Properties

Label 504.2.bs.a.257.1
Level $504$
Weight $2$
Character 504.257
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(257,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bs (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.1
Character \(\chi\) \(=\) 504.257
Dual form 504.2.bs.a.353.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.70597 + 0.299447i) q^{3} +(1.76701 + 3.06054i) q^{5} +(1.34480 + 2.27849i) q^{7} +(2.82066 - 1.02170i) q^{9} +O(q^{10})\) \(q+(-1.70597 + 0.299447i) q^{3} +(1.76701 + 3.06054i) q^{5} +(1.34480 + 2.27849i) q^{7} +(2.82066 - 1.02170i) q^{9} +(-1.57467 - 0.909136i) q^{11} +(-2.78280 - 1.60665i) q^{13} +(-3.93093 - 4.69207i) q^{15} +(2.93434 + 5.08242i) q^{17} +(-2.09143 - 1.20749i) q^{19} +(-2.97648 - 3.48434i) q^{21} +(-3.33308 + 1.92436i) q^{23} +(-3.74462 + 6.48586i) q^{25} +(-4.50602 + 2.58762i) q^{27} +(5.79110 - 3.34349i) q^{29} +4.87236i q^{31} +(2.95858 + 1.07943i) q^{33} +(-4.59714 + 8.14192i) q^{35} +(-0.905385 + 1.56817i) q^{37} +(5.22847 + 1.90759i) q^{39} +(-5.03152 + 8.71485i) q^{41} +(-2.36232 - 4.09166i) q^{43} +(8.11107 + 6.82742i) q^{45} +7.92151 q^{47} +(-3.38302 + 6.12823i) q^{49} +(-6.52781 - 7.79178i) q^{51} +(-2.26517 + 1.30780i) q^{53} -6.42580i q^{55} +(3.92950 + 1.43366i) q^{57} +8.81208 q^{59} -10.3744i q^{61} +(6.12115 + 5.05287i) q^{63} -11.3558i q^{65} -7.84769 q^{67} +(5.10990 - 4.28098i) q^{69} -7.62952i q^{71} +(-11.7672 + 6.79382i) q^{73} +(4.44602 - 12.1860i) q^{75} +(-0.0461599 - 4.81048i) q^{77} -1.84254 q^{79} +(6.91227 - 5.76372i) q^{81} +(5.61748 + 9.72975i) q^{83} +(-10.3700 + 17.9613i) q^{85} +(-8.87824 + 7.43803i) q^{87} +(6.89792 - 11.9475i) q^{89} +(-0.0815748 - 8.50119i) q^{91} +(-1.45902 - 8.31210i) q^{93} -8.53455i q^{95} +(13.7982 - 7.96638i) q^{97} +(-5.37048 - 0.955533i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75} - 36 q^{77} - 12 q^{79} - 24 q^{87} + 18 q^{89} + 6 q^{91} + 16 q^{93} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.70597 + 0.299447i −0.984942 + 0.172886i
\(4\) 0 0
\(5\) 1.76701 + 3.06054i 0.790229 + 1.36872i 0.925825 + 0.377952i \(0.123372\pi\)
−0.135596 + 0.990764i \(0.543295\pi\)
\(6\) 0 0
\(7\) 1.34480 + 2.27849i 0.508287 + 0.861188i
\(8\) 0 0
\(9\) 2.82066 1.02170i 0.940221 0.340565i
\(10\) 0 0
\(11\) −1.57467 0.909136i −0.474781 0.274115i 0.243458 0.969911i \(-0.421718\pi\)
−0.718239 + 0.695796i \(0.755052\pi\)
\(12\) 0 0
