698\) 0 0
\(699\) 3.58695 + 6.21278i 0.135671 + 0.234989i
\(700\) 0 0
\(701\) −3.52834 + 6.11126i −0.133263 + 0.230819i −0.924933 0.380131i \(-0.875879\pi\)
0.791669 + 0.610950i \(0.209212\pi\)
\(702\) 0 0
\(703\) 1.85621 + 12.8032i 0.0700083 + 0.482883i
\(704\) 0 0
\(705\) 4.74974 8.22680i 0.178886 0.309839i
\(706\) 0 0
\(707\) 3.72606 + 6.45372i 0.140133 + 0.242717i
\(708\) 0 0
\(709\) 15.6948 + 27.1842i 0.589432 + 1.02093i 0.994307 + 0.106554i \(0.0339816\pi\)
−0.404875 + 0.914372i \(0.632685\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.57093 + 2.72092i 0.0588316 + 0.101899i
\(714\) 0 0
\(715\) −24.3597 −0.911000
\(716\) 0 0
\(717\) 3.69165 6.39413i 0.137867 0.238793i
\(718\) 0 0
\(719\) 3.08612 5.34531i 0.115093 0.199347i −0.802724 0.596351i \(-0.796617\pi\)
0.917817 + 0.397004i \(0.129950\pi\)
\(720\) 0 0
\(721\) 40.4463 1.50630
\(722\) 0 0
\(723\) −21.2266 −0.789426
\(724\) 0 0
\(725\) −26.5086 + 45.9143i −0.984506 + 1.70521i
\(726\) 0 0
\(727\) −5.78062 + 10.0123i −0.214391 + 0.371336i −0.953084 0.302706i \(-0.902110\pi\)
0.738693 + 0.674042i \(0.235443\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −4.81817 8.34531i −0.178206 0.308663i
\(732\) 0 0
\(733\) 29.4784 1.08881 0.544404 0.838823i \(-0.316756\pi\)
0.544404 + 0.838823i \(0.316756\pi\)
\(734\) 0 0
\(735\) 41.4400 + 71.7761i 1.52854 + 2.64750i
\(736\) 0 0
\(737\) 7.97342 + 13.8104i 0.293705 + 0.508712i
\(738\) 0 0
\(739\) −13.2523 + 22.9537i −0.487495 + 0.844366i −0.999897 0.0143798i \(-0.995423\pi\)
0.512402 + 0.858746i \(0.328756\pi\)
\(740\) 0 0
\(741\) −1.33299 9.19433i −0.0489687 0.337762i
\(742\) 0 0
\(743\) −15.2792 + 26.4643i −0.560538 + 0.970880i 0.436911 + 0.899504i \(0.356072\pi\)
−0.997449 + 0.0713758i \(0.977261\pi\)
\(744\) 0 0
\(745\) −4.61528 7.99389i −0.169091 0.292874i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 66.5104 2.43024
\(750\) 0 0
\(751\) −19.2377 33.3207i −0.701994 1.21589i −0.967766 0.251853i \(-0.918960\pi\)
0.265772 0.964036i \(-0.414373\pi\)
\(752\) 0 0
\(753\) 8.71630 0.317639
\(754\) 0 0
\(755\) 28.7998 49.8828i 1.04813 1.81542i
\(756\) 0 0
\(757\) −20.6399 + 35.7494i −0.750171 + 1.29933i 0.197569 + 0.980289i \(0.436695\pi\)
−0.947740 + 0.319045i \(0.896638\pi\)
\(758\) 0 0
\(759\) −31.0976 −1.12877
\(760\) 0 0
\(761\) −48.8325 −1.77018 −0.885088 0.465424i \(-0.845902\pi\)
−0.885088 + 0.465424i \(0.845902\pi\)
\(762\) 0 0
\(763\) 42.9446 74.3823i 1.55470 2.69282i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 17.9980 0.649872
\(768\) 0 0
\(769\) −27.0049 46.7739i −0.973823 1.68671i −0.683766 0.729702i \(-0.739659\pi\)
−0.290057 0.957009i \(-0.593674\pi\)
\(770\) 0 0
\(771\) −11.4367 −0.411881
\(772\) 0 0
\(773\) 5.13129 + 8.88765i 0.184560 + 0.319667i 0.943428 0.331577i \(-0.107581\pi\)
−0.758868 + 0.651244i \(0.774247\pi\)
\(774\) 0 0
\(775\) −5.20936 9.02288i −0.187126 0.324112i
