Properties

Label 784.2.bb.b.111.5
Level $784$
Weight $2$
Character 784.111
Analytic conductor $6.260$
Analytic rank $0$
Dimension $120$
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] = [784,2,Mod(111,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bb (of order \(14\), degree \(6\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 111.5
Character \(\chi\) \(=\) 784.111
Dual form 784.2.bb.b.671.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.31050 - 1.64332i) q^{3} +(-0.883194 + 0.704324i) q^{5} +(2.18004 + 1.49914i) q^{7} +(-0.315513 + 1.38235i) q^{9} +O(q^{10})\) \(q+(-1.31050 - 1.64332i) q^{3} +(-0.883194 + 0.704324i) q^{5} +(2.18004 + 1.49914i) q^{7} +(-0.315513 + 1.38235i) q^{9} +(1.37781 - 0.314477i) q^{11} +(-5.18274 + 1.18293i) q^{13} +(2.31485 + 0.528350i) q^{15} +(-3.36325 + 6.98386i) q^{17} +2.00755 q^{19} +(-0.393391 - 5.54712i) q^{21} +(-0.914165 - 1.89828i) q^{23} +(-0.828645 + 3.63053i) q^{25} +(-2.99606 + 1.44283i) q^{27} +(4.94415 + 2.38098i) q^{29} +9.76273 q^{31} +(-2.32241 - 1.85206i) q^{33} +(-2.98128 + 0.211427i) q^{35} +(2.81066 + 1.35354i) q^{37} +(8.73591 + 6.96666i) q^{39} +(-0.681339 + 0.543349i) q^{41} +(-3.89477 - 3.10598i) q^{43} +(-0.694965 - 1.44311i) q^{45} +(2.75189 + 12.0568i) q^{47} +(2.50517 + 6.53637i) q^{49} +(15.8842 - 3.62547i) q^{51} +(5.52933 - 2.66278i) q^{53} +(-0.995384 + 1.24817i) q^{55} +(-2.63090 - 3.29905i) q^{57} +(-1.75057 + 2.19514i) q^{59} +(2.33760 - 4.85407i) q^{61} +(-2.76017 + 2.54059i) q^{63} +(3.74420 - 4.69508i) q^{65} +6.28419i q^{67} +(-1.92147 + 3.98997i) q^{69} +(-1.36050 - 2.82510i) q^{71} +(12.8418 + 2.93106i) q^{73} +(7.05205 - 3.39609i) q^{75} +(3.47514 + 1.37996i) q^{77} -10.3351i q^{79} +(10.1298 + 4.87827i) q^{81} +(-2.14760 + 9.40925i) q^{83} +(-1.94850 - 8.53693i) q^{85} +(-2.56662 - 11.2451i) q^{87} +(-13.5995 - 3.10399i) q^{89} +(-13.0720 - 5.19081i) q^{91} +(-12.7941 - 16.0433i) q^{93} +(-1.77306 + 1.41397i) q^{95} +4.89447i q^{97} +2.00385i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 120 q - 24 q^{9} - 14 q^{17} + 16 q^{21} + 40 q^{25} + 32 q^{29} - 62 q^{37} - 28 q^{41} - 60 q^{49} + 14 q^{53} - 34 q^{57} - 112 q^{61} - 32 q^{65} + 112 q^{69} + 42 q^{73} + 66 q^{77} - 44 q^{81} - 12 q^{85} + 28 q^{89} - 58 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{14}\right)\)

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
\(3\) −1.31050 1.64332i −0.756618 0.948769i 0.243157 0.969987i \(-0.421817\pi\)
−0.999775 + 0.0212178i \(0.993246\pi\)
\(4\) 0 0
\(5\) −0.883194 + 0.704324i −0.394977 + 0.314983i −0.800759 0.598986i \(-0.795570\pi\)
0.405783 + 0.913970i \(0.366999\pi\)
\(6\) 0 0
\(7\) 2.18004 + 1.49914i 0.823979 + 0.566621i
\(8\) 0 0
\(9\) −0.315513 + 1.38235i −0.105171 + 0.460784i
\(10\) 0 0
\(11\) 1.37781 0.314477i 0.415427 0.0948184i −0.00969862 0.999953i \(-0.503087\pi\)
0.425125 + 0.905135i \(0.360230\pi\)
\(12\) 0 0
\(13\) −5.18274 + 1.18293i −1.43743 + 0.328085i −0.869069 0.494691i \(-0.835281\pi\)
−0.568365 + 0.822776i \(0.692424\pi\)
\(14\) 0 0
\(15\) 2.31485 + 0.528350i 0.597693 + 0.136419i
\(16\) 0 0
\(17\) −3.36325 + 6.98386i −0.815708 + 1.69384i −0.100486 + 0.994939i \(0.532040\pi\)
−0.715223 + 0.698897i \(0.753675\pi\)
\(18\) 0 0
\(19\) 2.00755 0.460565 0.230282 0.973124i \(-0.426035\pi\)
0.230282 + 0.973124i \(0.426035\pi\)
\(20\) 0 0
\(21\) −0.393391 5.54712i −0.0858450 1.21048i
\(22\) 0 0
\(23\) −0.914165 1.89828i −0.190617 0.395820i 0.783655 0.621196i \(-0.213353\pi\)
−0.974272 + 0.225377i \(0.927639\pi\)
\(24\) 0 0
\(25\) −0.828645 + 3.63053i −0.165729 + 0.726106i
\(26\) 0 0
\(27\) −2.99606 + 1.44283i −0.576593 + 0.277672i
\(28\) 0 0
\(29\) 4.94415 + 2.38098i 0.918106 + 0.442137i 0.832395 0.554183i \(-0.186969\pi\)
0.0857115 + 0.996320i \(0.472684\pi\)
\(30\) 0 0
\(31\) 9.76273 1.75344 0.876719 0.481003i \(-0.159727\pi\)
0.876719 + 0.481003i \(0.159727\pi\)
\(32\) 0 0
\(33\) −2.32241 1.85206i −0.404280 0.322403i
\(34\) 0 0
\(35\) −2.98128 + 0.211427i −0.503928 + 0.0357376i
\(36\) 0 0
\(37\) 2.81066 + 1.35354i 0.462070 + 0.222521i 0.650411 0.759582i \(-0.274597\pi\)
−0.188341 + 0.982104i \(0.560311\pi\)
\(38\) 0 0
\(39\) 8.73591 + 6.96666i 1.39887 + 1.11556i
\(40\) 0 0
\(41\) −0.681339 + 0.543349i −0.106407 + 0.0848569i −0.675248 0.737591i \(-0.735963\pi\)
0.568841 + 0.822448i \(0.307392\pi\)
\(42\) 0 0
\(43\) −3.89477 3.10598i −0.593947 0.473657i 0.279786 0.960062i \(-0.409736\pi\)
−0.873733 + 0.486405i \(0.838308\pi\)
\(44\) 0 0
\(45\) −0.694965 1.44311i −0.103599 0.215126i
\(46\) 0 0
\(47\) 2.75189 + 12.0568i 0.401404 + 1.75866i 0.621723 + 0.783237i \(0.286433\pi\)
−0.220319 + 0.975428i \(0.570710\pi\)
\(48\) 0 0
\(49\) 2.50517 + 6.53637i 0.357882 + 0.933767i
\(50\) 0 0
\(51\) 15.8842 3.62547i 2.22424 0.507668i
\(52\) 0 0
\(53\) 5.52933 2.66278i 0.759511 0.365761i −0.0137031 0.999906i \(-0.504362\pi\)
0.773214 + 0.634145i \(0.218648\pi\)
\(54\) 0 0
\(55\) −0.995384 + 1.24817i −0.134218 + 0.168303i
\(56\) 0 0
\(57\) −2.63090 3.29905i −0.348472 0.436970i
\(58\) 0 0
\(59\) −1.75057 + 2.19514i −0.227904 + 0.285783i −0.882615 0.470097i \(-0.844219\pi\)
0.654711 + 0.755880i \(0.272790\pi\)
\(60\) 0 0
\(61\) 2.33760 4.85407i 0.299298 0.621500i −0.696034 0.718009i \(-0.745054\pi\)
0.995332 + 0.0965096i \(0.0307678\pi\)
\(62\) 0 0
\(63\) −2.76017 + 2.54059i −0.347749 + 0.320084i
\(64\) 0 0
\(65\) 3.74420 4.69508i 0.464411 0.582353i
\(66\) 0 0
\(67\) 6.28419i 0.767736i 0.923388 + 0.383868i \(0.125408\pi\)
