Properties

Label 1512.2.c.f.757.18
Level $1512$
Weight $2$
Character 1512.757
Analytic conductor $12.073$
Analytic rank $0$
Dimension $24$
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] = [1512,2,Mod(757,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.757");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (of order \(2\), degree \(1\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 757.18
Character \(\chi\) \(=\) 1512.757
Dual form 1512.2.c.f.757.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.796644 + 1.16849i) q^{2} +(-0.730716 + 1.86173i) q^{4} -2.52623i q^{5} +1.00000 q^{7} +(-2.75753 + 0.629309i) q^{8} +O(q^{10})\) \(q+(0.796644 + 1.16849i) q^{2} +(-0.730716 + 1.86173i) q^{4} -2.52623i q^{5} +1.00000 q^{7} +(-2.75753 + 0.629309i) q^{8} +(2.95186 - 2.01251i) q^{10} +5.70432i q^{11} +3.06046i q^{13} +(0.796644 + 1.16849i) q^{14} +(-2.93211 - 2.72080i) q^{16} +5.49926 q^{17} -7.28486i q^{19} +(4.70317 + 1.84596i) q^{20} +(-6.66541 + 4.54431i) q^{22} -0.539732 q^{23} -1.38184 q^{25} +(-3.57610 + 2.43809i) q^{26} +(-0.730716 + 1.86173i) q^{28} +8.35116i q^{29} +6.74848 q^{31} +(0.843365 - 5.59363i) q^{32} +(4.38095 + 6.42580i) q^{34} -2.52623i q^{35} +10.2711i q^{37} +(8.51226 - 5.80345i) q^{38} +(1.58978 + 6.96615i) q^{40} -5.58370 q^{41} -3.98332i q^{43} +(-10.6199 - 4.16824i) q^{44} +(-0.429975 - 0.630669i) q^{46} -4.83840 q^{47} +1.00000 q^{49} +(-1.10083 - 1.61466i) q^{50} +(-5.69776 - 2.23632i) q^{52} +11.1320i q^{53} +14.4104 q^{55} +(-2.75753 + 0.629309i) q^{56} +(-9.75821 + 6.65291i) q^{58} +10.1647i q^{59} +4.14075i q^{61} +(5.37614 + 7.88550i) q^{62} +(7.20794 - 3.47068i) q^{64} +7.73142 q^{65} +5.96348i q^{67} +(-4.01839 + 10.2382i) q^{68} +(2.95186 - 2.01251i) q^{70} +4.93014 q^{71} +8.66798 q^{73} +(-12.0016 + 8.18239i) q^{74} +(13.5625 + 5.32317i) q^{76} +5.70432i q^{77} +12.0532 q^{79} +(-6.87336 + 7.40718i) q^{80} +(-4.44823 - 6.52448i) q^{82} -5.83393i q^{83} -13.8924i q^{85} +(4.65445 - 3.17329i) q^{86} +(-3.58978 - 15.7298i) q^{88} -9.28810 q^{89} +3.06046i q^{91} +(0.394391 - 1.00484i) q^{92} +(-3.85449 - 5.65360i) q^{94} -18.4032 q^{95} +12.3500 q^{97} +(0.796644 + 1.16849i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{7} + 20 q^{10} - 4 q^{16} + 4 q^{22} - 24 q^{25} - 16 q^{31} + 4 q^{34} + 12 q^{40} - 52 q^{46} + 24 q^{49} + 12 q^{52} - 8 q^{55} - 28 q^{58} + 24 q^{64} + 20 q^{70} - 24 q^{76} + 32 q^{79} + 44 q^{82} - 60 q^{88} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.796644 + 1.16849i 0.563313 + 0.826244i
\(3\) 0 0
\(4\) −0.730716 + 1.86173i −0.365358 + 0.930867i
\(5\) 2.52623i 1.12976i −0.825172 0.564882i \(-0.808922\pi\)
0.825172 0.564882i \(-0.191078\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.75753 + 0.629309i −0.974934 + 0.222494i
\(9\) 0 0
\(10\) 2.95186 2.01251i 0.933461 0.636410i
\(11\) 5.70432i 1.71992i 0.510364 + 0.859958i \(0.329511\pi\)
−0.510364 + 0.859958i \(0.670489\pi\)
\(12\) 0 0
\(13\) 3.06046i 0.848818i 0.905471 + 0.424409i \(0.139518\pi\)
−0.905471 + 0.424409i \(0.860482\pi\)
\(14\) 0.796644 + 1.16849i 0.212912 + 0.312291i
\(15\) 0 0
\(16\) −2.93211 2.72080i −0.733027 0.680199i
\(17\) 5.49926 1.33377 0.666883 0.745163i \(-0.267628\pi\)
0.666883 + 0.745163i \(0.267628\pi\)
\(18\) 0 0
\(19\) 7.28486i 1.67126i −0.549291 0.835631i \(-0.685102\pi\)
0.549291 0.835631i \(-0.314898\pi\)
\(20\) 4.70317 + 1.84596i 1.05166 + 0.412768i
\(21\) 0 0
\(22\) −6.66541 + 4.54431i −1.42107 + 0.968851i
\(23\) −0.539732 −0.112542 −0.0562710 0.998416i \(-0.517921\pi\)
−0.0562710 + 0.998416i \(0.517921\pi\)
\(24\) 0 0
\(25\) −1.38184 −0.276367
\(26\) −3.57610 + 2.43809i −0.701331 + 0.478150i
\(27\) 0 0
\(28\) −0.730716 + 1.86173i −0.138092 + 0.351835i
\(29\) 8.35116i 1.55077i 0.631488 + 0.775386i \(0.282445\pi\)
−0.631488 + 0.775386i \(0.717555\pi\)
\(30\) 0 0
\(31\) 6.74848 1.21206 0.606032 0.795441i \(-0.292761\pi\)
0.606032 + 0.795441i \(0.292761\pi\)
\(32\) 0.843365 5.59363i 0.149087 0.988824i
\(33\) 0 0
\(34\) 4.38095 + 6.42580i 0.751327 + 1.10202i
\(35\) 2.52623i 0.427011i
\(36\) 0 0
\(37\) 10.2711i 1.68855i 0.535908 + 0.844277i \(0.319969\pi\)
−0.535908 + 0.844277i \(0.680031\pi\)
\(38\) 8.51226 5.80345i 1.38087 0.941443i
\(39\) 0 0
\(40\) 1.58978 + 6.96615i 0.251366 + 1.10145i
\(41\) −5.58370 −0.872028 −0.436014 0.899940i \(-0.643610\pi\)
−0.436014 + 0.899940i \(0.643610\pi\)
\(42\) 0 0
\(43\) 3.98332i 0.607451i −0.952760 0.303725i \(-0.901770\pi\)
0.952760 0.303725i \(-0.0982305\pi\)
\(44\) −10.6199 4.16824i −1.60101 0.628385i
\(45\) 0 0
\(46\) −0.429975 0.630669i −0.0633963 0.0929871i
\(47\) −4.83840 −0.705754 −0.352877 0.935670i \(-0.614797\pi\)
−0.352877 + 0.935670i \(0.614797\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.10083 1.61466i −0.155681 0.228347i
\(51\) 0 0
\(52\) −5.69776 2.23632i −0.790137 0.310122i
\(53\) 11.1320i 1.52910i 0.644562 + 0.764552i \(0.277040\pi\)
−0.644562 + 0.764552i \(0.722960\pi\)
\(54\) 0 0
\(55\) 14.4104 1.94310
\(56\) −2.75753 + 0.629309i −0.368490 + 0.0840950i
\(57\) 0 0
\(58\) −9.75821 + 6.65291i −1.28132 + 0.873569i
\(59\) 10.1647i 1.32333i 0.749798 + 0.661666i \(0.230150\pi\)
−0.749798 + 0.661666i \(0.769850\pi\)
\(60\) 0 0
\(61\) 4.14075i 0.530169i 0.964225 + 0.265084i \(0.0853998\pi\)
−0.964225 + 0.265084i \(0.914600\pi\)
\(62\) 5.37614 + 7.88550i 0.682770 + 1.00146i
\(63\) 0 0
\(64\) 7.20794 3.47068i 0.900993 0.433835i
\(65\) 7.73142 0.958964
\(66\) 0 0
\(67\) 5.96348i 0.728555i 0.931290 + 0.364278i \(0.118684\pi\)