\(13\) −2.78280 1.60665i −0.771809 0.445604i 0.0617107 0.998094i \(-0.480344\pi\)
−0.833519 + 0.552490i \(0.813678\pi\)
\(14\) 0 0
\(15\) −3.93093 4.69207i −1.01496 1.21149i
\(16\) 0 0
\(17\) 2.93434 + 5.08242i 0.711682 + 1.23267i 0.964225 + 0.265083i \(0.0853995\pi\)
−0.252544 + 0.967585i \(0.581267\pi\)
\(18\) 0 0
\(19\) −2.09143 1.20749i −0.479807 0.277017i 0.240529 0.970642i \(-0.422679\pi\)
−0.720336 + 0.693625i \(0.756012\pi\)
\(20\) 0 0
\(21\) −2.97648 3.48434i −0.649520 0.760344i
\(22\) 0 0
\(23\) −3.33308 + 1.92436i −0.694996 + 0.401256i −0.805481 0.592622i \(-0.798093\pi\)
0.110485 + 0.993878i \(0.464760\pi\)
\(24\) 0 0
\(25\) −3.74462 + 6.48586i −0.748923 + 1.29717i
\(26\) 0 0
\(27\) −4.50602 + 2.58762i −0.867184 + 0.497988i
\(28\) 0 0
\(29\) 5.79110 3.34349i 1.07538 0.620871i 0.145734 0.989324i \(-0.453446\pi\)
0.929647 + 0.368452i \(0.120112\pi\)
\(30\) 0 0
\(31\) 4.87236i 0.875102i 0.899193 + 0.437551i \(0.144154\pi\)
−0.899193 + 0.437551i \(0.855846\pi\)
\(32\) 0 0
\(33\) 2.95858 + 1.07943i 0.515022 + 0.187904i
\(34\) 0 0
\(35\) −4.59714 + 8.14192i −0.777059 + 1.37624i
\(36\) 0 0
\(37\) −0.905385 + 1.56817i −0.148844 + 0.257806i −0.930801 0.365527i \(-0.880889\pi\)
0.781956 + 0.623333i \(0.214222\pi\)
\(38\) 0 0
\(39\) 5.22847 + 1.90759i 0.837225 + 0.305459i
\(40\) 0 0
\(41\) −5.03152 + 8.71485i −0.785792 + 1.36103i 0.142733 + 0.989761i \(0.454411\pi\)
−0.928525 + 0.371270i \(0.878922\pi\)
\(42\) 0 0
\(43\) −2.36232 4.09166i −0.360251 0.623973i 0.627751 0.778414i \(-0.283976\pi\)
−0.988002 + 0.154441i \(0.950642\pi\)
\(44\) 0 0
\(45\) 8.11107 + 6.82742i 1.20913 + 1.01777i
\(46\) 0 0
\(47\) 7.92151 1.15547 0.577736 0.816224i \(-0.303936\pi\)
0.577736 + 0.816224i \(0.303936\pi\)
\(48\) 0 0
\(49\) −3.38302 + 6.12823i −0.483289 + 0.875461i
\(50\) 0 0
\(51\) −6.52781 7.79178i −0.914076 1.09107i
\(52\) 0 0
\(53\) −2.26517 + 1.30780i −0.311146 + 0.179640i −0.647439 0.762117i \(-0.724160\pi\)
0.336293 + 0.941757i \(0.390827\pi\)
\(54\) 0 0
\(55\) 6.42580i 0.866454i
\(56\) 0 0
\(57\) 3.92950 + 1.43366i 0.520474 + 0.189893i
\(58\) 0 0
\(59\) 8.81208 1.14724 0.573618 0.819123i \(-0.305540\pi\)
0.573618 + 0.819123i \(0.305540\pi\)
\(60\) 0 0
\(61\) 10.3744i 1.32830i −0.747599 0.664151i \(-0.768793\pi\)
0.747599 0.664151i \(-0.231207\pi\)
\(62\) 0 0
\(63\) 6.12115 + 5.05287i 0.771193 + 0.636602i
\(64\) 0 0
\(65\) 11.3558i 1.40852i
\(66\) 0 0
\(67\) −7.84769 −0.958748 −0.479374 0.877611i \(-0.659136\pi\)