−0.923388 + 0.383868i \(0.874592\pi\)
\(68\) 0 0
\(69\) −1.92147 + 3.98997i −0.231317 + 0.480336i
\(70\) 0 0
\(71\) −1.36050 2.82510i −0.161461 0.335278i 0.804505 0.593946i \(-0.202431\pi\)
−0.965966 + 0.258668i \(0.916716\pi\)
\(72\) 0 0
\(73\) 12.8418 + 2.93106i 1.50302 + 0.343055i 0.893264 0.449533i \(-0.148409\pi\)
0.609758 + 0.792588i \(0.291267\pi\)
\(74\) 0 0
\(75\) 7.05205 3.39609i 0.814301 0.392146i
\(76\) 0 0
\(77\) 3.47514 + 1.37996i 0.396029 + 0.157261i
\(78\) 0 0
\(79\) 10.3351i 1.16279i −0.813622 0.581395i \(-0.802507\pi\)
0.813622 0.581395i \(-0.197493\pi\)
\(80\) 0 0
\(81\) 10.1298 + 4.87827i 1.12554 + 0.542030i
\(82\) 0 0
\(83\) −2.14760 + 9.40925i −0.235730 + 1.03280i 0.709067 + 0.705141i \(0.249116\pi\)
−0.944796 + 0.327658i \(0.893741\pi\)
\(84\) 0 0
\(85\) −1.94850 8.53693i −0.211344 0.925960i
\(86\) 0 0
\(87\) −2.56662 11.2451i −0.275170 1.20560i
\(88\) 0 0
\(89\) −13.5995 3.10399i −1.44154 0.329023i −0.570933 0.820997i \(-0.693418\pi\)
−0.870610 + 0.491974i \(0.836275\pi\)
\(90\) 0 0
\(91\) −13.0720 5.19081i −1.37031 0.544145i
\(92\) 0 0
\(93\) −12.7941 16.0433i −1.32668 1.66361i
\(94\) 0 0
\(95\) −1.77306 + 1.41397i −0.181912 + 0.145070i
\(96\) 0 0
\(97\) 4.89447i 0.496958i 0.968637 + 0.248479i \(0.0799308\pi\)
−0.968637 + 0.248479i \(0.920069\pi\)
\(98\) 0 0
\(99\) 2.00385i 0.201394i
\(100\) 0 0
\(101\) −12.9356 + 10.3158i −1.28714 + 1.02646i −0.289544 + 0.957165i \(0.593504\pi\)
−0.997596 + 0.0692955i \(0.977925\pi\)
\(102\) 0 0
\(103\) 3.29632 + 4.13345i 0.324796 + 0.407281i 0.917243 0.398328i \(-0.130410\pi\)
−0.592447 + 0.805609i \(0.701838\pi\)
\(104\) 0 0
\(105\) 4.25441 + 4.62211i 0.415188 + 0.451072i
\(106\) 0 0
\(107\) −12.0461 2.74945i −1.16454 0.265800i −0.403797 0.914848i \(-0.632310\pi\)
−0.760747 + 0.649049i \(0.775167\pi\)
\(108\) 0 0
\(109\) 0.990406 + 4.33925i 0.0948637 + 0.415625i 0.999954 0.00959447i \(-0.00305406\pi\)
−0.905090 + 0.425219i \(0.860197\pi\)
\(110\) 0 0
\(111\) −1.45908 6.39263i −0.138489 0.606762i
\(112\) 0 0
\(113\) −0.554152 + 2.42790i −0.0521303 + 0.228398i −0.994282 0.106790i \(-0.965943\pi\)
0.942151 + 0.335188i \(0.108800\pi\)
\(114\) 0 0
\(115\) 2.14439 + 1.03269i 0.199966 + 0.0962984i
\(116\) 0 0
\(117\) 7.53761i 0.696852i
\(118\) 0 0
\(119\) −17.8018 + 10.1831i −1.63189 + 0.933487i
\(120\) 0 0
\(121\) −8.11118 + 3.90614i −0.737380 + 0.355104i
\(122\) 0 0
\(123\) 1.78579 + 0.407595i 0.161019 + 0.0367516i
\(124\) 0 0
\(125\) −4.27589 8.87899i −0.382448 0.794161i
\(126\) 0 0
\(127\) 5.39015 11.1928i 0.478299 0.993197i −0.512605 0.858624i \(-0.671320\pi\)
0.990904 0.134573i \(-0.0429662\pi\)
\(128\) 0 0
\(129\) 10.4707i 0.921896i
\(130\) 0 0
\(131\) 6.01964 7.54838i 0.525938 0.659505i −0.445920 0.895073i \(-0.647123\pi\)
0.971858 + 0.235567i \(0.0756948\pi\)
\(132\) 0 0
\(133\) 4.37656 + 3.00960i 0.379496 + 0.260965i
\(134\) 0 0
\(135\) 1.62989 3.38450i 0.140278 0.291291i
\(136\) 0 0
\(137\) −6.75603 + 8.47179i −0.577206 + 0.723794i −0.981634 0.190776i \(-0.938899\pi\)
0.404427 + 0.914570i \(0.367471\pi\)
\(138\) 0 0
\(139\) −7.06030 8.85333i −0.598846 0.750930i 0.386351 0.922352i \(-0.373735\pi\)
−0.985198 + 0.171422i \(0.945164\pi\)
\(140\) 0 0
\(141\) 16.2068 20.3227i 1.36486 1.71148i
\(142\) 0 0
\(143\) −6.76885 + 3.25971i −0.566040 + 0.272590i
\(144\) 0 0
\(145\) −6.04363 + 1.37942i −0.501896 + 0.114554i
\(146\) 0 0
\(147\) 7.45829 12.6827i 0.615149 1.04605i
\(148\) 0 0
\(149\) 2.64512 + 11.5890i 0.216697 + 0.949410i 0.959900 + 0.280344i \(0.0904485\pi\)
−0.743203 + 0.669066i \(0.766694\pi\)
\(150\) 0 0
\(151\) −0.130043 0.270038i −0.0105828 0.0219754i 0.895610 0.444839i \(-0.146739\pi\)
−0.906193 + 0.422864i \(0.861025\pi\)
\(152\) 0 0
\(153\) −8.59301 6.85270i −0.694704 0.554008i
\(154\) 0 0
\(155\) −8.62239 + 6.87613i −0.692567 + 0.552304i
\(156\) 0 0
\(157\) 1.39102 + 1.10930i 0.111016 + 0.0885321i 0.677424 0.735592i \(-0.263096\pi\)
−0.566409 + 0.824125i \(0.691668\pi\)
\(158\) 0 0
\(159\) −11.6220 5.59685i −0.921683 0.443859i
\(160\) 0 0
\(161\) 0.852869 5.50880i 0.0672155 0.434154i
\(162\) 0 0
\(163\) 2.68761 + 2.14330i 0.210510 + 0.167876i 0.723069 0.690776i \(-0.242731\pi\)
−0.512559 + 0.858652i \(0.671302\pi\)
\(164\) 0 0
\(165\) 3.35559 0.261233
\(166\) 0 0
\(167\) −6.87138 3.30908i −0.531724 0.256065i 0.148702 0.988882i \(-0.452491\pi\)
−0.680425 + 0.732817i \(0.738205\pi\)
\(168\) 0 0
\(169\) 13.7489 6.62112i 1.05761 0.509317i
\(170\) 0 0
\(171\) −0.633410 + 2.77515i −0.0484381 + 0.212221i
\(172\) 0 0
\(173\) −7.15031 14.8478i −0.543628 1.12885i −0.974072 0.226238i \(-0.927357\pi\)
0.430444 0.902617i \(-0.358357\pi\)
\(174\) 0 0
\(175\) −7.24914 + 6.67246i −0.547984 + 0.504390i
\(176\) 0 0
\(177\) 5.90143 0.443578
\(178\) 0 0
\(179\) −6.55426 + 13.6101i −0.489888 + 1.01726i 0.498722 + 0.866762i \(0.333803\pi\)
−0.988610 + 0.150501i \(0.951911\pi\)
\(180\) 0 0
\(181\) −17.0580 3.89337i −1.26791 0.289392i −0.464902 0.885362i \(-0.653910\pi\)
−0.803007 + 0.595970i \(0.796768\pi\)
\(182\) 0 0
\(183\) −11.0402 + 2.51985i −0.816114 + 0.186273i
\(184\) 0 0
\(185\) −3.43570 + 0.784175i −0.252597 + 0.0576537i
\(186\) 0 0
\(187\) −2.43767 + 10.6801i −0.178260 + 0.781008i
\(188\) 0 0
\(189\) −8.69454 1.34608i −0.632435 0.0979132i
\(190\) 0 0
\(191\) −0.384152 + 0.306351i −0.0277963 + 0.0221668i −0.637289 0.770625i \(-0.719944\pi\)
0.609493 + 0.792791i \(0.291373\pi\)
\(192\) 0 0
\(193\) 3.60979 + 4.52653i 0.259838 + 0.325827i 0.894589 0.446890i \(-0.147469\pi\)