−0.931290 + 0.364278i \(0.881316\pi\)
\(68\) −4.01839 + 10.2382i −0.487302 + 1.24156i
\(69\) 0 0
\(70\) 2.95186 2.01251i 0.352815 0.240541i
\(71\) 4.93014 0.585100 0.292550 0.956250i \(-0.405496\pi\)
0.292550 + 0.956250i \(0.405496\pi\)
\(72\) 0 0
\(73\) 8.66798 1.01451 0.507255 0.861796i \(-0.330660\pi\)
0.507255 + 0.861796i \(0.330660\pi\)
\(74\) −12.0016 + 8.18239i −1.39516 + 0.951183i
\(75\) 0 0
\(76\) 13.5625 + 5.32317i 1.55572 + 0.610609i
\(77\) 5.70432i 0.650067i
\(78\) 0 0
\(79\) 12.0532 1.35609 0.678044 0.735021i \(-0.262828\pi\)
0.678044 + 0.735021i \(0.262828\pi\)
\(80\) −6.87336 + 7.40718i −0.768465 + 0.828148i
\(81\) 0 0
\(82\) −4.44823 6.52448i −0.491224 0.720508i
\(83\) 5.83393i 0.640357i −0.947357 0.320178i \(-0.896257\pi\)
0.947357 0.320178i \(-0.103743\pi\)
\(84\) 0 0
\(85\) 13.8924i 1.50684i
\(86\) 4.65445 3.17329i 0.501902 0.342185i
\(87\) 0 0
\(88\) −3.58978 15.7298i −0.382672 1.67681i
\(89\) −9.28810 −0.984537 −0.492268 0.870444i \(-0.663832\pi\)
−0.492268 + 0.870444i \(0.663832\pi\)
\(90\) 0 0
\(91\) 3.06046i 0.320823i
\(92\) 0.394391 1.00484i 0.0411181 0.104762i
\(93\) 0 0
\(94\) −3.85449 5.65360i −0.397560 0.583125i
\(95\) −18.4032 −1.88813
\(96\) 0 0
\(97\) 12.3500 1.25395 0.626974 0.779040i \(-0.284293\pi\)
0.626974 + 0.779040i \(0.284293\pi\)
\(98\) 0.796644 + 1.16849i 0.0804732 + 0.118035i
\(99\) 0 0
\(100\) 1.00973 2.57261i 0.100973 0.257261i
\(101\) 9.33719i 0.929085i 0.885551 + 0.464542i \(0.153781\pi\)
−0.885551 + 0.464542i \(0.846219\pi\)
\(102\) 0 0
\(103\) −7.28802 −0.718110 −0.359055 0.933316i \(-0.616901\pi\)
−0.359055 + 0.933316i \(0.616901\pi\)
\(104\) −1.92597 8.43930i −0.188857 0.827541i
\(105\) 0 0
\(106\) −13.0076 + 8.86828i −1.26341 + 0.861364i
\(107\) 7.31179i 0.706858i −0.935461 0.353429i \(-0.885016\pi\)
0.935461 0.353429i \(-0.114984\pi\)
\(108\) 0 0
\(109\) 15.9330i 1.52611i −0.646336 0.763053i \(-0.723700\pi\)
0.646336 0.763053i \(-0.276300\pi\)
\(110\) 11.4800 + 16.8384i 1.09457 + 1.60548i
\(111\) 0 0
\(112\) −2.93211 2.72080i −0.277058 0.257091i
\(113\) 6.91459 0.650470 0.325235 0.945633i \(-0.394557\pi\)
0.325235 + 0.945633i \(0.394557\pi\)
\(114\) 0 0
\(115\) 1.36349i 0.127146i
\(116\) −15.5476 6.10233i −1.44356 0.566587i
\(117\) 0 0
\(118\) −11.8773 + 8.09766i −1.09340 + 0.745450i
\(119\) 5.49926 0.504116
\(120\) 0 0
\(121\) −21.5393 −1.95811
\(122\) −4.83840 + 3.29870i −0.438049 + 0.298651i
\(123\) 0 0
\(124\) −4.93122 + 12.5639i −0.442837 + 1.12827i
\(125\) 9.14031i 0.817534i
\(126\) 0 0
\(127\) −17.6103 −1.56267 −0.781333 0.624115i \(-0.785460\pi\)
−0.781333 + 0.624115i \(0.785460\pi\)
\(128\) 9.79760 + 5.65748i 0.865994 + 0.500055i
\(129\) 0 0
\(130\) 6.15919 + 9.03405i 0.540196 + 0.792338i
\(131\) 10.7551i 0.939675i −0.882753 0.469838i \(-0.844312\pi\)
0.882753 0.469838i \(-0.155688\pi\)
\(132\) 0 0
\(133\) 7.28486i 0.631678i
\(134\) −6.96824 + 4.75077i −0.601964 + 0.410404i
\(135\) 0 0
\(136\) −15.1644 + 3.46073i −1.30033 + 0.296755i
\(137\) 1.56413 0.133633 0.0668164 0.997765i \(-0.478716\pi\)
0.0668164 + 0.997765i \(0.478716\pi\)
\(138\) 0 0
\(139\) 11.2728i 0.956149i −0.878319 0.478074i \(-0.841335\pi\)
0.878319 0.478074i \(-0.158665\pi\)
\(140\) 4.70317 + 1.84596i 0.397490 + 0.156012i
\(141\) 0 0
\(142\) 3.92757 + 5.76080i 0.329594 + 0.483436i
\(143\) −17.4578 −1.45990
\(144\) 0 0
\(145\) 21.0970 1.75201
\(146\) 6.90529 + 10.1284i 0.571486 + 0.838233i
\(147\) 0 0
\(148\) −19.1220 7.50523i −1.57182 0.616926i
\(149\) 6.48749i 0.531476i −0.964045 0.265738i \(-0.914384\pi\)
0.964045 0.265738i \(-0.0856156\pi\)
\(150\) 0 0
\(151\) −0.383325 −0.0311945 −0.0155973 0.999878i \(-0.504965\pi\)
−0.0155973 + 0.999878i \(0.504965\pi\)
\(152\) 4.58443 + 20.0882i 0.371846 + 1.62937i
\(153\) 0 0
\(154\) −6.66541 + 4.54431i −0.537114 + 0.366191i
\(155\) 17.0482i 1.36935i
\(156\) 0 0
\(157\) 8.66658i 0.691669i −0.938296 0.345834i \(-0.887596\pi\)
0.938296 0.345834i \(-0.112404\pi\)
\(158\) 9.60210 + 14.0840i 0.763902 + 1.12046i
\(159\) 0 0
\(160\) −14.1308 2.13053i −1.11714 0.168433i
\(161\) −0.539732 −0.0425369
\(162\) 0 0
\(163\) 25.0771i 1.96419i −0.188382 0.982096i \(-0.560324\pi\)
0.188382 0.982096i \(-0.439676\pi\)
\(164\) 4.08010 10.3954i 0.318602 0.811742i
\(165\) 0 0
\(166\) 6.81686 4.64756i 0.529091 0.360721i
\(167\) −8.75411 −0.677414 −0.338707 0.940892i \(-0.609989\pi\)
−0.338707 + 0.940892i \(0.609989\pi\)
\(168\) 0 0
\(169\) 3.63361 0.279508
\(170\) 16.2330 11.0673i 1.24502 0.848822i
\(171\) 0 0
\(172\) 7.41588 + 2.91067i 0.565456 + 0.221937i
\(173\) 9.29760i 0.706883i −0.935457 0.353442i \(-0.885011\pi\)
0.935457 0.353442i \(-0.114989\pi\)
\(174\) 0 0
\(175\) −1.38184 −0.104457
\(176\) 15.5203 16.7257i 1.16989 1.26075i
\(177\) 0 0
\(178\) −7.39931 10.8530i −0.554602 0.813467i
\(179\) 7.65495i 0.572158i −0.958206 0.286079i \(-0.907648\pi\)
0.958206 0.286079i \(-0.0923520\pi\)
\(180\) 0 0
\(181\) 2.50424i 0.186138i 0.995660 + 0.0930692i \(0.0296678\pi\)
−0.995660 + 0.0930692i \(0.970332\pi\)
\(182\) −3.57610 + 2.43809i −0.265078 + 0.180724i
\(183\) 0 0
\(184\) 1.48833 0.339658i 0.109721 0.0250400i
\(185\) 25.9471 1.90767
\(186\) 0 0
\(187\) 31.3695i 2.29397i
\(188\) 3.53550 9.00782i 0.257853 0.656963i
\(189\) 0 0
\(190\) −14.6608 21.5039i −1.06361 1.56006i
\(191\) −3.20627 −0.231997 −0.115999 0.993249i \(-0.537007\pi\)
−0.115999 + 0.993249i \(0.537007\pi\)
\(192\) 0 0
\(193\) 10.9544 0.788518 0.394259 0.918999i \(-0.371001\pi\)
0.394259 + 0.918999i \(0.371001\pi\)
\(194\) 9.83853 + 14.4308i 0.706365 + 1.03607i