\(776\) 0 0
\(777\) 11.0246 19.0952i 0.395507 0.685038i
\(778\) 0 0
\(779\) −21.8503 8.71772i −0.782868 0.312345i
\(780\) 0 0
\(781\) −19.0887 + 33.0626i −0.683047 + 1.18307i
\(782\) 0 0
\(783\) 10.9081 + 18.8934i 0.389823 + 0.675193i
\(784\) 0 0
\(785\) −20.0764 34.7733i −0.716557 1.24111i
\(786\) 0 0
\(787\) 41.0061 1.46171 0.730855 0.682533i \(-0.239122\pi\)
0.730855 + 0.682533i \(0.239122\pi\)
\(788\) 0 0
\(789\) 13.1313 + 22.7441i 0.467486 + 0.809710i
\(790\) 0 0
\(791\) 9.56723 0.340171
\(792\) 0 0
\(793\) 5.04435 8.73707i 0.179130 0.310262i
\(794\) 0 0
\(795\) 10.0698 17.4414i 0.357139 0.618583i
\(796\) 0 0
\(797\) 26.7483 0.947474 0.473737 0.880666i \(-0.342905\pi\)
0.473737 + 0.880666i \(0.342905\pi\)
\(798\) 0 0
\(799\) 4.22007 0.149295
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −22.1532 + 38.3705i −0.781769 + 1.35406i
\(804\) 0 0
\(805\) 68.5744 2.41693
\(806\) 0 0
\(807\) 10.4136 + 18.0370i 0.366578 + 0.634931i
\(808\) 0 0
\(809\) −7.61385 −0.267689 −0.133844 0.991002i \(-0.542732\pi\)
−0.133844 + 0.991002i \(0.542732\pi\)
\(810\) 0 0
\(811\) −21.1498 36.6325i −0.742670 1.28634i −0.951275 0.308342i \(-0.900226\pi\)
0.208605 0.978000i \(-0.433108\pi\)
\(812\) 0 0
\(813\) −18.2978 31.6927i −0.641731 1.11151i
\(814\) 0 0
\(815\) 1.18214 2.04753i 0.0414086 0.0717218i
\(816\) 0 0
\(817\) 12.0764 + 4.81817i 0.422499 + 0.168566i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.2429 + 17.7412i 0.357479 + 0.619171i 0.987539 0.157375i \(-0.0503031\pi\)
−0.630060 + 0.776546i \(0.716970\pi\)
\(822\) 0 0
\(823\) −15.2395 26.3956i −0.531216 0.920094i −0.999336 0.0364287i \(-0.988402\pi\)
0.468120 0.883665i \(-0.344932\pi\)
\(824\) 0 0
\(825\) 103.123 3.59029
\(826\) 0 0
\(827\) 7.84970 + 13.5961i 0.272961 + 0.472782i 0.969619 0.244622i \(-0.0786637\pi\)
−0.696658 + 0.717404i \(0.745330\pi\)
\(828\) 0 0
\(829\) 47.1271 1.63679 0.818396 0.574655i \(-0.194864\pi\)
0.818396 + 0.574655i \(0.194864\pi\)
\(830\) 0 0
\(831\) 14.3508 24.8564i 0.497825 0.862259i
\(832\) 0 0
\(833\) −18.4094 + 31.8860i −0.637847 + 1.10478i
\(834\) 0 0
\(835\) −30.6428 −1.06044
\(836\) 0 0
\(837\) −4.28722 −0.148188
\(838\) 0 0
\(839\) −9.57536 + 16.5850i −0.330578 + 0.572578i −0.982625 0.185600i \(-0.940577\pi\)
0.652047 + 0.758178i \(0.273910\pi\)
\(840\) 0 0
\(841\) 5.68620 9.84879i 0.196076 0.339613i
\(842\) 0 0
\(843\) −19.7403 −0.679891
\(844\) 0 0
\(845\) −24.1116 41.7625i −0.829464 1.43667i
\(846\) 0 0
\(847\) 48.1698 1.65513
\(848\) 0 0
\(849\) −1.50000 2.59808i −0.0514799 0.0891657i
\(850\) 0 0
\(851\) −5.65095 9.78773i −0.193712 0.335519i
\(852\) 0 0
\(853\) 27.0123 46.7867i 0.924884 1.60195i 0.133137 0.991098i \(-0.457495\pi\)
0.791747 0.610849i \(-0.209172\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.0635 19.1626i 0.377923 0.654583i −0.612837 0.790210i \(-0.709972\pi\)
0.990760 + 0.135627i \(0.0433049\pi\)