−0.634751 + 0.772717i \(0.718897\pi\)
\(194\) 0 0
\(195\) −12.6223 −0.903901
\(196\) 0 0
\(197\) 18.3913 1.31033 0.655163 0.755488i \(-0.272600\pi\)
0.655163 + 0.755488i \(0.272600\pi\)
\(198\) 0 0
\(199\) −13.6929 17.1704i −0.970665 1.21717i −0.976129 0.217190i \(-0.930311\pi\)
0.00546475 0.999985i \(-0.498261\pi\)
\(200\) 0 0
\(201\) 10.3269 8.23543i 0.728404 0.580883i
\(202\) 0 0
\(203\) 7.20905 + 12.6026i 0.505976 + 0.884529i
\(204\) 0 0
\(205\) 0.219060 0.959766i 0.0152998 0.0670330i
\(206\) 0 0
\(207\) 2.91253 0.664766i 0.202435 0.0462044i
\(208\) 0 0
\(209\) 2.76604 0.631330i 0.191331 0.0436700i
\(210\) 0 0
\(211\) 24.3607 + 5.56018i 1.67706 + 0.382779i 0.952047 0.305953i \(-0.0989751\pi\)
0.725016 + 0.688732i \(0.241832\pi\)
\(212\) 0 0
\(213\) −2.85960 + 5.93802i −0.195937 + 0.406867i
\(214\) 0 0
\(215\) 5.62746 0.383789
\(216\) 0 0
\(217\) 21.2832 + 14.6357i 1.44480 + 0.993534i
\(218\) 0 0
\(219\) −12.0126 24.9443i −0.811734 1.68558i
\(220\) 0 0
\(221\) 9.16946 40.1740i 0.616805 2.70240i
\(222\) 0 0
\(223\) 15.4983 7.46358i 1.03784 0.499798i 0.164230 0.986422i \(-0.447486\pi\)
0.873612 + 0.486624i \(0.161772\pi\)
\(224\) 0 0
\(225\) −4.75723 2.29096i −0.317148 0.152731i
\(226\) 0 0
\(227\) −0.767670 −0.0509521 −0.0254760 0.999675i \(-0.508110\pi\)
−0.0254760 + 0.999675i \(0.508110\pi\)
\(228\) 0 0
\(229\) 14.4532 + 11.5260i 0.955092 + 0.761660i 0.971214 0.238210i \(-0.0765606\pi\)
−0.0161221 + 0.999870i \(0.505132\pi\)
\(230\) 0 0
\(231\) −2.28646 7.51919i −0.150438 0.494726i
\(232\) 0 0
\(233\) −14.6669 7.06322i −0.960862 0.462727i −0.113380 0.993552i \(-0.536168\pi\)
−0.847481 + 0.530825i \(0.821882\pi\)
\(234\) 0 0
\(235\) −10.9223 8.71028i −0.712495 0.568196i
\(236\) 0 0
\(237\) −16.9838 + 13.5442i −1.10322 + 0.879788i
\(238\) 0 0
\(239\) 5.56441 + 4.43747i 0.359932 + 0.287036i 0.786713 0.617319i \(-0.211781\pi\)
−0.426781 + 0.904355i \(0.640353\pi\)
\(240\) 0 0
\(241\) 0.257898 + 0.535531i 0.0166127 + 0.0344966i 0.909110 0.416556i \(-0.136763\pi\)
−0.892497 + 0.451053i \(0.851049\pi\)
\(242\) 0 0
\(243\) −3.03871 13.3135i −0.194934 0.854060i
\(244\) 0 0
\(245\) −6.81628 4.00843i −0.435476 0.256089i
\(246\) 0 0
\(247\) −10.4046 + 2.37479i −0.662031 + 0.151104i
\(248\) 0 0
\(249\) 18.2768 8.80165i 1.15825 0.557782i
\(250\) 0 0
\(251\) 8.97401 11.2531i 0.566434 0.710286i −0.413298 0.910596i \(-0.635623\pi\)
0.979733 + 0.200309i \(0.0641947\pi\)
\(252\) 0 0
\(253\) −1.85652 2.32800i −0.116718 0.146360i
\(254\) 0 0
\(255\) −11.4754 + 14.3896i −0.718615 + 0.901115i
\(256\) 0 0
\(257\) 7.68198 15.9518i 0.479189 0.995046i −0.511549 0.859254i \(-0.670928\pi\)
0.990737 0.135791i \(-0.0433577\pi\)
\(258\) 0 0
\(259\) 4.09822 + 7.16435i 0.254651 + 0.445171i
\(260\) 0 0
\(261\) −4.85130 + 6.08334i −0.300288 + 0.376549i
\(262\) 0 0
\(263\) 1.63281i 0.100683i 0.998732 + 0.0503416i \(0.0160310\pi\)
−0.998732 + 0.0503416i \(0.983969\pi\)
\(264\) 0 0
\(265\) −3.00801 + 6.24619i −0.184780 + 0.383700i
\(266\) 0 0
\(267\) 12.7213 + 26.4160i 0.778531 + 1.61664i
\(268\) 0 0
\(269\) 2.36730 + 0.540321i 0.144337 + 0.0329439i 0.294079 0.955781i \(-0.404987\pi\)
−0.149742 + 0.988725i \(0.547844\pi\)
\(270\) 0 0
\(271\) 1.12411 0.541345i 0.0682850 0.0328843i −0.399430 0.916764i \(-0.630792\pi\)
0.467715 + 0.883880i \(0.345078\pi\)
\(272\) 0 0
\(273\) 8.60069 + 28.2840i 0.520537 + 1.71182i
\(274\) 0 0
\(275\) 5.26278i 0.317358i
\(276\) 0 0
\(277\) 23.4147 + 11.2759i 1.40685 + 0.677505i 0.974539 0.224219i \(-0.0719829\pi\)
0.432314 + 0.901723i \(0.357697\pi\)
\(278\) 0 0
\(279\) −3.08027 + 13.4955i −0.184411 + 0.807957i
\(280\) 0 0
\(281\) −2.26857 9.93924i −0.135331 0.592925i −0.996425 0.0844793i \(-0.973077\pi\)
0.861094 0.508446i \(-0.169780\pi\)
\(282\) 0 0
\(283\) 2.40148 + 10.5216i 0.142753 + 0.625442i 0.994789 + 0.101958i \(0.0325107\pi\)
−0.852036 + 0.523484i \(0.824632\pi\)
\(284\) 0 0
\(285\) 4.64720 + 1.06069i 0.275276 + 0.0628300i
\(286\) 0 0
\(287\) −2.29990 + 0.163105i −0.135759 + 0.00962777i
\(288\) 0 0
\(289\) −26.8635 33.6858i −1.58021 1.98152i
\(290\) 0 0
\(291\) 8.04317 6.41421i 0.471499 0.376008i
\(292\) 0 0
\(293\) 0.437380i 0.0255520i 0.999918 + 0.0127760i \(0.00406684\pi\)
−0.999918 + 0.0127760i \(0.995933\pi\)
\(294\) 0 0
\(295\) 3.17170i 0.184664i
\(296\) 0 0
\(297\) −3.67428 + 2.93014i −0.213203 + 0.170024i
\(298\) 0 0
\(299\) 6.98342 + 8.75693i 0.403861 + 0.506426i
\(300\) 0 0
\(301\) −3.83448 12.6100i −0.221016 0.726826i
\(302\) 0 0
\(303\) 33.9042 + 7.73842i 1.94775 + 0.444561i
\(304\) 0 0
\(305\) 1.35428 + 5.93351i 0.0775461 + 0.339752i
\(306\) 0 0
\(307\) −6.56510 28.7636i −0.374690 1.64162i −0.713417 0.700739i \(-0.752854\pi\)
0.338727 0.940885i \(-0.390004\pi\)
\(308\) 0 0
\(309\) 2.47274 10.8338i 0.140669 0.616312i
\(310\) 0 0
\(311\) −20.3742 9.81170i −1.15531 0.556370i −0.244689 0.969602i \(-0.578686\pi\)
−0.910626 + 0.413231i \(0.864400\pi\)
\(312\) 0 0
\(313\) 8.98244i 0.507717i −0.967241 0.253859i \(-0.918300\pi\)
0.967241 0.253859i \(-0.0816998\pi\)
\(314\) 0 0
\(315\) 0.648367 4.18789i 0.0365313 0.235961i
\(316\) 0 0
\(317\) −17.5066 + 8.43074i −0.983270 + 0.473518i −0.855228 0.518251i \(-0.826583\pi\)
−0.128041 + 0.991769i \(0.540869\pi\)
\(318\) 0 0
\(319\) 7.56089 + 1.72572i 0.423328 + 0.0966219i
\(320\) 0 0
\(321\) 11.2683 + 23.3988i 0.628933 + 1.30599i
\(322\) 0 0
\(323\) −6.75191 + 14.0205i −0.375686 + 0.780121i
\(324\) 0 0
\(325\) 19.7963i 1.09810i