\(195\) 0 0
\(196\) −0.730716 + 1.86173i −0.0521940 + 0.132981i
\(197\) 7.87749i 0.561248i 0.959818 + 0.280624i \(0.0905414\pi\)
−0.959818 + 0.280624i \(0.909459\pi\)
\(198\) 0 0
\(199\) 1.00753 0.0714217 0.0357109 0.999362i \(-0.488630\pi\)
0.0357109 + 0.999362i \(0.488630\pi\)
\(200\) 3.81046 0.869603i 0.269440 0.0614902i
\(201\) 0 0
\(202\) −10.9104 + 7.43842i −0.767651 + 0.523365i
\(203\) 8.35116i 0.586137i
\(204\) 0 0
\(205\) 14.1057i 0.985186i
\(206\) −5.80596 8.51594i −0.404520 0.593334i
\(207\) 0 0
\(208\) 8.32688 8.97359i 0.577365 0.622206i
\(209\) 41.5552 2.87443
\(210\) 0 0
\(211\) 16.2709i 1.12014i −0.828446 0.560069i \(-0.810775\pi\)
0.828446 0.560069i \(-0.189225\pi\)
\(212\) −20.7249 8.13436i −1.42339 0.558670i
\(213\) 0 0
\(214\) 8.54372 5.82490i 0.584037 0.398182i
\(215\) −10.0628 −0.686276
\(216\) 0 0
\(217\) 6.74848 0.458117
\(218\) 18.6175 12.6929i 1.26094 0.859674i
\(219\) 0 0
\(220\) −10.5299 + 26.8284i −0.709927 + 1.80877i
\(221\) 16.8302i 1.13212i
\(222\) 0 0
\(223\) 2.78339 0.186390 0.0931949 0.995648i \(-0.470292\pi\)
0.0931949 + 0.995648i \(0.470292\pi\)
\(224\) 0.843365 5.59363i 0.0563497 0.373740i
\(225\) 0 0
\(226\) 5.50847 + 8.07959i 0.366418 + 0.537447i
\(227\) 11.4440i 0.759565i 0.925076 + 0.379782i \(0.124001\pi\)
−0.925076 + 0.379782i \(0.875999\pi\)
\(228\) 0 0
\(229\) 4.54391i 0.300270i −0.988666 0.150135i \(-0.952029\pi\)
0.988666 0.150135i \(-0.0479708\pi\)
\(230\) −1.59322 + 1.08621i −0.105054 + 0.0716229i
\(231\) 0 0
\(232\) −5.25546 23.0286i −0.345038 1.51190i
\(233\) −16.0188 −1.04943 −0.524715 0.851278i \(-0.675828\pi\)
−0.524715 + 0.851278i \(0.675828\pi\)
\(234\) 0 0
\(235\) 12.2229i 0.797335i
\(236\) −18.9240 7.42752i −1.23185 0.483490i
\(237\) 0 0
\(238\) 4.38095 + 6.42580i 0.283975 + 0.416523i
\(239\) −19.2968 −1.24821 −0.624104 0.781341i \(-0.714536\pi\)
−0.624104 + 0.781341i \(0.714536\pi\)
\(240\) 0 0
\(241\) −22.1358 −1.42589 −0.712947 0.701218i \(-0.752640\pi\)
−0.712947 + 0.701218i \(0.752640\pi\)
\(242\) −17.1591 25.1683i −1.10303 1.61788i
\(243\) 0 0
\(244\) −7.70897 3.02571i −0.493516 0.193701i
\(245\) 2.52623i 0.161395i
\(246\) 0 0
\(247\) 22.2950 1.41860
\(248\) −18.6091 + 4.24688i −1.18168 + 0.269677i
\(249\) 0 0
\(250\) 10.6803 7.28158i 0.675483 0.460527i
\(251\) 7.94172i 0.501277i −0.968081 0.250638i \(-0.919360\pi\)
0.968081 0.250638i \(-0.0806405\pi\)
\(252\) 0 0
\(253\) 3.07881i 0.193563i
\(254\) −14.0292 20.5774i −0.880269 1.29114i
\(255\) 0 0
\(256\) 1.19452 + 15.9553i 0.0746576 + 0.997209i
\(257\) 7.02501 0.438208 0.219104 0.975701i \(-0.429687\pi\)
0.219104 + 0.975701i \(0.429687\pi\)
\(258\) 0 0
\(259\) 10.2711i 0.638213i
\(260\) −5.64947 + 14.3938i −0.350365 + 0.892668i
\(261\) 0 0
\(262\) 12.5671 8.56797i 0.776401 0.529331i
\(263\) 19.4695 1.20054 0.600270 0.799798i \(-0.295060\pi\)
0.600270 + 0.799798i \(0.295060\pi\)
\(264\) 0 0
\(265\) 28.1221 1.72753
\(266\) 8.51226 5.80345i 0.521920 0.355832i
\(267\) 0 0
\(268\) −11.1024 4.35761i −0.678188 0.266184i
\(269\) 15.4258i 0.940526i 0.882526 + 0.470263i \(0.155841\pi\)
−0.882526 + 0.470263i \(0.844159\pi\)
\(270\) 0 0
\(271\) 7.28198 0.442349 0.221174 0.975234i \(-0.429011\pi\)
0.221174 + 0.975234i \(0.429011\pi\)
\(272\) −16.1244 14.9624i −0.977686 0.907227i
\(273\) 0 0
\(274\) 1.24606 + 1.82766i 0.0752770 + 0.110413i
\(275\) 7.88244i 0.475329i
\(276\) 0 0
\(277\) 2.43449i 0.146275i −0.997322 0.0731373i \(-0.976699\pi\)
0.997322 0.0731373i \(-0.0233011\pi\)
\(278\) 13.1721 8.98043i 0.790012 0.538611i
\(279\) 0 0
\(280\) 1.58978 + 6.96615i 0.0950075 + 0.416307i
\(281\) 18.1555 1.08307 0.541534 0.840679i \(-0.317844\pi\)
0.541534 + 0.840679i \(0.317844\pi\)
\(282\) 0 0
\(283\) 31.2909i 1.86005i −0.367496 0.930025i \(-0.619785\pi\)
0.367496 0.930025i \(-0.380215\pi\)
\(284\) −3.60253 + 9.17862i −0.213771 + 0.544651i
\(285\) 0 0
\(286\) −13.9077 20.3992i −0.822378 1.20623i
\(287\) −5.58370 −0.329596
\(288\) 0 0
\(289\) 13.2418 0.778931
\(290\) 16.8068 + 24.6515i 0.986928 + 1.44759i
\(291\) 0 0
\(292\) −6.33383 + 16.1375i −0.370659 + 0.944374i
\(293\) 18.3419i 1.07155i −0.844362 0.535773i \(-0.820020\pi\)
0.844362 0.535773i \(-0.179980\pi\)
\(294\) 0 0
\(295\) 25.6784 1.49505
\(296\) −6.46368 28.3228i −0.375694 1.64623i
\(297\) 0 0
\(298\) 7.58053 5.16822i 0.439129 0.299387i
\(299\) 1.65183i 0.0955276i
\(300\) 0 0
\(301\) 3.98332i 0.229595i
\(302\) −0.305373 0.447909i −0.0175723 0.0257743i
\(303\) 0 0
\(304\) −19.8206 + 21.3600i −1.13679 + 1.22508i
\(305\) 10.4605 0.598966
\(306\) 0 0
\(307\) 24.5811i 1.40292i 0.712709 + 0.701460i \(0.247468\pi\)
−0.712709 + 0.701460i \(0.752532\pi\)
\(308\) −10.6199 4.16824i −0.605126 0.237507i
\(309\) 0 0
\(310\) 19.9206 13.5814i 1.13141 0.771370i
\(311\) 6.95251 0.394240 0.197120 0.980379i \(-0.436841\pi\)
0.197120 + 0.980379i \(0.436841\pi\)
\(312\) 0 0
\(313\) −9.81498 −0.554775 −0.277388 0.960758i \(-0.589469\pi\)
−0.277388 + 0.960758i \(0.589469\pi\)
\(314\) 10.1268 6.90418i 0.571487 0.389626i
\(315\) 0 0
\(316\) −8.80745 + 22.4398i −0.495458 + 1.26234i
\(317\) 6.69289i 0.375910i 0.982178 + 0.187955i \(0.0601860\pi\)
−0.982178 + 0.187955i \(0.939814\pi\)
\(318\) 0 0
\(319\) −47.6377 −2.66720
\(320\) −8.76773 18.2089i −0.490131 1.01791i
\(321\) 0 0
\(322\) −0.429975 0.630669i −0.0239616 0.0351458i
\(323\) 40.0613i 2.22907i
\(324\) 0 0
\(325\) 4.22905i 0.234586i
\(326\) 29.3022 19.9775i 1.62290 1.10645i
\(327\) 0 0
\(328\) 15.3972 3.51388i 0.850170 0.194021i
\(329\) −4.83840 −0.266750