\(858\) 0 0
\(859\) 9.04506 + 15.6665i 0.308613 + 0.534534i 0.978059 0.208327i \(-0.0668018\pi\)
−0.669446 + 0.742861i \(0.733468\pi\)
\(860\) 0 0
\(861\) 20.0475 + 34.7233i 0.683218 + 1.18337i
\(862\) 0 0
\(863\) 23.2070 0.789975 0.394988 0.918686i \(-0.370749\pi\)
0.394988 + 0.918686i \(0.370749\pi\)
\(864\) 0 0
\(865\) 41.0690 + 71.1336i 1.39639 + 2.41861i
\(866\) 0 0
\(867\) −11.3684 −0.386091
\(868\) 0 0
\(869\) 10.9222 18.9178i 0.370510 0.641743i
\(870\) 0 0
\(871\) 2.08099 3.60437i 0.0705115 0.122130i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −137.359 −4.64359
\(876\) 0 0
\(877\) −7.06723 + 12.2408i −0.238644 + 0.413343i −0.960325 0.278882i \(-0.910036\pi\)
0.721682 + 0.692225i \(0.243369\pi\)
\(878\) 0 0
\(879\) −3.84759 + 6.66422i −0.129776 + 0.224779i
\(880\) 0 0
\(881\) 49.4217 1.66506 0.832530 0.553981i \(-0.186892\pi\)
0.832530 + 0.553981i \(0.186892\pi\)
\(882\) 0 0
\(883\) 28.3596 + 49.1202i 0.954375 + 1.65303i 0.735791 + 0.677209i \(0.236811\pi\)
0.218585 + 0.975818i \(0.429856\pi\)
\(884\) 0 0
\(885\) −106.361 −3.57529
\(886\) 0 0
\(887\) −25.2325 43.7040i −0.847224 1.46744i −0.883675 0.468100i \(-0.844939\pi\)
0.0364508 0.999335i \(-0.488395\pi\)
\(888\) 0 0
\(889\) 10.2872 + 17.8180i 0.345022 + 0.597596i
\(890\) 0 0
\(891\) 21.2172 36.7492i 0.710802 1.23115i
\(892\) 0 0
\(893\) −4.47166 + 3.52508i −0.149639 + 0.117962i
\(894\) 0 0
\(895\) −37.1986 + 64.4298i −1.24341 + 2.15365i
\(896\) 0 0
\(897\) 4.05809 + 7.02882i 0.135496 + 0.234685i
\(898\) 0 0
\(899\) −1.73205 3.00000i −0.0577671 0.100056i
\(900\) 0 0
\(901\) 8.94685 0.298063
\(902\) 0 0
\(903\) −11.0800 19.1911i −0.368720 0.638641i
\(904\) 0 0
\(905\) 78.0739 2.59527
\(906\) 0 0
\(907\) 5.12878 8.88330i 0.170298 0.294965i −0.768226 0.640179i \(-0.778860\pi\)
0.938524 + 0.345214i \(0.112194\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.9971 −0.629402 −0.314701 0.949191i \(-0.601904\pi\)
−0.314701 + 0.949191i \(0.601904\pi\)
\(912\) 0 0
\(913\) −34.5488 −1.14340
\(914\) 0 0
\(915\) −29.8101 + 51.6325i −0.985490 + 1.70692i
\(916\) 0 0
\(917\) −15.0338 + 26.0393i −0.496459 + 0.859893i
\(918\) 0 0
\(919\) 26.5866 0.877010 0.438505 0.898729i \(-0.355508\pi\)
0.438505 + 0.898729i \(0.355508\pi\)
\(920\) 0 0
\(921\) −26.7872 46.3968i −0.882669 1.52883i
\(922\) 0 0
\(923\) 9.96393 0.327967
\(924\) 0 0
\(925\) 18.7392 + 32.4572i 0.616140 + 1.06719i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.7306 18.5859i 0.352058 0.609782i −0.634552 0.772880i \(-0.718815\pi\)
0.986610 + 0.163098i \(0.0521487\pi\)
\(930\) 0 0
\(931\) −7.12787 49.1646i −0.233606 1.61130i
\(932\) 0 0
\(933\) 25.9057 44.8699i 0.848114 1.46898i
\(934\) 0 0
\(935\) 31.9756 + 55.3833i 1.04571 + 1.81123i
\(936\) 0 0
\(937\) 10.6207 + 18.3956i 0.346964 + 0.600959i 0.985709 0.168460i \(-0.0538793\pi\)
−0.638745 + 0.769419i \(0.720546\pi\)