−0.479374 + 0.877611i \(0.659136\pi\)
\(68\) 0 0
\(69\) 5.10990 4.28098i 0.615159 0.515369i
\(70\) 0 0
\(71\) 7.62952i 0.905458i −0.891648 0.452729i \(-0.850451\pi\)
0.891648 0.452729i \(-0.149549\pi\)
\(72\) 0 0
\(73\) −11.7672 + 6.79382i −1.37725 + 0.795156i −0.991828 0.127584i \(-0.959278\pi\)
−0.385423 + 0.922740i \(0.625945\pi\)
\(74\) 0 0
\(75\) 4.44602 12.1860i 0.513383 1.40712i
\(76\) 0 0
\(77\) −0.0461599 4.81048i −0.00526041 0.548205i
\(78\) 0 0
\(79\) −1.84254 −0.207301 −0.103651 0.994614i \(-0.533052\pi\)
−0.103651 + 0.994614i \(0.533052\pi\)
\(80\) 0 0
\(81\) 6.91227 5.76372i 0.768031 0.640413i
\(82\) 0 0
\(83\) 5.61748 + 9.72975i 0.616598 + 1.06798i 0.990102 + 0.140351i \(0.0448230\pi\)
−0.373504 + 0.927629i \(0.621844\pi\)
\(84\) 0 0
\(85\) −10.3700 + 17.9613i −1.12478 + 1.94818i
\(86\) 0 0
\(87\) −8.87824 + 7.43803i −0.951848 + 0.797440i
\(88\) 0 0
\(89\) 6.89792 11.9475i 0.731178 1.26644i −0.225202 0.974312i \(-0.572304\pi\)
0.956380 0.292125i \(-0.0943624\pi\)
\(90\) 0 0
\(91\) −0.0815748 8.50119i −0.00855137 0.891167i
\(92\) 0 0
\(93\) −1.45902 8.31210i −0.151293 0.861925i
\(94\) 0 0
\(95\) 8.53455i 0.875627i
\(96\) 0 0
\(97\) 13.7982 7.96638i 1.40099 0.808863i 0.406497 0.913652i \(-0.366750\pi\)
0.994494 + 0.104789i \(0.0334168\pi\)
\(98\) 0 0
\(99\) −5.37048 0.955533i −0.539753 0.0960346i
\(100\) 0 0
\(101\) 5.56250 9.63453i 0.553489 0.958671i −0.444530 0.895764i \(-0.646629\pi\)
0.998019 0.0629076i \(-0.0200373\pi\)
\(102\) 0 0
\(103\) 0.522237 0.301514i 0.0514576 0.0297091i −0.474050 0.880498i \(-0.657209\pi\)
0.525508 + 0.850789i \(0.323875\pi\)
\(104\) 0 0
\(105\) 5.40451 15.2665i 0.527426 1.48985i
\(106\) 0 0
\(107\) 11.3342 + 6.54381i 1.09572 + 0.632614i 0.935094 0.354401i \(-0.115315\pi\)
0.160627 + 0.987015i \(0.448649\pi\)
\(108\) 0 0
\(109\) −1.37061 2.37396i −0.131280 0.227384i 0.792890 0.609365i \(-0.208575\pi\)
−0.924170 + 0.381981i \(0.875242\pi\)
\(110\) 0 0
\(111\) 1.07497 2.94637i 0.102032 0.279657i
\(112\) 0 0
\(113\) 6.65125 + 3.84010i 0.625697 + 0.361246i 0.779084 0.626920i \(-0.215685\pi\)
−0.153387 + 0.988166i \(0.549018\pi\)
\(114\) 0 0
\(115\) −11.7792 6.80070i −1.09841 0.634168i
\(116\) 0 0
\(117\) −9.49083 1.68864i −0.877428 0.156115i
\(118\) 0 0
\(119\) −7.63415 + 13.5207i −0.699821 + 1.23944i
\(120\) 0 0
\(121\) −3.84694 6.66310i −0.349722 0.605736i
\(122\) 0 0
\(123\) 5.97398 16.3739i 0.538656 1.47639i
\(124\) 0 0
\(125\) −8.79697 −0.786825