\(326\) 0 0
\(327\) 5.83284 7.31415i 0.322557 0.404473i
\(328\) 0 0
\(329\) −12.0756 + 30.4098i −0.665748 + 1.67655i
\(330\) 0 0
\(331\) −1.66278 + 3.45280i −0.0913946 + 0.189783i −0.941655 0.336579i \(-0.890730\pi\)
0.850261 + 0.526362i \(0.176444\pi\)
\(332\) 0 0
\(333\) −2.75788 + 3.45827i −0.151131 + 0.189512i
\(334\) 0 0
\(335\) −4.42610 5.55016i −0.241824 0.303238i
\(336\) 0 0
\(337\) −14.2172 + 17.8278i −0.774462 + 0.971144i −0.999995 0.00309116i \(-0.999016\pi\)
0.225534 + 0.974235i \(0.427587\pi\)
\(338\) 0 0
\(339\) 4.71603 2.27112i 0.256139 0.123350i
\(340\) 0 0
\(341\) 13.4512 3.07016i 0.728425 0.166258i
\(342\) 0 0
\(343\) −4.33753 + 18.0052i −0.234205 + 0.972187i
\(344\) 0 0
\(345\) −1.11320 4.87725i −0.0599327 0.262582i
\(346\) 0 0
\(347\) 0.473626 + 0.983495i 0.0254256 + 0.0527968i 0.913295 0.407299i \(-0.133529\pi\)
−0.887869 + 0.460096i \(0.847815\pi\)
\(348\) 0 0
\(349\) 15.3887 + 12.2721i 0.823741 + 0.656911i 0.941829 0.336092i \(-0.109105\pi\)
−0.118088 + 0.993003i \(0.537677\pi\)
\(350\) 0 0
\(351\) 13.8211 11.0219i 0.737714 0.588307i
\(352\) 0 0
\(353\) 25.4385 + 20.2865i 1.35396 + 1.07974i 0.988872 + 0.148769i \(0.0475310\pi\)
0.365084 + 0.930975i \(0.381040\pi\)
\(354\) 0 0
\(355\) 3.19137 + 1.53688i 0.169380 + 0.0815692i
\(356\) 0 0
\(357\) 40.0634 + 15.9090i 2.12038 + 0.841992i
\(358\) 0 0
\(359\) 23.6266 + 18.8416i 1.24697 + 0.994422i 0.999675 + 0.0254811i \(0.00811177\pi\)
0.247291 + 0.968941i \(0.420460\pi\)
\(360\) 0 0
\(361\) −14.9697 −0.787880
\(362\) 0 0
\(363\) 17.0487 + 8.21024i 0.894827 + 0.430926i
\(364\) 0 0
\(365\) −13.4062 + 6.45611i −0.701715 + 0.337928i
\(366\) 0 0
\(367\) −0.409354 + 1.79350i −0.0213681 + 0.0936198i −0.984488 0.175454i \(-0.943861\pi\)
0.963119 + 0.269074i \(0.0867177\pi\)
\(368\) 0 0
\(369\) −0.536130 1.11328i −0.0279098 0.0579553i
\(370\) 0 0
\(371\) 16.0460 + 2.48424i 0.833069 + 0.128975i
\(372\) 0 0
\(373\) −22.0289 −1.14061 −0.570307 0.821432i \(-0.693176\pi\)
−0.570307 + 0.821432i \(0.693176\pi\)
\(374\) 0 0
\(375\) −8.98742 + 18.6626i −0.464108 + 0.963731i
\(376\) 0 0
\(377\) −28.4408 6.49143i −1.46478 0.334325i
\(378\) 0 0
\(379\) 15.0539 3.43596i 0.773267 0.176493i 0.182359 0.983232i \(-0.441627\pi\)
0.590908 + 0.806739i \(0.298770\pi\)
\(380\) 0 0
\(381\) −25.4571 + 5.81041i −1.30420 + 0.297676i
\(382\) 0 0
\(383\) 1.04751 4.58942i 0.0535251 0.234509i −0.941089 0.338160i \(-0.890195\pi\)
0.994614 + 0.103651i \(0.0330526\pi\)
\(384\) 0 0
\(385\) −4.04116 + 1.22885i −0.205957 + 0.0626280i
\(386\) 0 0
\(387\) 5.52241 4.40397i 0.280720 0.223867i
\(388\) 0 0
\(389\) −7.63767 9.57733i −0.387245 0.485590i 0.549554 0.835458i \(-0.314798\pi\)
−0.936799 + 0.349868i \(0.886226\pi\)
\(390\) 0 0
\(391\) 16.3319 0.825941
\(392\) 0 0
\(393\) −20.2931 −1.02365
\(394\) 0 0
\(395\) 7.27926 + 9.12790i 0.366259 + 0.459275i
\(396\) 0 0
\(397\) −13.5331 + 10.7923i −0.679205 + 0.541648i −0.901202 0.433399i \(-0.857314\pi\)
0.221997 + 0.975047i \(0.428743\pi\)
\(398\) 0 0
\(399\) −0.789755 11.1362i −0.0395372 0.557505i
\(400\) 0 0
\(401\) −2.64332 + 11.5811i −0.132001 + 0.578334i 0.865056 + 0.501675i \(0.167283\pi\)
−0.997057 + 0.0766593i \(0.975575\pi\)
\(402\) 0 0
\(403\) −50.5977 + 11.5486i −2.52045 + 0.575277i
\(404\) 0 0
\(405\) −12.3825 + 2.82622i −0.615291 + 0.140436i
\(406\) 0 0
\(407\) 4.29823 + 0.981043i 0.213055 + 0.0486285i
\(408\) 0 0
\(409\) −9.21686 + 19.1390i −0.455744 + 0.946363i 0.538838 + 0.842410i \(0.318864\pi\)
−0.994582 + 0.103953i \(0.966851\pi\)
\(410\) 0 0
\(411\) 22.7756 1.12344
\(412\) 0 0
\(413\) −7.10712 + 2.16116i −0.349719 + 0.106344i
\(414\) 0 0
\(415\) −4.73041 9.82280i −0.232207 0.482182i
\(416\) 0 0
\(417\) −5.29630 + 23.2046i −0.259361 + 1.13633i
\(418\) 0 0
\(419\) 10.1126 4.86999i 0.494034 0.237914i −0.170241 0.985402i \(-0.554455\pi\)
0.664275 + 0.747488i \(0.268740\pi\)
\(420\) 0 0
\(421\) 12.2597 + 5.90397i 0.597502 + 0.287742i 0.708087 0.706125i \(-0.249558\pi\)
−0.110586 + 0.993867i \(0.535273\pi\)
\(422\) 0 0
\(423\) −17.5350 −0.852581
\(424\) 0 0
\(425\) −22.5682 17.9975i −1.09472 0.873008i
\(426\) 0 0
\(427\) 12.3730 7.07770i 0.598770 0.342514i
\(428\) 0 0
\(429\) 14.2273 + 6.85152i 0.686901 + 0.330794i
\(430\) 0 0
\(431\) −1.95681 1.56050i −0.0942562 0.0751668i 0.575228 0.817993i \(-0.304913\pi\)
−0.669484 + 0.742826i \(0.733485\pi\)
\(432\) 0 0
\(433\) 29.0486 23.1655i 1.39599 1.11326i 0.417102 0.908860i \(-0.363046\pi\)
0.978887 0.204404i \(-0.0655257\pi\)
\(434\) 0 0
\(435\) 10.1870 + 8.12386i 0.488429 + 0.389509i
\(436\) 0 0
\(437\) −1.83524 3.81091i −0.0877913 0.182301i
\(438\) 0 0
\(439\) −8.98456 39.3639i −0.428810 1.87874i −0.475308 0.879819i \(-0.657664\pi\)
0.0464989 0.998918i \(-0.485194\pi\)
\(440\) 0 0
\(441\) −9.82598 + 1.40072i −0.467904 + 0.0667012i
\(442\) 0 0
\(443\) −32.3401 + 7.38141i −1.53652 + 0.350702i −0.905257 0.424864i \(-0.860322\pi\)
−0.631267 + 0.775566i \(0.717465\pi\)
\(444\) 0 0
\(445\) 14.1972 6.83701i 0.673012 0.324105i
\(446\) 0 0
\(447\) 15.5780 19.5342i 0.736814 0.923936i
\(448\) 0 0
\(449\) 10.7080 + 13.4274i 0.505340 + 0.633677i 0.967425 0.253159i \(-0.0814697\pi\)
−0.462084 + 0.886836i \(0.652898\pi\)
\(450\) 0 0
\(451\) −0.767887 + 0.962900i −0.0361584 + 0.0453412i
\(452\) 0 0
\(453\) −0.273336 + 0.567588i −0.0128424 + 0.0266676i
\(454\) 0 0
\(455\) 15.2011 4.62241i 0.712639 0.216702i
\(456\) 0 0
\(457\) 22.5001 28.2142i 1.05251 1.31980i 0.106982 0.994261i \(-0.465881\pi\)