\(330\) 0 0
\(331\) 2.10916i 0.115930i 0.998319 + 0.0579651i \(0.0184612\pi\)
−0.998319 + 0.0579651i \(0.981539\pi\)
\(332\) 10.8612 + 4.26294i 0.596087 + 0.233959i
\(333\) 0 0
\(334\) −6.97391 10.2291i −0.381596 0.559709i
\(335\) 15.0651 0.823096
\(336\) 0 0
\(337\) −5.53954 −0.301758 −0.150879 0.988552i \(-0.548210\pi\)
−0.150879 + 0.988552i \(0.548210\pi\)
\(338\) 2.89469 + 4.24582i 0.157451 + 0.230942i
\(339\) 0 0
\(340\) 25.8639 + 10.1514i 1.40267 + 0.550536i
\(341\) 38.4955i 2.08465i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.50674 + 10.9841i 0.135154 + 0.592224i
\(345\) 0 0
\(346\) 10.8641 7.40688i 0.584058 0.398196i
\(347\) 9.63790i 0.517389i 0.965959 + 0.258695i \(0.0832924\pi\)
−0.965959 + 0.258695i \(0.916708\pi\)
\(348\) 0 0
\(349\) 8.04565i 0.430674i −0.976540 0.215337i \(-0.930915\pi\)
0.976540 0.215337i \(-0.0690850\pi\)
\(350\) −1.10083 1.61466i −0.0588420 0.0863070i
\(351\) 0 0
\(352\) 31.9079 + 4.81082i 1.70070 + 0.256418i
\(353\) −27.2775 −1.45184 −0.725918 0.687782i \(-0.758585\pi\)
−0.725918 + 0.687782i \(0.758585\pi\)
\(354\) 0 0
\(355\) 12.4547i 0.661025i
\(356\) 6.78696 17.2920i 0.359708 0.916473i
\(357\) 0 0
\(358\) 8.94470 6.09827i 0.472742 0.322304i
\(359\) 7.36413 0.388664 0.194332 0.980936i \(-0.437746\pi\)
0.194332 + 0.980936i \(0.437746\pi\)
\(360\) 0 0
\(361\) −34.0692 −1.79312
\(362\) −2.92616 + 1.99499i −0.153796 + 0.104854i
\(363\) 0 0
\(364\) −5.69776 2.23632i −0.298644 0.117215i
\(365\) 21.8973i 1.14616i
\(366\) 0 0
\(367\) 21.8746 1.14185 0.570923 0.821003i \(-0.306585\pi\)
0.570923 + 0.821003i \(0.306585\pi\)
\(368\) 1.58255 + 1.46850i 0.0824963 + 0.0765510i
\(369\) 0 0
\(370\) 20.6706 + 30.3188i 1.07461 + 1.57620i
\(371\) 11.1320i 0.577947i
\(372\) 0 0
\(373\) 8.99106i 0.465539i −0.972532 0.232770i \(-0.925221\pi\)
0.972532 0.232770i \(-0.0747788\pi\)
\(374\) −36.6548 + 24.9903i −1.89538 + 1.29222i
\(375\) 0 0
\(376\) 13.3420 3.04485i 0.688063 0.157026i
\(377\) −25.5584 −1.31632
\(378\) 0 0
\(379\) 28.9165i 1.48534i 0.669656 + 0.742671i \(0.266442\pi\)
−0.669656 + 0.742671i \(0.733558\pi\)
\(380\) 13.4475 34.2619i 0.689844 1.75760i
\(381\) 0 0
\(382\) −2.55425 3.74647i −0.130687 0.191686i
\(383\) −12.5404 −0.640786 −0.320393 0.947285i \(-0.603815\pi\)
−0.320393 + 0.947285i \(0.603815\pi\)
\(384\) 0 0
\(385\) 14.4104 0.734423
\(386\) 8.72680 + 12.8001i 0.444182 + 0.651509i
\(387\) 0 0
\(388\) −9.02431 + 22.9924i −0.458140 + 1.16726i
\(389\) 7.50374i 0.380455i −0.981740 0.190227i \(-0.939077\pi\)
0.981740 0.190227i \(-0.0609225\pi\)
\(390\) 0 0
\(391\) −2.96813 −0.150105
\(392\) −2.75753 + 0.629309i −0.139276 + 0.0317849i
\(393\) 0 0
\(394\) −9.20474 + 6.27556i −0.463728 + 0.316158i
\(395\) 30.4491i 1.53206i
\(396\) 0 0
\(397\) 27.6639i 1.38841i 0.719778 + 0.694205i \(0.244244\pi\)
−0.719778 + 0.694205i \(0.755756\pi\)
\(398\) 0.802641 + 1.17728i 0.0402327 + 0.0590117i
\(399\) 0 0
\(400\) 4.05170 + 3.75970i 0.202585 + 0.187985i
\(401\) 35.8414 1.78983 0.894916 0.446234i \(-0.147235\pi\)
0.894916 + 0.446234i \(0.147235\pi\)
\(402\) 0 0
\(403\) 20.6534i 1.02882i
\(404\) −17.3834 6.82283i −0.864855 0.339449i
\(405\) 0 0
\(406\) −9.75821 + 6.65291i −0.484292 + 0.330178i
\(407\) −58.5894 −2.90417
\(408\) 0 0
\(409\) 18.4056 0.910097 0.455048 0.890467i \(-0.349622\pi\)
0.455048 + 0.890467i \(0.349622\pi\)
\(410\) −16.4823 + 11.2372i −0.814004 + 0.554968i
\(411\) 0 0
\(412\) 5.32547 13.5684i 0.262367 0.668465i
\(413\) 10.1647i 0.500173i
\(414\) 0 0
\(415\) −14.7378 −0.723452
\(416\) 17.1191 + 2.58108i 0.839331 + 0.126548i
\(417\) 0 0
\(418\) 33.1047 + 48.5566i 1.61920 + 2.37498i
\(419\) 0.0173724i 0.000848699i −1.00000 0.000424349i \(-0.999865\pi\)
1.00000 0.000424349i \(-0.000135075\pi\)
\(420\) 0 0
\(421\) 1.03437i 0.0504121i −0.999682 0.0252060i \(-0.991976\pi\)
0.999682 0.0252060i \(-0.00802418\pi\)
\(422\) 19.0124 12.9622i 0.925507 0.630988i
\(423\) 0 0
\(424\) −7.00550 30.6970i −0.340217 1.49078i
\(425\) −7.59908 −0.368609
\(426\) 0 0
\(427\) 4.14075i 0.200385i
\(428\) 13.6126 + 5.34284i 0.657991 + 0.258256i
\(429\) 0 0
\(430\) −8.01646 11.7582i −0.386588 0.567031i
\(431\) −10.1392 −0.488390 −0.244195 0.969726i \(-0.578524\pi\)
−0.244195 + 0.969726i \(0.578524\pi\)
\(432\) 0 0
\(433\) −33.5806 −1.61378 −0.806890 0.590701i \(-0.798851\pi\)
−0.806890 + 0.590701i \(0.798851\pi\)
\(434\) 5.37614 + 7.88550i 0.258063 + 0.378516i
\(435\) 0 0
\(436\) 29.6630 + 11.6425i 1.42060 + 0.557575i
\(437\) 3.93188i 0.188087i
\(438\) 0 0
\(439\) −3.92098 −0.187138 −0.0935692 0.995613i \(-0.529828\pi\)
−0.0935692 + 0.995613i \(0.529828\pi\)
\(440\) −39.7372 + 9.06861i −1.89439 + 0.432329i
\(441\) 0 0
\(442\) −19.6659 + 13.4077i −0.935411 + 0.637740i
\(443\) 34.6306i 1.64535i −0.568514 0.822674i \(-0.692481\pi\)
0.568514 0.822674i \(-0.307519\pi\)
\(444\) 0 0
\(445\) 23.4639i 1.11229i
\(446\) 2.21737 + 3.25235i 0.104996 + 0.154003i
\(447\) 0 0
\(448\) 7.20794 3.47068i 0.340543 0.163974i
\(449\) 17.7862 0.839383 0.419691 0.907667i \(-0.362138\pi\)
0.419691 + 0.907667i \(0.362138\pi\)
\(450\) 0 0
\(451\) 31.8512i 1.49982i
\(452\) −5.05260 + 12.8731i −0.237654 + 0.605501i
\(453\) 0 0
\(454\) −13.3721 + 9.11679i −0.627586 + 0.427872i
\(455\) 7.73142 0.362454
\(456\) 0 0
\(457\) 11.5852 0.541933 0.270966 0.962589i \(-0.412657\pi\)
0.270966 + 0.962589i \(0.412657\pi\)
\(458\) 5.30949 3.61988i 0.248096 0.169146i
\(459\) 0 0
\(460\) −2.53845 0.996322i −0.118356 0.0464538i
\(461\) 12.5948i 0.586596i −0.956021 0.293298i \(-0.905247\pi\)