\(938\) 0 0
\(939\) −10.9453 −0.357185
\(940\) 0 0
\(941\) −20.5759 35.6384i −0.670754 1.16178i −0.977691 0.210050i \(-0.932637\pi\)
0.306937 0.951730i \(-0.400696\pi\)
\(942\) 0 0
\(943\) 20.5517 0.669256
\(944\) 0 0
\(945\) −46.7867 + 81.0370i −1.52197 + 2.63613i
\(946\) 0 0
\(947\) −21.2872 + 36.8705i −0.691740 + 1.19813i 0.279527 + 0.960138i \(0.409822\pi\)
−0.971267 + 0.237991i \(0.923511\pi\)
\(948\) 0 0
\(949\) 11.5635 0.375368
\(950\) 0 0
\(951\) −25.5814 −0.829535
\(952\) 0 0
\(953\) −25.3261 + 43.8661i −0.820394 + 1.42096i 0.0849960 + 0.996381i \(0.472912\pi\)
−0.905390 + 0.424582i \(0.860421\pi\)
\(954\) 0 0
\(955\) 29.0882 50.3823i 0.941273 1.63033i
\(956\) 0 0
\(957\) 34.2872 1.10835
\(958\) 0 0
\(959\) −38.1610 66.0968i −1.23228 2.13438i
\(960\) 0 0
\(961\) −30.3192 −0.978040
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.44878 14.6337i −0.271976 0.471076i
\(966\) 0 0
\(967\) 2.02814 3.51283i 0.0652205 0.112965i −0.831571 0.555418i \(-0.812558\pi\)
0.896792 + 0.442453i \(0.145892\pi\)
\(968\) 0 0
\(969\) −19.1542 + 15.0995i −0.615321 + 0.485067i
\(970\) 0 0
\(971\) 12.9085 22.3581i 0.414253 0.717507i −0.581097 0.813834i \(-0.697376\pi\)
0.995350 + 0.0963273i \(0.0307096\pi\)
\(972\) 0 0
\(973\) 33.0796 + 57.2955i 1.06048 + 1.83681i
\(974\) 0 0
\(975\) −13.4571 23.3083i −0.430972 0.746465i
\(976\) 0 0
\(977\) −21.8828 −0.700093 −0.350046 0.936732i \(-0.613834\pi\)
−0.350046 + 0.936732i \(0.613834\pi\)
\(978\) 0 0
\(979\) −26.4756 45.8571i −0.846165 1.46560i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.4765 42.3946i 0.780680 1.35218i −0.150866 0.988554i \(-0.548206\pi\)
0.931546 0.363624i \(-0.118461\pi\)
\(984\) 0 0
\(985\) −6.23055 + 10.7916i −0.198522 + 0.343850i
\(986\) 0 0
\(987\) 9.70460 0.308901
\(988\) 0 0
\(989\) −11.3587 −0.361184
\(990\) 0 0
\(991\) −0.185139 + 0.320670i −0.00588112 + 0.0101864i −0.868951 0.494898i \(-0.835205\pi\)
0.863070 + 0.505085i \(0.168539\pi\)
\(992\) 0 0
\(993\) 0.487678 0.844683i 0.0154760 0.0268052i
\(994\) 0 0
\(995\) −63.0842 −1.99990
\(996\) 0 0
\(997\) −7.66280 13.2724i −0.242683 0.420340i 0.718794 0.695223i \(-0.244694\pi\)
−0.961478 + 0.274883i \(0.911361\pi\)
\(998\) 0 0
\(999\) 15.4220 0.487932
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 1216.2.i.r.961.3 12
4.3 odd 2 inner 1216.2.i.r.961.6 12
8.3 odd 2 608.2.i.f.353.1 12
8.5 even 2 608.2.i.f.353.4 yes 12
19.7 even 3 inner 1216.2.i.r.577.3 12
76.7 odd 6 inner 1216.2.i.r.577.6 12
152.45 even 6 608.2.i.f.577.4 yes 12
152.83 odd 6 608.2.i.f.577.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.i.f.353.1 12 8.3 odd 2
608.2.i.f.353.4 yes 12 8.5 even 2
608.2.i.f.577.1 yes 12 152.83 odd 6
608.2.i.f.577.4 yes 12 152.45 even 6
1216.2.i.r.577.3 12 19.7 even 3 inner
1216.2.i.r.577.6 12 76.7 odd 6 inner
1216.2.i.r.961.3 12 1.1 even 1 trivial
1216.2.i.r.961.6 12 4.3 odd 2 inner