0.945527 0.325544i \(-0.105547\pi\)
\(458\) 0 0
\(459\) 25.7767i 1.20315i
\(460\) 0 0
\(461\) −7.28244 + 15.1222i −0.339177 + 0.704309i −0.998885 0.0472136i \(-0.984966\pi\)
0.659708 + 0.751522i \(0.270680\pi\)
\(462\) 0 0
\(463\) −1.38682 2.87977i −0.0644511 0.133834i 0.866255 0.499602i \(-0.166521\pi\)
−0.930706 + 0.365768i \(0.880806\pi\)
\(464\) 0 0
\(465\) 22.5993 + 5.15814i 1.04802 + 0.239203i
\(466\) 0 0
\(467\) 1.30490 0.628409i 0.0603838 0.0290793i −0.403448 0.915002i \(-0.632188\pi\)
0.463832 + 0.885923i \(0.346474\pi\)
\(468\) 0 0
\(469\) −9.42086 + 13.6998i −0.435015 + 0.632598i
\(470\) 0 0
\(471\) 3.73964i 0.172313i
\(472\) 0 0
\(473\) −6.34303 3.05464i −0.291653 0.140453i
\(474\) 0 0
\(475\) −1.66355 + 7.28849i −0.0763289 + 0.334419i
\(476\) 0 0
\(477\) 1.93633 + 8.48362i 0.0886585 + 0.388438i
\(478\) 0 0
\(479\) 5.39019 + 23.6160i 0.246284 + 1.07904i 0.935178 + 0.354179i \(0.115239\pi\)
−0.688894 + 0.724862i \(0.741903\pi\)
\(480\) 0 0
\(481\) −16.1681 3.69026i −0.737201 0.168261i
\(482\) 0 0
\(483\) −10.1704 + 5.81776i −0.462769 + 0.264717i
\(484\) 0 0
\(485\) −3.44729 4.32277i −0.156534 0.196287i
\(486\) 0 0
\(487\) 33.2132 26.4866i 1.50503 1.20022i 0.583383 0.812197i \(-0.301729\pi\)
0.921649 0.388025i \(-0.126843\pi\)
\(488\) 0 0
\(489\) 7.22539i 0.326743i
\(490\) 0 0
\(491\) 8.14568i 0.367610i 0.982963 + 0.183805i \(0.0588414\pi\)
−0.982963 + 0.183805i \(0.941159\pi\)
\(492\) 0 0
\(493\) −33.2569 + 26.5215i −1.49781 + 1.19447i
\(494\) 0 0
\(495\) −1.41136 1.76979i −0.0634358 0.0795460i
\(496\) 0 0
\(497\) 1.26927 8.19841i 0.0569347 0.367749i
\(498\) 0 0
\(499\) 28.6155 + 6.53130i 1.28101 + 0.292381i 0.808264 0.588820i \(-0.200407\pi\)
0.472742 + 0.881201i \(0.343264\pi\)
\(500\) 0 0
\(501\) 3.56708 + 15.6284i 0.159366 + 0.698226i
\(502\) 0 0
\(503\) −3.94965 17.3046i −0.176106 0.771572i −0.983404 0.181428i \(-0.941928\pi\)
0.807298 0.590144i \(-0.200929\pi\)
\(504\) 0 0
\(505\) 4.15899 18.2217i 0.185072 0.810855i
\(506\) 0 0
\(507\) −28.8986 13.9168i −1.28343 0.618067i
\(508\) 0 0
\(509\) 2.45317i 0.108735i −0.998521 0.0543674i \(-0.982686\pi\)
0.998521 0.0543674i \(-0.0173142\pi\)
\(510\) 0 0
\(511\) 23.6017 + 25.6415i 1.04408 + 1.13431i
\(512\) 0 0
\(513\) −6.01476 + 2.89656i −0.265558 + 0.127886i
\(514\) 0 0
\(515\) −5.82257 1.32896i −0.256573 0.0585612i
\(516\) 0 0
\(517\) 7.58317 + 15.7466i 0.333508 + 0.692536i
\(518\) 0 0
\(519\) −15.0291 + 31.2082i −0.659704 + 1.36989i
\(520\) 0 0
\(521\) 40.1331i 1.75827i −0.476577 0.879133i \(-0.658123\pi\)
0.476577 0.879133i \(-0.341877\pi\)
\(522\) 0 0
\(523\) 14.6480 18.3680i 0.640511 0.803175i −0.350556 0.936542i \(-0.614007\pi\)
0.991067 + 0.133367i \(0.0425788\pi\)
\(524\) 0 0
\(525\) 20.4650 + 3.16837i 0.893165 + 0.138279i
\(526\) 0 0
\(527\) −32.8345 + 68.1816i −1.43029 + 2.97004i
\(528\) 0 0
\(529\) 11.5725 14.5114i 0.503151 0.630932i
\(530\) 0 0
\(531\) −2.48213 3.11250i −0.107715 0.135071i
\(532\) 0 0
\(533\) 2.88846 3.62201i 0.125113 0.156887i
\(534\) 0 0
\(535\) 12.5756 6.05608i 0.543690 0.261827i
\(536\) 0 0
\(537\) 30.9550 7.06528i 1.33581 0.304889i
\(538\) 0 0
\(539\) 5.50720 + 8.21808i 0.237212 + 0.353978i
\(540\) 0 0
\(541\) 0.919204 + 4.02730i 0.0395197 + 0.173147i 0.990836 0.135068i \(-0.0431254\pi\)
−0.951317 + 0.308215i \(0.900268\pi\)
\(542\) 0 0
\(543\) 15.9564 + 33.1339i 0.684757 + 1.42191i
\(544\) 0 0
\(545\) −3.93096 3.13484i −0.168384 0.134282i
\(546\) 0 0
\(547\) −2.03753 + 1.62488i −0.0871185 + 0.0694747i −0.666081 0.745880i \(-0.732029\pi\)
0.578962 + 0.815354i \(0.303458\pi\)
\(548\) 0 0
\(549\) 5.97249 + 4.76290i 0.254900 + 0.203276i
\(550\) 0 0
\(551\) 9.92566 + 4.77995i 0.422847 + 0.203633i
\(552\) 0 0
\(553\) 15.4937 22.5310i 0.658861 0.958114i
\(554\) 0 0
\(555\) 5.79113 + 4.61827i 0.245820 + 0.196035i
\(556\) 0 0
\(557\) 32.6655 1.38408 0.692040 0.721859i \(-0.256712\pi\)
0.692040 + 0.721859i \(0.256712\pi\)
\(558\) 0 0
\(559\) 23.8597 + 11.4902i 1.00916 + 0.485986i
\(560\) 0 0
\(561\) 20.7454 9.99046i 0.875871 0.421797i
\(562\) 0 0
\(563\) −9.32661 + 40.8626i −0.393070 + 1.72215i 0.260666 + 0.965429i \(0.416058\pi\)
−0.653736 + 0.756723i \(0.726799\pi\)
\(564\) 0 0
\(565\) −1.22060 2.53461i −0.0513512 0.106632i
\(566\) 0 0
\(567\) 14.7703 + 25.8208i 0.620293 + 1.08437i
\(568\) 0 0
\(569\) 4.88304 0.204708 0.102354 0.994748i \(-0.467363\pi\)
0.102354 + 0.994748i \(0.467363\pi\)
\(570\) 0 0
\(571\) −3.54974 + 7.37110i −0.148552 + 0.308471i −0.961945 0.273243i \(-0.911904\pi\)
0.813393 + 0.581714i \(0.197618\pi\)
\(572\) 0 0
\(573\) 1.00686 + 0.229810i 0.0420623 + 0.00960046i
\(574\) 0 0
\(575\) 7.64930 1.74590i 0.318998 0.0728091i
\(576\) 0 0
\(577\) −11.9450 + 2.72637i −0.497278 + 0.113500i −0.463802 0.885939i \(-0.653515\pi\)
−0.0334761 + 0.999440i \(0.510658\pi\)
\(578\) 0 0
\(579\) 2.70789 11.8640i 0.112536 0.493053i
\(580\) 0 0
\(581\) −18.7876 + 17.2930i −0.779442 + 0.717435i
\(582\) 0 0
\(583\) 6.78100 5.40767i 0.280840 0.223963i
\(584\) 0 0
\(585\) 5.30892 + 6.65717i 0.219497 + 0.275240i
\(586\) 0 0
\(587\) −40.7289 −1.68106 −0.840532 0.541762i \(-0.817757\pi\)
−0.840532 + 0.541762i \(0.817757\pi\)
\(588\) 0 0
\(589\) 19.5992 0.807572
\(590\) 0 0
\(591\) −24.1018 30.2227i −0.991416 1.24320i
\(592\) 0 0
\(593\) −3.10605 + 2.47699i −0.127550 + 0.101718i −0.685187 0.728367i \(-0.740280\pi\)
0.557637 + 0.830085i \(0.311708\pi\)
\(594\) 0 0
\(595\) 8.55022 21.5319i 0.350525 0.882723i
\(596\) 0 0
\(597\) −10.2718 + 45.0036i −0.420396 + 1.84187i