0.956021 0.293298i \(-0.0947529\pi\)
\(462\) 0 0
\(463\) −3.16107 −0.146907 −0.0734537 0.997299i \(-0.523402\pi\)
−0.0734537 + 0.997299i \(0.523402\pi\)
\(464\) 22.7218 24.4865i 1.05483 1.13676i
\(465\) 0 0
\(466\) −12.7613 18.7178i −0.591157 0.867085i
\(467\) 20.8974i 0.967015i 0.875340 + 0.483507i \(0.160637\pi\)
−0.875340 + 0.483507i \(0.839363\pi\)
\(468\) 0 0
\(469\) 5.96348i 0.275368i
\(470\) −14.2823 + 9.73732i −0.658794 + 0.449149i
\(471\) 0 0
\(472\) −6.39674 28.0295i −0.294434 1.29016i
\(473\) 22.7221 1.04476
\(474\) 0 0
\(475\) 10.0665i 0.461882i
\(476\) −4.01839 + 10.2382i −0.184183 + 0.469265i
\(477\) 0 0
\(478\) −15.3727 22.5481i −0.703131 1.03132i
\(479\) −28.2766 −1.29199 −0.645996 0.763341i \(-0.723558\pi\)
−0.645996 + 0.763341i \(0.723558\pi\)
\(480\) 0 0
\(481\) −31.4341 −1.43327
\(482\) −17.6344 25.8654i −0.803224 1.17814i
\(483\) 0 0
\(484\) 15.7391 40.1004i 0.715412 1.82274i
\(485\) 31.1988i 1.41667i
\(486\) 0 0
\(487\) −26.8879 −1.21841 −0.609203 0.793014i \(-0.708511\pi\)
−0.609203 + 0.793014i \(0.708511\pi\)
\(488\) −2.60581 11.4182i −0.117960 0.516879i
\(489\) 0 0
\(490\) 2.95186 2.01251i 0.133352 0.0909158i
\(491\) 6.00142i 0.270840i −0.990788 0.135420i \(-0.956762\pi\)
0.990788 0.135420i \(-0.0432384\pi\)
\(492\) 0 0
\(493\) 45.9252i 2.06837i
\(494\) 17.7612 + 26.0514i 0.799114 + 1.17211i
\(495\) 0 0
\(496\) −19.7873 18.3613i −0.888475 0.824445i
\(497\) 4.93014 0.221147
\(498\) 0 0
\(499\) 18.3124i 0.819777i 0.912136 + 0.409888i \(0.134432\pi\)
−0.912136 + 0.409888i \(0.865568\pi\)
\(500\) 17.0168 + 6.67897i 0.761016 + 0.298693i
\(501\) 0 0
\(502\) 9.27978 6.32672i 0.414177 0.282375i
\(503\) 6.95874 0.310275 0.155137 0.987893i \(-0.450418\pi\)
0.155137 + 0.987893i \(0.450418\pi\)
\(504\) 0 0
\(505\) 23.5879 1.04965
\(506\) 3.59754 2.45271i 0.159930 0.109036i
\(507\) 0 0
\(508\) 12.8682 32.7858i 0.570932 1.45463i
\(509\) 15.2426i 0.675618i −0.941215 0.337809i \(-0.890314\pi\)
0.941215 0.337809i \(-0.109686\pi\)
\(510\) 0 0
\(511\) 8.66798 0.383449
\(512\) −17.6920 + 14.1065i −0.781882 + 0.623426i
\(513\) 0 0
\(514\) 5.59644 + 8.20862i 0.246848 + 0.362067i
\(515\) 18.4112i 0.811295i
\(516\) 0 0
\(517\) 27.5998i 1.21384i
\(518\) −12.0016 + 8.18239i −0.527320 + 0.359513i
\(519\) 0 0
\(520\) −21.3196 + 4.86545i −0.934927 + 0.213364i
\(521\) 13.0764 0.572887 0.286444 0.958097i \(-0.407527\pi\)
0.286444 + 0.958097i \(0.407527\pi\)
\(522\) 0 0
\(523\) 23.9123i 1.04561i 0.852452 + 0.522806i \(0.175115\pi\)
−0.852452 + 0.522806i \(0.824885\pi\)
\(524\) 20.0231 + 7.85891i 0.874713 + 0.343318i
\(525\) 0 0
\(526\) 15.5103 + 22.7498i 0.676279 + 0.991939i
\(527\) 37.1116 1.61661
\(528\) 0 0
\(529\) −22.7087 −0.987334
\(530\) 22.4033 + 32.8603i 0.973138 + 1.42736i
\(531\) 0 0
\(532\) 13.5625 + 5.32317i 0.588008 + 0.230789i
\(533\) 17.0887i 0.740193i
\(534\) 0 0
\(535\) −18.4713 −0.798583
\(536\) −3.75287 16.4445i −0.162099 0.710293i
\(537\) 0 0
\(538\) −18.0248 + 12.2889i −0.777104 + 0.529810i
\(539\) 5.70432i 0.245702i
\(540\) 0 0
\(541\) 13.9717i 0.600689i 0.953831 + 0.300345i \(0.0971017\pi\)
−0.953831 + 0.300345i \(0.902898\pi\)
\(542\) 5.80115 + 8.50889i 0.249181 + 0.365488i
\(543\) 0 0
\(544\) 4.63788 30.7608i 0.198847 1.31886i
\(545\) −40.2504 −1.72414
\(546\) 0 0
\(547\) 21.5760i 0.922522i −0.887264 0.461261i \(-0.847397\pi\)
0.887264 0.461261i \(-0.152603\pi\)
\(548\) −1.14294 + 2.91200i −0.0488238 + 0.124394i
\(549\) 0 0
\(550\) 9.21051 6.27950i 0.392738 0.267759i
\(551\) 60.8371 2.59175
\(552\) 0 0
\(553\) 12.0532 0.512553
\(554\) 2.84467 1.93942i 0.120858 0.0823983i
\(555\) 0 0
\(556\) 20.9870 + 8.23723i 0.890047 + 0.349337i
\(557\) 0.679074i 0.0287733i −0.999897 0.0143866i \(-0.995420\pi\)
0.999897 0.0143866i \(-0.00457957\pi\)
\(558\) 0 0
\(559\) 12.1908 0.515615
\(560\) −6.87336 + 7.40718i −0.290452 + 0.313010i
\(561\) 0 0
\(562\) 14.4635 + 21.2145i 0.610106 + 0.894878i
\(563\) 15.8429i 0.667698i −0.942626 0.333849i \(-0.891652\pi\)
0.942626 0.333849i \(-0.108348\pi\)
\(564\) 0 0
\(565\) 17.4678i 0.734877i
\(566\) 36.5629 24.9277i 1.53686 1.04779i
\(567\) 0 0
\(568\) −13.5950 + 3.10258i −0.570434 + 0.130182i
\(569\) 3.24267 0.135940 0.0679698 0.997687i \(-0.478348\pi\)
0.0679698 + 0.997687i \(0.478348\pi\)
\(570\) 0 0
\(571\) 1.38251i 0.0578563i −0.999581 0.0289281i \(-0.990791\pi\)
0.999581 0.0289281i \(-0.00920939\pi\)
\(572\) 12.7567 32.5018i 0.533385 1.35897i
\(573\) 0 0
\(574\) −4.44823 6.52448i −0.185665 0.272326i
\(575\) 0.745822 0.0311029
\(576\) 0 0
\(577\) −7.57598 −0.315392 −0.157696 0.987488i \(-0.550407\pi\)
−0.157696 + 0.987488i \(0.550407\pi\)
\(578\) 10.5490 + 15.4729i 0.438782 + 0.643587i
\(579\) 0 0
\(580\) −15.4159 + 39.2769i −0.640110 + 1.63089i
\(581\) 5.83393i 0.242032i
\(582\) 0 0
\(583\) −63.5007 −2.62993
\(584\) −23.9022 + 5.45484i −0.989080 + 0.225723i
\(585\) 0 0
\(586\) 21.4323 14.6120i 0.885358 0.603615i
\(587\) 5.03970i 0.208011i 0.994577 + 0.104005i \(0.0331659\pi\)
−0.994577 + 0.104005i \(0.966834\pi\)
\(588\) 0 0
\(589\) 49.1618i 2.02568i
\(590\) 20.4565 + 30.0048i 0.842183 + 1.23528i
\(591\) 0 0
\(592\) 27.9455 30.1159i 1.14855 1.23776i
\(593\) −7.94121 −0.326106 −0.163053 0.986617i \(-0.552134\pi\)
−0.163053 + 0.986617i \(0.552134\pi\)
\(594\) 0 0
\(595\) 13.8924i 0.569532i
\(596\) 12.0780 + 4.74051i 0.494733 + 0.194179i
\(597\) 0 0
\(598\) 1.93014 1.31592i 0.0789291 0.0538119i
\(599\) −33.0131 −1.34888 −0.674439 0.738331i \(-0.735614\pi\)
−0.674439 + 0.738331i \(0.735614\pi\)
\(600\) 0 0