\(126\) 0 0
\(127\) 17.9246 1.59055 0.795277 0.606246i \(-0.207325\pi\)
0.795277 + 0.606246i \(0.207325\pi\)
\(128\) 0 0
\(129\) 5.25529 + 6.27286i 0.462702 + 0.552295i
\(130\) 0 0
\(131\) 8.48534 + 14.6970i 0.741367 + 1.28409i 0.951873 + 0.306493i \(0.0991556\pi\)
−0.210506 + 0.977593i \(0.567511\pi\)
\(132\) 0 0
\(133\) −0.0613082 6.38913i −0.00531609 0.554008i
\(134\) 0 0
\(135\) −15.8817 9.21852i −1.36688 0.793404i
\(136\) 0 0
\(137\) 4.81631 + 2.78070i 0.411485 + 0.237571i 0.691428 0.722446i \(-0.256982\pi\)
−0.279943 + 0.960017i \(0.590315\pi\)
\(138\) 0 0
\(139\) 12.4503 + 7.18817i 1.05602 + 0.609693i 0.924328 0.381598i \(-0.124626\pi\)
0.131690 + 0.991291i \(0.457960\pi\)
\(140\) 0 0
\(141\) −13.5139 + 2.37208i −1.13807 + 0.199765i
\(142\) 0 0
\(143\) 2.92132 + 5.05988i 0.244293 + 0.423129i
\(144\) 0 0
\(145\) 20.4658 + 11.8159i 1.69959 + 0.981261i
\(146\) 0 0
\(147\) 3.93626 11.4676i 0.324657 0.945832i
\(148\) 0 0
\(149\) −5.56086 + 3.21056i −0.455563 + 0.263019i −0.710177 0.704023i \(-0.751385\pi\)
0.254614 + 0.967043i \(0.418052\pi\)
\(150\) 0 0
\(151\) −7.99627 + 13.8499i −0.650727 + 1.12709i 0.332220 + 0.943202i \(0.392202\pi\)
−0.982947 + 0.183890i \(0.941131\pi\)
\(152\) 0 0
\(153\) 13.4695 + 11.3378i 1.08894 + 0.916607i
\(154\) 0 0
\(155\) −14.9121 + 8.60949i −1.19777 + 0.691531i
\(156\) 0 0
\(157\) 13.9271i 1.11151i 0.831348 + 0.555753i \(0.187570\pi\)
−0.831348 + 0.555753i \(0.812430\pi\)
\(158\) 0 0
\(159\) 3.47270 2.90937i 0.275403 0.230728i
\(160\) 0 0
\(161\) −8.86696 5.00652i −0.698814 0.394569i
\(162\) 0 0
\(163\) 7.79498 13.5013i 0.610550 1.05750i −0.380598 0.924740i \(-0.624282\pi\)
0.991148 0.132762i \(-0.0423847\pi\)
\(164\) 0 0
\(165\) 1.92419 + 10.9622i 0.149798 + 0.853407i
\(166\) 0 0
\(167\) 6.58728 11.4095i 0.509739 0.882893i −0.490198 0.871611i \(-0.663075\pi\)
0.999936 0.0112821i \(-0.00359127\pi\)
\(168\) 0 0
\(169\) −1.33736 2.31638i −0.102874 0.178183i
\(170\) 0 0
\(171\) −7.13291 1.26911i −0.545467 0.0970513i
\(172\) 0 0
\(173\) −11.2080 −0.852132 −0.426066 0.904692i \(-0.640101\pi\)
−0.426066 + 0.904692i \(0.640101\pi\)
\(174\) 0 0
\(175\) −19.8137 + 0.190127i −1.49778 + 0.0143722i
\(176\) 0 0
\(177\) −15.0331 + 2.63875i −1.12996 + 0.198341i
\(178\) 0 0
\(179\) −9.22783 + 5.32769i −0.689721 + 0.398210i −0.803507 0.595295i \(-0.797035\pi\)
0.113787 + 0.993505i \(0.463702\pi\)
\(180\) 0 0
\(181\) 11.8897i 0.883754i −0.897076 0.441877i \(-0.854313\pi\)