\(598\) 0 0
\(599\) 2.26110 0.516081i 0.0923860 0.0210865i −0.176078 0.984376i \(-0.556341\pi\)
0.268464 + 0.963290i \(0.413484\pi\)
\(600\) 0 0
\(601\) 31.5657 7.20466i 1.28759 0.293884i 0.476692 0.879070i \(-0.341836\pi\)
0.810899 + 0.585186i \(0.198979\pi\)
\(602\) 0 0
\(603\) −8.68697 1.98274i −0.353761 0.0807435i
\(604\) 0 0
\(605\) 4.41256 9.16278i 0.179396 0.372520i
\(606\) 0 0
\(607\) 2.78007 0.112839 0.0564197 0.998407i \(-0.482032\pi\)
0.0564197 + 0.998407i \(0.482032\pi\)
\(608\) 0 0
\(609\) 11.2626 28.3625i 0.456383 1.14931i
\(610\) 0 0
\(611\) −28.5246 59.2320i −1.15398 2.39627i
\(612\) 0 0
\(613\) 1.49305 6.54148i 0.0603037 0.264208i −0.935784 0.352573i \(-0.885307\pi\)
0.996088 + 0.0883647i \(0.0281641\pi\)
\(614\) 0 0
\(615\) −1.86428 + 0.897789i −0.0751750 + 0.0362024i
\(616\) 0 0
\(617\) −36.3188 17.4902i −1.46214 0.704130i −0.477485 0.878640i \(-0.658452\pi\)
−0.984655 + 0.174510i \(0.944166\pi\)
\(618\) 0 0
\(619\) 23.8051 0.956807 0.478404 0.878140i \(-0.341216\pi\)
0.478404 + 0.878140i \(0.341216\pi\)
\(620\) 0 0
\(621\) 5.47780 + 4.36840i 0.219816 + 0.175298i
\(622\) 0 0
\(623\) −24.9941 27.1543i −1.00137 1.08792i
\(624\) 0 0
\(625\) −6.74544 3.24843i −0.269818 0.129937i
\(626\) 0 0
\(627\) −4.66237 3.71812i −0.186197 0.148487i
\(628\) 0 0
\(629\) −18.9059 + 15.0770i −0.753829 + 0.601158i
\(630\) 0 0
\(631\) 13.9141 + 11.0961i 0.553912 + 0.441730i 0.860015 0.510268i \(-0.170454\pi\)
−0.306104 + 0.951998i \(0.599025\pi\)
\(632\) 0 0
\(633\) −22.7876 47.3190i −0.905728 1.88076i
\(634\) 0 0
\(635\) 3.12278 + 13.6818i 0.123924 + 0.542946i
\(636\) 0 0
\(637\) −20.7157 30.9129i −0.820786 1.22481i
\(638\) 0 0
\(639\) 4.33454 0.989331i 0.171472 0.0391373i
\(640\) 0 0
\(641\) 23.7689 11.4465i 0.938815 0.452109i 0.0990637 0.995081i \(-0.468415\pi\)
0.839751 + 0.542972i \(0.182701\pi\)
\(642\) 0 0
\(643\) −25.3736 + 31.8175i −1.00064 + 1.25476i −0.0337830 + 0.999429i \(0.510756\pi\)
−0.966854 + 0.255330i \(0.917816\pi\)
\(644\) 0 0
\(645\) −7.37479 9.24769i −0.290382 0.364127i
\(646\) 0 0
\(647\) 22.9231 28.7447i 0.901200 1.13007i −0.0897665 0.995963i \(-0.528612\pi\)
0.990967 0.134107i \(-0.0428165\pi\)
\(648\) 0 0
\(649\) −1.72163 + 3.57501i −0.0675800 + 0.140331i
\(650\) 0 0
\(651\) −3.84057 54.1551i −0.150524 2.12250i
\(652\) 0 0
\(653\) 27.3146 34.2514i 1.06890 1.34036i 0.131812 0.991275i \(-0.457920\pi\)
0.937090 0.349087i \(-0.113508\pi\)
\(654\) 0 0
\(655\) 10.9065i 0.426151i
\(656\) 0 0
\(657\) −8.10353 + 16.8271i −0.316149 + 0.656490i
\(658\) 0 0
\(659\) −7.57366 15.7269i −0.295028 0.612632i 0.699785 0.714354i \(-0.253279\pi\)
−0.994813 + 0.101722i \(0.967565\pi\)
\(660\) 0 0
\(661\) 20.6580 + 4.71504i 0.803501 + 0.183394i 0.604500 0.796605i \(-0.293373\pi\)
0.199001 + 0.979999i \(0.436230\pi\)
\(662\) 0 0
\(663\) −78.0352 + 37.5798i −3.03064 + 1.45948i
\(664\) 0 0
\(665\) −5.98508 + 0.424451i −0.232092 + 0.0164595i
\(666\) 0 0
\(667\) 11.5620i 0.447683i
\(668\) 0 0
\(669\) −32.5755 15.6876i −1.25944 0.606516i
\(670\) 0 0
\(671\) 1.69428 7.42312i 0.0654069 0.286566i
\(672\) 0 0
\(673\) 5.41507 + 23.7250i 0.208736 + 0.914531i 0.965410 + 0.260738i \(0.0839659\pi\)
−0.756674 + 0.653792i \(0.773177\pi\)
\(674\) 0 0
\(675\) −2.75556 12.0729i −0.106061 0.464686i
\(676\) 0 0
\(677\) 32.1885 + 7.34681i 1.23710 + 0.282361i 0.790551 0.612397i \(-0.209794\pi\)
0.446552 + 0.894757i \(0.352652\pi\)
\(678\) 0 0
\(679\) −7.33749 + 10.6702i −0.281587 + 0.409483i
\(680\) 0 0
\(681\) 1.00603 + 1.26153i 0.0385513 + 0.0483418i
\(682\) 0 0
\(683\) −22.4240 + 17.8825i −0.858029 + 0.684255i −0.950252 0.311483i \(-0.899174\pi\)
0.0922227 + 0.995738i \(0.470603\pi\)
\(684\) 0 0
\(685\) 12.2407i 0.467692i
\(686\) 0 0
\(687\) 38.8560i 1.48245i
\(688\) 0 0
\(689\) −25.5072 + 20.3413i −0.971747 + 0.774942i
\(690\) 0 0
\(691\) 0.792354 + 0.993581i 0.0301426 + 0.0377976i 0.796674 0.604409i \(-0.206591\pi\)
−0.766532 + 0.642206i \(0.778019\pi\)
\(692\) 0 0
\(693\) −3.00404 + 4.36847i −0.114114 + 0.165945i
\(694\) 0 0
\(695\) 12.4712 + 2.84648i 0.473061 + 0.107973i
\(696\) 0 0
\(697\) −1.50316 6.58580i −0.0569364 0.249455i
\(698\) 0 0
\(699\) 7.61392 + 33.3587i 0.287985 + 1.26174i
\(700\) 0 0
\(701\) −8.34281 + 36.5522i −0.315104 + 1.38056i 0.530924 + 0.847420i \(0.321845\pi\)
−0.846027 + 0.533140i \(0.821012\pi\)
\(702\) 0 0
\(703\) 5.64256 + 2.71731i 0.212813 + 0.102485i
\(704\) 0 0
\(705\) 29.3637i 1.10590i
\(706\) 0 0
\(707\) −43.6650 + 3.09664i −1.64219 + 0.116461i
\(708\) 0 0
\(709\) 9.97338 4.80293i 0.374558 0.180378i −0.237128 0.971479i \(-0.576206\pi\)
0.611686 + 0.791101i \(0.290492\pi\)
\(710\) 0 0
\(711\) 14.2868 + 3.26086i 0.535795 + 0.122292i
\(712\) 0 0
\(713\) −8.92475 18.5324i −0.334235 0.694045i
\(714\) 0 0
\(715\) 3.68232 7.64642i 0.137711 0.285960i
\(716\) 0 0
\(717\) 14.9594i 0.558669i
\(718\) 0 0
\(719\) −17.2653 + 21.6501i −0.643889 + 0.807411i −0.991483 0.130234i \(-0.958427\pi\)
0.347594 + 0.937645i \(0.386999\pi\)
\(720\) 0 0
\(721\) 0.989501 + 13.9527i 0.0368509 + 0.519627i
\(722\) 0 0
\(723\) 0.542072 1.12562i 0.0201599 0.0418624i
\(724\) 0 0
\(725\) −12.7412 + 15.9769i −0.473195 + 0.593367i
\(726\) 0 0
\(727\) −19.4938 24.4445i −0.722987 0.906597i 0.275516 0.961296i \(-0.411151\pi\)
−0.998503 + 0.0546997i \(0.982580\pi\)
\(728\) 0 0
\(729\) 3.13416 3.93011i 0.116080 0.145560i
\(730\) 0 0
\(731\) 34.7908 16.7544i 1.28678 0.619683i
\(732\) 0 0