\(601\) 40.4038 1.64810 0.824052 0.566513i \(-0.191708\pi\)
0.824052 + 0.566513i \(0.191708\pi\)
\(602\) 4.65445 3.17329i 0.189701 0.129334i
\(603\) 0 0
\(604\) 0.280101 0.713648i 0.0113972 0.0290379i
\(605\) 54.4131i 2.21221i
\(606\) 0 0
\(607\) 0.544975 0.0221199 0.0110599 0.999939i \(-0.496479\pi\)
0.0110599 + 0.999939i \(0.496479\pi\)
\(608\) −40.7489 6.14380i −1.65258 0.249164i
\(609\) 0 0
\(610\) 8.33328 + 12.2229i 0.337405 + 0.494892i
\(611\) 14.8077i 0.599056i
\(612\) 0 0
\(613\) 2.75425i 0.111243i −0.998452 0.0556215i \(-0.982286\pi\)
0.998452 0.0556215i \(-0.0177140\pi\)
\(614\) −28.7227 + 19.5824i −1.15915 + 0.790283i
\(615\) 0 0
\(616\) −3.58978 15.7298i −0.144636 0.633773i
\(617\) 32.6720 1.31533 0.657664 0.753312i \(-0.271545\pi\)
0.657664 + 0.753312i \(0.271545\pi\)
\(618\) 0 0
\(619\) 10.4789i 0.421183i 0.977574 + 0.210591i \(0.0675389\pi\)
−0.977574 + 0.210591i \(0.932461\pi\)
\(620\) 31.7392 + 12.4574i 1.27468 + 0.500301i
\(621\) 0 0
\(622\) 5.53867 + 8.12390i 0.222081 + 0.325739i
\(623\) −9.28810 −0.372120
\(624\) 0 0
\(625\) −29.9997 −1.19999
\(626\) −7.81904 11.4687i −0.312512 0.458380i
\(627\) 0 0
\(628\) 16.1349 + 6.33281i 0.643852 + 0.252707i
\(629\) 56.4832i 2.25213i
\(630\) 0 0
\(631\) −7.94128 −0.316138 −0.158069 0.987428i \(-0.550527\pi\)
−0.158069 + 0.987428i \(0.550527\pi\)
\(632\) −33.2370 + 7.58518i −1.32210 + 0.301722i
\(633\) 0 0
\(634\) −7.82055 + 5.33185i −0.310594 + 0.211755i
\(635\) 44.4878i 1.76544i
\(636\) 0 0
\(637\) 3.06046i 0.121260i
\(638\) −37.9503 55.6639i −1.50247 2.20376i
\(639\) 0 0
\(640\) 14.2921 24.7510i 0.564944 0.978369i
\(641\) 17.5222 0.692084 0.346042 0.938219i \(-0.387525\pi\)
0.346042 + 0.938219i \(0.387525\pi\)
\(642\) 0 0
\(643\) 34.7998i 1.37237i −0.727426 0.686186i \(-0.759284\pi\)
0.727426 0.686186i \(-0.240716\pi\)
\(644\) 0.394391 1.00484i 0.0155412 0.0395962i
\(645\) 0 0
\(646\) 46.8111 31.9146i 1.84176 1.25566i
\(647\) 24.4054 0.959474 0.479737 0.877412i \(-0.340732\pi\)
0.479737 + 0.877412i \(0.340732\pi\)
\(648\) 0 0
\(649\) −57.9827 −2.27602
\(650\) 4.94158 3.36905i 0.193825 0.132145i
\(651\) 0 0
\(652\) 46.6869 + 18.3243i 1.82840 + 0.717633i
\(653\) 20.9374i 0.819345i −0.912233 0.409673i \(-0.865643\pi\)
0.912233 0.409673i \(-0.134357\pi\)
\(654\) 0 0
\(655\) −27.1698 −1.06161
\(656\) 16.3720 + 15.1921i 0.639220 + 0.593153i
\(657\) 0 0
\(658\) −3.85449 5.65360i −0.150264 0.220400i
\(659\) 27.6061i 1.07538i 0.843143 + 0.537690i \(0.180703\pi\)
−0.843143 + 0.537690i \(0.819297\pi\)
\(660\) 0 0
\(661\) 31.2679i 1.21618i 0.793868 + 0.608090i \(0.208064\pi\)
−0.793868 + 0.608090i \(0.791936\pi\)
\(662\) −2.46453 + 1.68025i −0.0957865 + 0.0653049i
\(663\) 0 0
\(664\) 3.67134 + 16.0872i 0.142476 + 0.624305i
\(665\) −18.4032 −0.713647
\(666\) 0 0
\(667\) 4.50739i 0.174527i
\(668\) 6.39677 16.2978i 0.247498 0.630582i
\(669\) 0 0
\(670\) 12.0015 + 17.6034i 0.463660 + 0.680078i
\(671\) −23.6201 −0.911846
\(672\) 0 0
\(673\) 18.7023 0.720923 0.360461 0.932774i \(-0.382619\pi\)
0.360461 + 0.932774i \(0.382619\pi\)
\(674\) −4.41304 6.47287i −0.169984 0.249326i
\(675\) 0 0
\(676\) −2.65514 + 6.76481i −0.102121 + 0.260185i
\(677\) 1.58757i 0.0610152i 0.999535 + 0.0305076i \(0.00971238\pi\)
−0.999535 + 0.0305076i \(0.990288\pi\)
\(678\) 0 0
\(679\) 12.3500 0.473948
\(680\) 8.74260 + 38.3087i 0.335264 + 1.46907i
\(681\) 0 0
\(682\) −44.9814 + 30.6672i −1.72243 + 1.17431i
\(683\) 34.1798i 1.30785i −0.756557 0.653927i \(-0.773120\pi\)
0.756557 0.653927i \(-0.226880\pi\)
\(684\) 0 0
\(685\) 3.95135i 0.150974i
\(686\) 0.796644 + 1.16849i 0.0304160 + 0.0446130i
\(687\) 0 0
\(688\) −10.8378 + 11.6795i −0.413187 + 0.445278i
\(689\) −34.0691 −1.29793
\(690\) 0 0
\(691\) 39.5718i 1.50538i −0.658373 0.752692i \(-0.728755\pi\)
0.658373 0.752692i \(-0.271245\pi\)
\(692\) 17.3097 + 6.79390i 0.658014 + 0.258265i
\(693\) 0 0
\(694\) −11.2617 + 7.67798i −0.427490 + 0.291452i
\(695\) −28.4778 −1.08022
\(696\) 0 0
\(697\) −30.7062 −1.16308
\(698\) 9.40123 6.40952i 0.355842 0.242604i
\(699\) 0 0
\(700\) 1.00973 2.57261i 0.0381642 0.0972356i
\(701\) 30.2500i 1.14253i 0.820767 + 0.571263i \(0.193546\pi\)
−0.820767 + 0.571263i \(0.806454\pi\)
\(702\) 0 0
\(703\) 74.8233 2.82202
\(704\) 19.7978 + 41.1164i 0.746159 + 1.54963i
\(705\) 0 0
\(706\) −21.7305 31.8734i −0.817837 1.19957i
\(707\) 9.33719i 0.351161i
\(708\) 0 0
\(709\) 51.5735i 1.93688i 0.249244 + 0.968441i \(0.419818\pi\)
−0.249244 + 0.968441i \(0.580182\pi\)
\(710\) 14.5531 9.92194i 0.546168 0.372364i
\(711\) 0 0
\(712\) 25.6122 5.84509i 0.959858 0.219054i
\(713\) −3.64237 −0.136408
\(714\) 0 0
\(715\) 44.1025i 1.64934i
\(716\) 14.2515 + 5.59359i 0.532603 + 0.209042i
\(717\) 0 0
\(718\) 5.86659 + 8.60488i 0.218939 + 0.321131i
\(719\) 11.2028 0.417795 0.208898 0.977938i \(-0.433012\pi\)
0.208898 + 0.977938i \(0.433012\pi\)
\(720\) 0 0
\(721\) −7.28802 −0.271420
\(722\) −27.1411 39.8094i −1.01009 1.48155i
\(723\) 0 0
\(724\) −4.66222 1.82989i −0.173270 0.0680072i
\(725\) 11.5399i 0.428583i
\(726\) 0 0
\(727\) −0.814724 −0.0302164 −0.0151082 0.999886i \(-0.504809\pi\)
−0.0151082 + 0.999886i \(0.504809\pi\)
\(728\) −1.92597 8.43930i −0.0713813 0.312781i
\(729\) 0 0
\(730\) 25.5867 17.4444i 0.947005 0.645645i
\(731\) 21.9053i 0.810197i
\(732\) 0 0
\(733\) 12.7283i 0.470131i −0.971980 0.235066i \(-0.924470\pi\)
0.971980 0.235066i \(-0.0755305\pi\)
\(734\) 17.4263 + 25.5602i 0.643217 + 0.943444i
\(735\) 0 0
\(736\) −0.455191 + 3.01906i −0.0167786 + 0.111284i