0.897076 0.441877i \(-0.145687\pi\)
\(182\) 0 0
\(183\) 3.10658 + 17.6984i 0.229645 + 1.30830i
\(184\) 0 0
\(185\) −6.39928 −0.470484
\(186\) 0 0
\(187\) 10.6709i 0.780330i
\(188\) 0 0
\(189\) −11.9556 6.78708i −0.869639 0.493688i
\(190\) 0 0
\(191\) 0.223967i 0.0162057i −0.999967 0.00810283i \(-0.997421\pi\)
0.999967 0.00810283i \(-0.00257924\pi\)
\(192\) 0 0
\(193\) 26.0745 1.87688 0.938442 0.345437i \(-0.112269\pi\)
0.938442 + 0.345437i \(0.112269\pi\)
\(194\) 0 0
\(195\) 3.40047 + 19.3727i 0.243513 + 1.38731i
\(196\) 0 0
\(197\) 1.56289i 0.111352i 0.998449 + 0.0556758i \(0.0177313\pi\)
−0.998449 + 0.0556758i \(0.982269\pi\)
\(198\) 0 0
\(199\) 13.4725 7.77832i 0.955037 0.551391i 0.0603949 0.998175i \(-0.480764\pi\)
0.894642 + 0.446784i \(0.147431\pi\)
\(200\) 0 0
\(201\) 13.3879 2.34997i 0.944311 0.165754i
\(202\) 0 0
\(203\) 15.4060 + 8.69863i 1.08129 + 0.610524i
\(204\) 0 0
\(205\) −35.5629 −2.48382
\(206\) 0 0
\(207\) −7.43540 + 8.83336i −0.516796 + 0.613961i
\(208\) 0 0
\(209\) 2.19554 + 3.80279i 0.151869 + 0.263045i
\(210\) 0 0
\(211\) 6.98216 12.0935i 0.480672 0.832548i −0.519082 0.854724i \(-0.673726\pi\)
0.999754 + 0.0221763i \(0.00705950\pi\)
\(212\) 0 0
\(213\) 2.28464 + 13.0157i 0.156541 + 0.891823i
\(214\) 0 0
\(215\) 8.34848 14.4600i 0.569361 0.986163i
\(216\) 0 0
\(217\) −11.1016 + 6.55236i −0.753628 + 0.444803i
\(218\) 0 0
\(219\) 18.0402 15.1137i 1.21904 1.02129i
\(220\) 0 0
\(221\) 18.8578i 1.26851i
\(222\) 0 0
\(223\) 21.4242 12.3693i 1.43467 0.828307i 0.437198 0.899366i \(-0.355971\pi\)
0.997472 + 0.0710586i \(0.0226377\pi\)
\(224\) 0 0
\(225\) −3.93571 + 22.1203i −0.262381 + 1.47469i
\(226\) 0 0
\(227\) −3.12076 + 5.40532i −0.207132 + 0.358764i −0.950810 0.309775i \(-0.899746\pi\)
0.743678 + 0.668538i \(0.233080\pi\)
\(228\) 0 0
\(229\) −13.7329 + 7.92870i −0.907496 + 0.523943i −0.879625 0.475668i \(-0.842206\pi\)
−0.0278714 + 0.999612i \(0.508873\pi\)
\(230\) 0 0
\(231\) 1.51923 + 8.19270i 0.0999581 + 0.539040i
\(232\) 0 0
\(233\) −2.94574 1.70072i −0.192982 0.111418i 0.400396 0.916342i \(-0.368873\pi\)
−0.593378 + 0.804924i \(0.702206\pi\)
\(234\) 0 0
\(235\) 13.9974 + 24.2441i 0.913087 + 1.58151i
\(236\) 0 0
\(237\) 3.14331 0.551742i 0.204180 0.0358395i
\(238\) 0 0
\(239\) −16.3482 9.43864i −1.05748 0.610535i −0.132744 0.991150i \(-0.542379\pi\)
−0.924733 + 0.380615i \(0.875712\pi\)
\(240\) 0 0