\(733\) −21.0375 + 4.80167i −0.777037 + 0.177354i −0.592607 0.805492i \(-0.701901\pi\)
−0.184430 + 0.982846i \(0.559044\pi\)
\(734\) 0 0
\(735\) 2.34562 + 16.4544i 0.0865195 + 0.606928i
\(736\) 0 0
\(737\) 1.97623 + 8.65844i 0.0727955 + 0.318938i
\(738\) 0 0
\(739\) −15.9905 33.2047i −0.588221 1.22145i −0.956499 0.291737i \(-0.905767\pi\)
0.368278 0.929716i \(-0.379948\pi\)
\(740\) 0 0
\(741\) 17.5378 + 13.9859i 0.644268 + 0.513787i
\(742\) 0 0
\(743\) 29.5773 23.5871i 1.08509 0.865328i 0.0936102 0.995609i \(-0.470159\pi\)
0.991477 + 0.130281i \(0.0415878\pi\)
\(744\) 0 0
\(745\) −10.4986 8.37234i −0.384638 0.306739i
\(746\) 0 0
\(747\) −12.3293 5.93748i −0.451106 0.217241i
\(748\) 0 0
\(749\) −22.1393 24.0527i −0.808952 0.878868i
\(750\) 0 0
\(751\) −9.74296 7.76975i −0.355526 0.283522i 0.429397 0.903116i \(-0.358726\pi\)
−0.784923 + 0.619593i \(0.787298\pi\)
\(752\) 0 0
\(753\) −30.2528 −1.10247
\(754\) 0 0
\(755\) 0.305048 + 0.146903i 0.0111018 + 0.00534636i
\(756\) 0 0
\(757\) −18.2906 + 8.80828i −0.664783 + 0.320142i −0.735668 0.677342i \(-0.763132\pi\)
0.0708857 + 0.997484i \(0.477417\pi\)
\(758\) 0 0
\(759\) −1.39267 + 6.10169i −0.0505507 + 0.221477i
\(760\) 0 0
\(761\) −7.74795 16.0888i −0.280863 0.583218i 0.712041 0.702138i \(-0.247771\pi\)
−0.992904 + 0.118920i \(0.962057\pi\)
\(762\) 0 0
\(763\) −4.34601 + 10.9445i −0.157336 + 0.396218i
\(764\) 0 0
\(765\) 12.4158 0.448895
\(766\) 0 0
\(767\) 6.47604 13.4476i 0.233836 0.485566i
\(768\) 0 0
\(769\) −8.18332 1.86779i −0.295098 0.0673542i 0.0724088 0.997375i \(-0.476931\pi\)
−0.367507 + 0.930021i \(0.619789\pi\)
\(770\) 0 0
\(771\) −36.2811 + 8.28092i −1.30663 + 0.298230i
\(772\) 0 0
\(773\) 37.9293 8.65712i 1.36422 0.311375i 0.523124 0.852257i \(-0.324767\pi\)
0.841099 + 0.540882i \(0.181909\pi\)
\(774\) 0 0
\(775\) −8.08984 + 35.4439i −0.290595 + 1.27318i
\(776\) 0 0
\(777\) 6.40258 16.1236i 0.229691 0.578430i
\(778\) 0 0
\(779\) −1.36782 + 1.09080i −0.0490074 + 0.0390821i
\(780\) 0 0
\(781\) −2.76294 3.46462i −0.0988658 0.123974i
\(782\) 0 0
\(783\) −18.2483 −0.652142
\(784\) 0 0
\(785\) −2.00985 −0.0717347
\(786\) 0 0
\(787\) −3.49660 4.38460i −0.124640 0.156294i 0.715596 0.698514i \(-0.246155\pi\)
−0.840236 + 0.542220i \(0.817584\pi\)
\(788\) 0 0
\(789\) 2.68322 2.13979i 0.0955251 0.0761787i
\(790\) 0 0
\(791\) −4.84783 + 4.46218i −0.172369 + 0.158657i
\(792\) 0 0
\(793\) −6.37315 + 27.9226i −0.226317 + 0.991560i
\(794\) 0 0
\(795\) 14.2065 3.24253i 0.503851 0.115001i
\(796\) 0 0
\(797\) 28.7879 6.57065i 1.01972 0.232744i 0.320194 0.947352i \(-0.396252\pi\)
0.699526 + 0.714608i \(0.253395\pi\)
\(798\) 0 0
\(799\) −93.4583 21.3312i −3.30632 0.754645i
\(800\) 0 0
\(801\) 8.58163 17.8199i 0.303217 0.629636i
\(802\) 0 0
\(803\) 18.6154 0.656923
\(804\) 0 0
\(805\) 3.12673 + 5.46604i 0.110203 + 0.192653i
\(806\) 0 0
\(807\) −2.21443 4.59831i −0.0779517 0.161868i
\(808\) 0 0
\(809\) 6.60772 28.9503i 0.232315 1.01784i −0.715399 0.698716i \(-0.753755\pi\)
0.947714 0.319122i \(-0.103388\pi\)
\(810\) 0 0
\(811\) −11.6795 + 5.62456i −0.410123 + 0.197505i −0.627558 0.778570i \(-0.715945\pi\)
0.217435 + 0.976075i \(0.430231\pi\)
\(812\) 0 0
\(813\) −2.36275 1.13784i −0.0828654 0.0399059i
\(814\) 0 0
\(815\) −3.88326 −0.136025
\(816\) 0 0
\(817\) −7.81897 6.23542i −0.273551 0.218150i
\(818\) 0 0
\(819\) 11.2999 16.4323i 0.394851 0.574191i
\(820\) 0 0
\(821\) 27.6957 + 13.3375i 0.966586 + 0.465483i 0.849471 0.527635i \(-0.176921\pi\)
0.117115 + 0.993118i \(0.462635\pi\)
\(822\) 0 0
\(823\) −2.34299 1.86847i −0.0816716 0.0651309i 0.581803 0.813330i \(-0.302347\pi\)
−0.663475 + 0.748199i \(0.730919\pi\)
\(824\) 0 0
\(825\) 8.64842 6.89689i 0.301099 0.240119i
\(826\) 0 0
\(827\) −24.7832 19.7639i −0.861796 0.687260i 0.0893499 0.996000i \(-0.471521\pi\)
−0.951146 + 0.308741i \(0.900092\pi\)
\(828\) 0 0
\(829\) 2.41687 + 5.01868i 0.0839413 + 0.174306i 0.938714 0.344696i \(-0.112018\pi\)
−0.854773 + 0.519002i \(0.826304\pi\)
\(830\) 0 0
\(831\) −12.1551 53.2549i −0.421655 1.84739i
\(832\) 0 0
\(833\) −54.0746 4.48766i −1.87357 0.155488i
\(834\) 0 0
\(835\) 8.39944 1.91712i 0.290674 0.0663445i
\(836\) 0 0
\(837\) −29.2498 + 14.0859i −1.01102 + 0.486881i
\(838\) 0 0
\(839\) 32.7727 41.0956i 1.13144 1.41878i 0.237045 0.971499i \(-0.423821\pi\)
0.894394 0.447281i \(-0.147607\pi\)
\(840\) 0 0
\(841\) 0.694385 + 0.870731i 0.0239443 + 0.0300252i
\(842\) 0 0
\(843\) −13.3604 + 16.7534i −0.460155 + 0.577016i
\(844\) 0 0
\(845\) −7.47954 + 15.5314i −0.257304 + 0.534297i
\(846\) 0 0
\(847\) −23.5386 3.64423i −0.808795 0.125217i
\(848\) 0 0
\(849\) 14.1431 17.7349i 0.485390 0.608660i
\(850\) 0 0
\(851\) 6.57280i 0.225313i
\(852\) 0 0
\(853\) 5.94410 12.3430i 0.203522 0.422618i −0.774078 0.633091i \(-0.781786\pi\)
0.977600 + 0.210473i \(0.0675003\pi\)
\(854\) 0 0
\(855\) −1.39518 2.89712i −0.0477142 0.0990795i
\(856\) 0 0
\(857\) 45.8488 + 10.4647i 1.56616 + 0.357467i 0.915634 0.402013i \(-0.131689\pi\)
0.650531 + 0.759480i \(0.274547\pi\)
\(858\) 0 0
\(859\) 16.7777 8.07973i 0.572449 0.275677i −0.125182 0.992134i \(-0.539952\pi\)
0.697631 + 0.716457i \(0.254237\pi\)
\(860\) 0 0
\(861\) 3.28206 + 3.56572i 0.111852 + 0.121519i
\(862\) 0 0
\(863\) 20.5249i 0.698676i −0.936997 0.349338i \(-0.886406\pi\)
0.936997 0.349338i \(-0.113594\pi\)
\(864\) 0 0
\(865\) 16.7727 + 8.07733i 0.570290 + 0.274637i
\(866\) 0 0
\(867\) −20.1518 + 88.2906i −0.684390 + 2.99851i
\(868\) 0 0
\(869\) −3.25015 14.2398i −0.110254 0.483054i
\(870\) 0 0