\(737\) −34.0176 −1.25305
\(738\) 0 0
\(739\) 9.13395i 0.335998i −0.985787 0.167999i \(-0.946269\pi\)
0.985787 0.167999i \(-0.0537305\pi\)
\(740\) −18.9599 + 48.3066i −0.696981 + 1.77578i
\(741\) 0 0
\(742\) −13.0076 + 8.86828i −0.477525 + 0.325565i
\(743\) 15.4112 0.565383 0.282692 0.959211i \(-0.408773\pi\)
0.282692 + 0.959211i \(0.408773\pi\)
\(744\) 0 0
\(745\) −16.3889 −0.600442
\(746\) 10.5059 7.16267i 0.384649 0.262244i
\(747\) 0 0
\(748\) −58.4017 22.9222i −2.13538 0.838119i
\(749\) 7.31179i 0.267167i
\(750\) 0 0
\(751\) 40.7526 1.48708 0.743542 0.668690i \(-0.233144\pi\)
0.743542 + 0.668690i \(0.233144\pi\)
\(752\) 14.1867 + 13.1643i 0.517337 + 0.480053i
\(753\) 0 0
\(754\) −20.3609 29.8646i −0.741501 1.08760i
\(755\) 0.968366i 0.0352424i
\(756\) 0 0
\(757\) 26.0085i 0.945296i 0.881251 + 0.472648i \(0.156702\pi\)
−0.881251 + 0.472648i \(0.843298\pi\)
\(758\) −33.7885 + 23.0362i −1.22726 + 0.836712i
\(759\) 0 0
\(760\) 50.7475 11.5813i 1.84080 0.420099i
\(761\) 10.3686 0.375861 0.187931 0.982182i \(-0.439822\pi\)
0.187931 + 0.982182i \(0.439822\pi\)
\(762\) 0 0
\(763\) 15.9330i 0.576814i
\(764\) 2.34287 5.96922i 0.0847620 0.215959i
\(765\) 0 0
\(766\) −9.99027 14.6533i −0.360963 0.529446i
\(767\) −31.1086 −1.12327
\(768\) 0 0
\(769\) −31.7525 −1.14502 −0.572512 0.819896i \(-0.694031\pi\)
−0.572512 + 0.819896i \(0.694031\pi\)
\(770\) 11.4800 + 16.8384i 0.413710 + 0.606813i
\(771\) 0 0
\(772\) −8.00459 + 20.3943i −0.288091 + 0.734006i
\(773\) 16.3614i 0.588478i 0.955732 + 0.294239i \(0.0950662\pi\)
−0.955732 + 0.294239i \(0.904934\pi\)
\(774\) 0 0
\(775\) −9.32530 −0.334975
\(776\) −34.0554 + 7.77194i −1.22252 + 0.278997i
\(777\) 0 0
\(778\) 8.76801 5.97781i 0.314348 0.214315i
\(779\) 40.6765i 1.45739i
\(780\) 0 0
\(781\) 28.1231i 1.00632i
\(782\) −2.36454 3.46821i −0.0845558 0.124023i
\(783\) 0 0
\(784\) −2.93211 2.72080i −0.104718 0.0971713i
\(785\) −21.8938 −0.781422
\(786\) 0 0
\(787\) 25.4664i 0.907781i −0.891058 0.453890i \(-0.850036\pi\)
0.891058 0.453890i \(-0.149964\pi\)
\(788\) −14.6658 5.75621i −0.522448 0.205057i
\(789\) 0 0
\(790\) 35.5793 24.2571i 1.26586 0.863029i
\(791\) 6.91459 0.245854
\(792\) 0 0
\(793\) −12.6726 −0.450016
\(794\) −32.3248 + 22.0383i −1.14716 + 0.782108i
\(795\) 0 0
\(796\) −0.736216 + 1.87575i −0.0260945 + 0.0664841i
\(797\) 27.4312i 0.971664i −0.874052 0.485832i \(-0.838517\pi\)
0.874052 0.485832i \(-0.161483\pi\)
\(798\) 0 0
\(799\) −26.6076 −0.941310
\(800\) −1.16539 + 7.72949i −0.0412028 + 0.273279i
\(801\) 0 0
\(802\) 28.5528 + 41.8801i 1.00824 + 1.47884i
\(803\) 49.4449i 1.74487i
\(804\) 0 0
\(805\) 1.36349i 0.0480566i
\(806\) −24.1332 + 16.4534i −0.850057 + 0.579548i
\(807\) 0 0
\(808\) −5.87598 25.7476i −0.206716 0.905796i
\(809\) −23.8507 −0.838546 −0.419273 0.907860i \(-0.637715\pi\)
−0.419273 + 0.907860i \(0.637715\pi\)
\(810\) 0 0
\(811\) 27.8123i 0.976622i 0.872670 + 0.488311i \(0.162387\pi\)
−0.872670 + 0.488311i \(0.837613\pi\)
\(812\) −15.5476 6.10233i −0.545615 0.214150i
\(813\) 0 0
\(814\) −46.6749 68.4609i −1.63596 2.39955i
\(815\) −63.3506 −2.21907
\(816\) 0 0
\(817\) −29.0179 −1.01521
\(818\) 14.6627 + 21.5066i 0.512669 + 0.751962i
\(819\) 0 0
\(820\) −26.2611 10.3073i −0.917077 0.359946i
\(821\) 53.9215i 1.88187i −0.338583 0.940936i \(-0.609948\pi\)
0.338583 0.940936i \(-0.390052\pi\)
\(822\) 0 0
\(823\) −11.4932 −0.400628 −0.200314 0.979732i \(-0.564196\pi\)
−0.200314 + 0.979732i \(0.564196\pi\)
\(824\) 20.0969 4.58642i 0.700110 0.159775i
\(825\) 0 0
\(826\) −11.8773 + 8.09766i −0.413265 + 0.281754i
\(827\) 0.946387i 0.0329091i −0.999865 0.0164546i \(-0.994762\pi\)
0.999865 0.0164546i \(-0.00523788\pi\)
\(828\) 0 0
\(829\) 42.9151i 1.49050i −0.666783 0.745252i \(-0.732329\pi\)
0.666783 0.745252i \(-0.267671\pi\)
\(830\) −11.7408 17.2210i −0.407530 0.597748i
\(831\) 0 0
\(832\) 10.6219 + 22.0596i 0.368247 + 0.764778i
\(833\) 5.49926 0.190538
\(834\) 0 0
\(835\) 22.1149i 0.765318i
\(836\) −30.3650 + 77.3647i −1.05020 + 2.67571i
\(837\) 0 0
\(838\) 0.0202994 0.0138396i 0.000701232 0.000478083i
\(839\) 31.2098 1.07748 0.538741 0.842471i \(-0.318900\pi\)
0.538741 + 0.842471i \(0.318900\pi\)
\(840\) 0 0
\(841\) −40.7419 −1.40489
\(842\) 1.20865 0.824024i 0.0416527 0.0283978i
\(843\) 0 0
\(844\) 30.2922 + 11.8894i 1.04270 + 0.409251i
\(845\) 9.17933i 0.315779i
\(846\) 0 0
\(847\) −21.5393 −0.740097
\(848\) 30.2880 32.6404i 1.04010 1.12087i
\(849\) 0 0
\(850\) −6.05376 8.87941i −0.207642 0.304561i
\(851\) 5.54363i 0.190033i
\(852\) 0 0
\(853\) 8.84389i 0.302809i 0.988472 + 0.151404i \(0.0483796\pi\)
−0.988472 + 0.151404i \(0.951620\pi\)
\(854\) −4.83840 + 3.29870i −0.165567 + 0.112879i
\(855\) 0 0
\(856\) 4.60138 + 20.1625i 0.157272 + 0.689140i
\(857\) 49.1414 1.67864 0.839319 0.543640i \(-0.182954\pi\)
0.839319 + 0.543640i \(0.182954\pi\)
\(858\) 0 0
\(859\) 43.8201i 1.49512i −0.664193 0.747561i \(-0.731225\pi\)
0.664193 0.747561i \(-0.268775\pi\)
\(860\) 7.35303 18.7342i 0.250736 0.638832i
\(861\) 0 0
\(862\) −8.07737 11.8476i −0.275116 0.403529i
\(863\) 31.5811 1.07503 0.537516 0.843253i \(-0.319363\pi\)
0.537516 + 0.843253i \(0.319363\pi\)
\(864\) 0 0
\(865\) −23.4879 −0.798612
\(866\) −26.7518 39.2384i −0.909063 1.33338i
\(867\) 0 0
\(868\) −4.93122 + 12.5639i −0.167377 + 0.426446i
\(869\) 68.7552i 2.33236i
\(870\) 0 0
\(871\) −18.2510 −0.618411
\(872\) 10.0268 + 43.9357i 0.339550 + 1.48785i
\(873\) 0 0
\(874\) −4.59434 + 3.13231i −0.155406 + 0.105952i
\(875\) 9.14031i 0.308999i
\(876\) 0 0