\(241\) −9.68610 5.59227i −0.623937 0.360230i 0.154463 0.987999i \(-0.450635\pi\)
−0.778400 + 0.627768i \(0.783968\pi\)
\(242\) 0 0
\(243\) −10.0662 + 11.9026i −0.645747 + 0.763552i
\(244\) 0 0
\(245\) −24.7335 + 0.474714i −1.58017 + 0.0303284i
\(246\) 0 0
\(247\) 3.88002 + 6.72039i 0.246880 + 0.427608i
\(248\) 0 0
\(249\) −12.4968 14.9165i −0.791952 0.945296i
\(250\) 0 0
\(251\) −15.4537 −0.975431 −0.487716 0.873003i \(-0.662170\pi\)
−0.487716 + 0.873003i \(0.662170\pi\)
\(252\) 0 0
\(253\) 6.99801 0.439961
\(254\) 0 0
\(255\) 12.3124 33.7468i 0.771032 2.11330i
\(256\) 0 0
\(257\) −5.32238 9.21864i −0.332001 0.575043i 0.650903 0.759161i \(-0.274390\pi\)
−0.982904 + 0.184118i \(0.941057\pi\)
\(258\) 0 0
\(259\) −4.79063 + 0.0459694i −0.297675 + 0.00285640i
\(260\) 0 0
\(261\) 12.9187 15.3476i 0.799648 0.949994i
\(262\) 0 0
\(263\) 9.60206 + 5.54375i 0.592088 + 0.341842i 0.765923 0.642933i \(-0.222282\pi\)
−0.173835 + 0.984775i \(0.555616\pi\)
\(264\) 0 0
\(265\) −8.00515 4.62178i −0.491753 0.283913i
\(266\) 0 0
\(267\) −8.18998 + 22.4477i −0.501218 + 1.37378i
\(268\) 0 0
\(269\) 3.14147 + 5.44119i 0.191539 + 0.331755i 0.945760 0.324865i \(-0.105319\pi\)
−0.754221 + 0.656620i \(0.771986\pi\)
\(270\) 0 0
\(271\) 0.954533 + 0.551100i 0.0579838 + 0.0334769i 0.528712 0.848802i \(-0.322675\pi\)
−0.470728 + 0.882278i \(0.656009\pi\)
\(272\) 0 0
\(273\) 2.68482 + 14.4783i 0.162493 + 0.876269i
\(274\) 0 0
\(275\) 11.7931 6.80873i 0.711149 0.410582i
\(276\) 0 0
\(277\) 7.68081 13.3036i 0.461495 0.799333i −0.537541 0.843238i \(-0.680647\pi\)
0.999036 + 0.0439049i \(0.0139798\pi\)
\(278\) 0 0
\(279\) 4.97807 + 13.7433i 0.298030 + 0.822790i
\(280\) 0 0
\(281\) 18.9133 10.9196i 1.12827 0.651407i 0.184771 0.982782i \(-0.440846\pi\)
0.943499 + 0.331374i \(0.107512\pi\)
\(282\) 0 0
\(283\) 11.0785i 0.658548i 0.944234 + 0.329274i \(0.106804\pi\)
−0.944234 + 0.329274i \(0.893196\pi\)
\(284\) 0 0
\(285\) 2.55565 + 14.5597i 0.151384 + 0.862441i
\(286\) 0 0
\(287\) −26.6231 + 0.255467i −1.57151 + 0.0150797i
\(288\) 0 0
\(289\) −8.72069 + 15.1047i −0.512982 + 0.888511i
\(290\) 0 0
\(291\) −21.1537 + 17.7222i −1.24005 + 1.03889i
\(292\) 0 0
\(293\) −12.6330 + 21.8810i −0.738027 + 1.27830i 0.215355 + 0.976536i \(0.430909\pi\)
−0.953382 + 0.301765i \(0.902424\pi\)
\(294\) 0 0
\(295\) 15.5710 + 26.9698i 0.906578 + 1.57024i
\(296\) 0 0
\(297\) 9.44800 + 0.0219345i 0.548228 + 0.00127277i
\(298\) 0 0
\(299\) 12.3671 0.715206
\(300\) 0 0