\(871\) −7.43373 32.5693i −0.251882 1.10357i
\(872\) 0 0
\(873\) −6.76589 1.54427i −0.228991 0.0522656i
\(874\) 0 0
\(875\) 3.98919 25.7667i 0.134859 0.871074i
\(876\) 0 0
\(877\) −30.4794 38.2199i −1.02922 1.29060i −0.956027 0.293277i \(-0.905254\pi\)
−0.0731879 0.997318i \(-0.523317\pi\)
\(878\) 0 0
\(879\) 0.718753 0.573186i 0.0242429 0.0193331i
\(880\) 0 0
\(881\) 34.7822i 1.17184i −0.810369 0.585920i \(-0.800733\pi\)
0.810369 0.585920i \(-0.199267\pi\)
\(882\) 0 0
\(883\) 32.3737i 1.08946i 0.838611 + 0.544730i \(0.183368\pi\)
−0.838611 + 0.544730i \(0.816632\pi\)
\(884\) 0 0
\(885\) −5.21211 + 4.15652i −0.175203 + 0.139720i
\(886\) 0 0
\(887\) 25.6918 + 32.2165i 0.862646 + 1.08172i 0.995884 + 0.0906399i \(0.0288912\pi\)
−0.133238 + 0.991084i \(0.542537\pi\)
\(888\) 0 0
\(889\) 28.5303 16.3201i 0.956874 0.547360i
\(890\) 0 0
\(891\) 15.4911 + 3.53575i 0.518972 + 0.118452i
\(892\) 0 0
\(893\) 5.52456 + 24.2047i 0.184872 + 0.809979i
\(894\) 0 0
\(895\) −3.79721 16.6366i −0.126927 0.556102i
\(896\) 0 0
\(897\) 5.23863 22.9519i 0.174913 0.766342i
\(898\) 0 0
\(899\) 48.2684 + 23.2449i 1.60984 + 0.775259i
\(900\) 0 0
\(901\) 47.5716i 1.58484i
\(902\) 0 0
\(903\) −15.6971 + 22.8266i −0.522366 + 0.759623i
\(904\) 0 0
\(905\) 17.8077 8.57573i 0.591948 0.285067i
\(906\) 0 0
\(907\) 15.6377 + 3.56920i 0.519241 + 0.118513i 0.474106 0.880468i \(-0.342771\pi\)
0.0451346 + 0.998981i \(0.485628\pi\)
\(908\) 0 0
\(909\) −10.1787 21.1363i −0.337607 0.701048i
\(910\) 0 0
\(911\) 5.87288 12.1952i 0.194577 0.404044i −0.780739 0.624858i \(-0.785157\pi\)
0.975316 + 0.220814i \(0.0708713\pi\)
\(912\) 0 0
\(913\) 13.6396i 0.451404i
\(914\) 0 0
\(915\) 7.97584 10.0014i 0.263673 0.330636i
\(916\) 0 0
\(917\) 24.4391 7.43154i 0.807051 0.245411i
\(918\) 0 0
\(919\) −11.5941 + 24.0755i −0.382455 + 0.794176i 0.617516 + 0.786559i \(0.288139\pi\)
−0.999971 + 0.00761797i \(0.997575\pi\)
\(920\) 0 0
\(921\) −38.6641 + 48.4832i −1.27403 + 1.59758i
\(922\) 0 0
\(923\) 10.3930 + 13.0324i 0.342089 + 0.428967i
\(924\) 0 0
\(925\) −7.24312 + 9.08259i −0.238152 + 0.298634i
\(926\) 0 0
\(927\) −6.75392 + 3.25252i −0.221828 + 0.106827i
\(928\) 0 0
\(929\) −7.85528 + 1.79292i −0.257723 + 0.0588237i −0.349430 0.936962i \(-0.613625\pi\)
0.0917069 + 0.995786i \(0.470768\pi\)
\(930\) 0 0
\(931\) 5.02927 + 13.1221i 0.164828 + 0.430060i
\(932\) 0 0
\(933\) 10.5767 + 46.3395i 0.346265 + 1.51709i
\(934\) 0 0
\(935\) −5.36933 11.1495i −0.175596 0.364629i
\(936\) 0 0
\(937\) −15.3113 12.2103i −0.500197 0.398894i 0.340630 0.940197i \(-0.389360\pi\)
−0.840827 + 0.541304i \(0.817931\pi\)
\(938\) 0 0
\(939\) −14.7610 + 11.7715i −0.481707 + 0.384148i
\(940\) 0 0
\(941\) 40.7957 + 32.5335i 1.32990 + 1.06056i 0.992887 + 0.119060i \(0.0379882\pi\)
0.337013 + 0.941500i \(0.390583\pi\)
\(942\) 0 0
\(943\) 1.65429 + 0.796663i 0.0538710 + 0.0259429i
\(944\) 0 0
\(945\) 8.62705 4.93492i 0.280638 0.160533i
\(946\) 0 0
\(947\) 9.84969 + 7.85486i 0.320072 + 0.255249i 0.770324 0.637653i \(-0.220094\pi\)
−0.450252 + 0.892901i \(0.648666\pi\)
\(948\) 0 0
\(949\) −70.0231 −2.27305
\(950\) 0 0
\(951\) 36.7968 + 17.7204i 1.19322 + 0.574624i
\(952\) 0 0
\(953\) −21.2346 + 10.2260i −0.687857 + 0.331254i −0.744955 0.667114i \(-0.767529\pi\)
0.0570989 + 0.998369i \(0.481815\pi\)
\(954\) 0 0
\(955\) 0.123511 0.541135i 0.00399671 0.0175107i
\(956\) 0 0
\(957\) −7.07264 14.6865i −0.228626 0.474747i
\(958\) 0 0
\(959\) −27.4288 + 8.34065i −0.885722 + 0.269334i
\(960\) 0 0
\(961\) 64.3109 2.07455
\(962\) 0 0
\(963\) 7.60143 15.7845i 0.244953 0.508649i
\(964\) 0 0
\(965\) −6.37629 1.45535i −0.205260 0.0468492i
\(966\) 0 0
\(967\) 52.8548 12.0638i 1.69970 0.387945i 0.740805 0.671720i \(-0.234444\pi\)
0.958891 + 0.283775i \(0.0915870\pi\)
\(968\) 0 0
\(969\) 31.8885 7.27834i 1.02441 0.233814i
\(970\) 0 0
\(971\) −4.22785 + 18.5234i −0.135678 + 0.594445i 0.860678 + 0.509150i \(0.170040\pi\)
−0.996356 + 0.0852945i \(0.972817\pi\)
\(972\) 0 0
\(973\) −2.11939 29.8850i −0.0679444 0.958069i
\(974\) 0 0
\(975\) −32.5316 + 25.9431i −1.04185 + 0.830844i
\(976\) 0 0
\(977\) −29.5013 36.9934i −0.943830 1.18353i −0.982872 0.184288i \(-0.941002\pi\)
0.0390426 0.999238i \(-0.487569\pi\)
\(978\) 0 0
\(979\) −19.7137 −0.630052
\(980\) 0 0
\(981\) −6.31087 −0.201490
\(982\) 0 0
\(983\) 18.3534 + 23.0145i 0.585383 + 0.734047i 0.983021 0.183495i \(-0.0587412\pi\)
−0.397637 + 0.917543i \(0.630170\pi\)
\(984\) 0 0
\(985\) −16.2431 + 12.9534i −0.517548 + 0.412730i
\(986\) 0 0
\(987\) 65.7980 20.0081i 2.09437 0.636864i
\(988\) 0 0
\(989\) −2.33556 + 10.2328i −0.0742665 + 0.325383i
\(990\) 0 0
\(991\) −8.30957 + 1.89661i −0.263962 + 0.0602476i −0.352454 0.935829i \(-0.614653\pi\)
0.0884914 + 0.996077i \(0.471795\pi\)
\(992\) 0 0
\(993\) 7.85311 1.79242i 0.249211 0.0568808i
\(994\) 0 0
\(995\) 24.1870 + 5.52052i 0.766779 + 0.175012i
\(996\) 0 0
\(997\) 0.724419 1.50427i 0.0229426 0.0476408i −0.889183 0.457552i \(-0.848726\pi\)
0.912125 + 0.409912i \(0.134440\pi\)
\(998\) 0 0
\(999\) −10.3739 −0.328214
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 784.2.bb.b.111.5 120
4.3 odd 2 inner 784.2.bb.b.111.16 yes 120
49.34 odd 14 inner 784.2.bb.b.671.16 yes 120
196.83 even 14 inner 784.2.bb.b.671.5 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
784.2.bb.b.111.5 120 1.1 even 1 trivial
784.2.bb.b.111.16 yes 120 4.3 odd 2 inner
784.2.bb.b.671.5 yes 120 196.83 even 14 inner
784.2.bb.b.671.16 yes 120 49.34 odd 14 inner