\(877\) 33.7451i 1.13949i −0.821821 0.569746i \(-0.807041\pi\)
0.821821 0.569746i \(-0.192959\pi\)
\(878\) −3.12363 4.58161i −0.105417 0.154622i
\(879\) 0 0
\(880\) −42.2529 39.2078i −1.42435 1.32170i
\(881\) −8.58523 −0.289244 −0.144622 0.989487i \(-0.546197\pi\)
−0.144622 + 0.989487i \(0.546197\pi\)
\(882\) 0 0
\(883\) 8.07348i 0.271694i 0.990730 + 0.135847i \(0.0433756\pi\)
−0.990730 + 0.135847i \(0.956624\pi\)
\(884\) −31.3334 12.2981i −1.05386 0.413631i
\(885\) 0 0
\(886\) 40.4653 27.5882i 1.35946 0.926845i
\(887\) 31.4397 1.05564 0.527821 0.849356i \(-0.323009\pi\)
0.527821 + 0.849356i \(0.323009\pi\)
\(888\) 0 0
\(889\) −17.6103 −0.590632
\(890\) −27.4172 + 18.6924i −0.919026 + 0.626569i
\(891\) 0 0
\(892\) −2.03387 + 5.18194i −0.0680990 + 0.173504i
\(893\) 35.2471i 1.17950i
\(894\) 0 0
\(895\) −19.3382 −0.646404
\(896\) 9.79760 + 5.65748i 0.327315 + 0.189003i
\(897\) 0 0
\(898\) 14.1693 + 20.7829i 0.472835 + 0.693535i
\(899\) 56.3577i 1.87963i
\(900\) 0 0
\(901\) 61.2180i 2.03947i
\(902\) 37.2177 25.3741i 1.23921 0.844865i
\(903\) 0 0
\(904\) −19.0672 + 4.35141i −0.634165 + 0.144726i
\(905\) 6.32628 0.210293
\(906\) 0 0
\(907\) 10.4183i 0.345932i 0.984928 + 0.172966i \(0.0553352\pi\)
−0.984928 + 0.172966i \(0.944665\pi\)
\(908\) −21.3057 8.36231i −0.707054 0.277513i
\(909\) 0 0
\(910\) 6.15919 + 9.03405i 0.204175 + 0.299476i
\(911\) −47.9123 −1.58741 −0.793703 0.608305i \(-0.791850\pi\)
−0.793703 + 0.608305i \(0.791850\pi\)
\(912\) 0 0
\(913\) 33.2786 1.10136
\(914\) 9.22928 + 13.5371i 0.305277 + 0.447768i
\(915\) 0 0
\(916\) 8.45955 + 3.32031i 0.279511 + 0.109706i
\(917\) 10.7551i 0.355164i
\(918\) 0 0
\(919\) −37.6472 −1.24187 −0.620933 0.783863i \(-0.713246\pi\)
−0.620933 + 0.783863i \(0.713246\pi\)
\(920\) −0.858055 3.75986i −0.0282892 0.123959i
\(921\) 0 0
\(922\) 14.7168 10.0335i 0.484672 0.330437i
\(923\) 15.0885i 0.496644i
\(924\) 0 0
\(925\) 14.1929i 0.466661i
\(926\) −2.51825 3.69366i −0.0827547 0.121381i
\(927\) 0 0
\(928\) 46.7133 + 7.04308i 1.53344 + 0.231200i
\(929\) −44.8135 −1.47028 −0.735141 0.677914i \(-0.762884\pi\)
−0.735141 + 0.677914i \(0.762884\pi\)
\(930\) 0 0
\(931\) 7.28486i 0.238752i
\(932\) 11.7052 29.8228i 0.383417 0.976880i
\(933\) 0 0
\(934\) −24.4183 + 16.6478i −0.798990 + 0.544732i
\(935\) 79.2466 2.59164
\(936\) 0 0
\(937\) 29.2431 0.955330 0.477665 0.878542i \(-0.341483\pi\)
0.477665 + 0.878542i \(0.341483\pi\)
\(938\) −6.96824 + 4.75077i −0.227521 + 0.155118i
\(939\) 0 0
\(940\) −22.7558 8.93148i −0.742213 0.291313i
\(941\) 11.7936i 0.384460i −0.981350 0.192230i \(-0.938428\pi\)
0.981350 0.192230i \(-0.0615719\pi\)
\(942\) 0 0
\(943\) 3.01371 0.0981398
\(944\) 27.6561 29.8040i 0.900130 0.970039i
\(945\) 0 0
\(946\) 18.1014 + 26.5505i 0.588529 + 0.863230i
\(947\) 14.5710i 0.473493i −0.971572 0.236746i \(-0.923919\pi\)
0.971572 0.236746i \(-0.0760810\pi\)
\(948\) 0 0
\(949\) 26.5280i 0.861134i
\(950\) −11.7626 + 8.01942i −0.381628 + 0.260184i
\(951\) 0 0
\(952\) −15.1644 + 3.46073i −0.491480 + 0.112163i
\(953\) 54.0151 1.74972 0.874860 0.484375i \(-0.160953\pi\)
0.874860 + 0.484375i \(0.160953\pi\)
\(954\) 0 0
\(955\) 8.09977i 0.262102i
\(956\) 14.1005 35.9256i 0.456043 1.16192i
\(957\) 0 0
\(958\) −22.5264 33.0408i −0.727795 1.06750i
\(959\) 1.56413 0.0505084
\(960\) 0 0
\(961\) 14.5420 0.469097
\(962\) −25.0418 36.7303i −0.807381 1.18423i
\(963\) 0 0
\(964\) 16.1750 41.2110i 0.520961 1.32732i
\(965\) 27.6735i 0.890840i
\(966\) 0 0
\(967\) 39.1750 1.25978 0.629892 0.776683i \(-0.283099\pi\)
0.629892 + 0.776683i \(0.283099\pi\)
\(968\) 59.3951 13.5548i 1.90903 0.435669i
\(969\) 0 0
\(970\) 36.4554 24.8544i 1.17051 0.798026i
\(971\) 2.39896i 0.0769864i 0.999259 + 0.0384932i \(0.0122558\pi\)
−0.999259 + 0.0384932i \(0.987744\pi\)
\(972\) 0 0
\(973\) 11.2728i 0.361390i
\(974\) −21.4201 31.4181i −0.686343 1.00670i
\(975\) 0 0
\(976\) 11.2661 12.1411i 0.360620 0.388628i
\(977\) −15.7447 −0.503717 −0.251858 0.967764i \(-0.581042\pi\)
−0.251858 + 0.967764i \(0.581042\pi\)
\(978\) 0 0
\(979\) 52.9823i 1.69332i
\(980\) 4.70317 + 1.84596i 0.150237 + 0.0589669i
\(981\) 0 0
\(982\) 7.01257 4.78100i 0.223780 0.152568i
\(983\) 14.5566 0.464285 0.232142 0.972682i \(-0.425426\pi\)
0.232142 + 0.972682i \(0.425426\pi\)
\(984\) 0 0
\(985\) 19.9004 0.634078
\(986\) −53.6629 + 36.5860i −1.70898 + 1.16514i
\(987\) 0 0
\(988\) −16.2913 + 41.5074i −0.518296 + 1.32053i
\(989\) 2.14993i 0.0683637i
\(990\) 0 0
\(991\) −23.2853 −0.739681 −0.369841 0.929095i \(-0.620588\pi\)
−0.369841 + 0.929095i \(0.620588\pi\)
\(992\) 5.69143 37.7485i 0.180703 1.19852i
\(993\) 0 0
\(994\) 3.92757 + 5.76080i 0.124575 + 0.182721i
\(995\) 2.54525i 0.0806897i
\(996\) 0 0
\(997\) 1.38107i 0.0437390i 0.999761 + 0.0218695i \(0.00696184\pi\)
−0.999761 + 0.0218695i \(0.993038\pi\)
\(998\) −21.3978 + 14.5885i −0.677335 + 0.461790i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1512.2.c.f.757.18 yes 24
3.2 odd 2 inner 1512.2.c.f.757.7 24
4.3 odd 2 6048.2.c.g.3025.6 24
8.3 odd 2 6048.2.c.g.3025.19 24
8.5 even 2 inner 1512.2.c.f.757.17 yes 24
12.11 even 2 6048.2.c.g.3025.20 24
24.5 odd 2 inner 1512.2.c.f.757.8 yes 24
24.11 even 2 6048.2.c.g.3025.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.7 24 3.2 odd 2 inner
1512.2.c.f.757.8 yes 24 24.5 odd 2 inner
1512.2.c.f.757.17 yes 24 8.5 even 2 inner
1512.2.c.f.757.18 yes 24 1.1 even 1 trivial
6048.2.c.g.3025.5 24 24.11 even 2
6048.2.c.g.3025.6 24 4.3 odd 2
6048.2.c.g.3025.19 24 8.3 odd 2
6048.2.c.g.3025.20 24 12.11 even 2