\(301\) 6.14596 10.8850i 0.354247 0.627401i
\(302\) 0 0
\(303\) −6.60442 + 18.1019i −0.379414 + 1.03993i
\(304\) 0 0
\(305\) 31.7512 18.3316i 1.81807 1.04966i
\(306\) 0 0
\(307\) 12.2163i 0.697219i 0.937268 + 0.348610i \(0.113346\pi\)
−0.937268 + 0.348610i \(0.886654\pi\)
\(308\) 0 0
\(309\) −0.800634 + 0.670756i −0.0455465 + 0.0381580i
\(310\) 0 0
\(311\) −9.48967 −0.538110 −0.269055 0.963125i \(-0.586711\pi\)
−0.269055 + 0.963125i \(0.586711\pi\)
\(312\) 0 0
\(313\) 13.5513i 0.765964i 0.923756 + 0.382982i \(0.125103\pi\)
−0.923756 + 0.382982i \(0.874897\pi\)
\(314\) 0 0
\(315\) −4.64842 + 27.6625i −0.261909 + 1.55861i
\(316\) 0 0
\(317\) 15.8839i 0.892131i −0.895000 0.446065i \(-0.852825\pi\)
0.895000 0.446065i \(-0.147175\pi\)
\(318\) 0 0
\(319\) −12.1588 −0.680761
\(320\) 0 0
\(321\) −21.2954 7.76954i −1.18859 0.433654i
\(322\) 0 0
\(323\) 14.1727i 0.788591i
\(324\) 0 0
\(325\) 20.8410 12.0326i 1.15605 0.667446i
\(326\) 0 0
\(327\) 3.04909 + 3.63948i 0.168615 + 0.201264i
\(328\) 0 0
\(329\) 10.6529 + 18.0491i 0.587311 + 0.995078i
\(330\) 0 0
\(331\) 10.9739 0.603183 0.301591 0.953437i \(-0.402482\pi\)
0.301591 + 0.953437i \(0.402482\pi\)
\(332\) 0 0
\(333\) −0.951590 + 5.34831i −0.0521468 + 0.293086i
\(334\) 0 0
\(335\) −13.8669 24.0182i −0.757630 1.31225i
\(336\) 0 0
\(337\) 2.80751 4.86275i 0.152935 0.264891i −0.779370 0.626564i \(-0.784461\pi\)
0.932305 + 0.361673i \(0.117794\pi\)
\(338\) 0 0
\(339\) −12.4967 4.55940i −0.678730 0.247632i
\(340\) 0 0
\(341\) 4.42964 7.67237i 0.239879 0.415482i
\(342\) 0 0
\(343\) −18.5126 + 0.533054i −0.999586 + 0.0287822i
\(344\) 0 0
\(345\) 22.1313 + 8.07455i 1.19151 + 0.434719i
\(346\) 0 0
\(347\) 4.82668i 0.259110i −0.991572 0.129555i \(-0.958645\pi\)
0.991572 0.129555i \(-0.0413549\pi\)
\(348\) 0 0
\(349\) −22.7167 + 13.1155i −1.21600 + 0.702056i −0.964059 0.265688i \(-0.914401\pi\)
−0.251937 + 0.967744i \(0.581068\pi\)
\(350\) 0 0
\(351\) 16.6967 + 0.0387633i 0.891206 + 0.00206903i
\(352\) 0 0
\(353\) −17.4957 + 30.3034i −0.931201 + 1.61289i −0.149930 + 0.988697i \(0.547905\pi\)
−0.781271 + 0.624192i \(0.785428\pi\)
\(354\) 0 0
\(355\) 23.3505 13.4814i 1.23931 0.715519i
\(356\) 0 0
\(357\) 8.97488 25.3519i 0.475001 1.34177i
\(358\) 0 0
\(359\) 21.6396 + 12.4936i 1.14209 + 0.659388i 0.946948 0.321386i \(-0.104149\pi\)
0.195146 + 0.980774i \(0.437482\pi\)
\(360\) 0 0
\(361\) −6.58394 11.4037i −0.346523 0.600196i
\(362\) 0 0
\(363\) 8.55801 + 10.2151i 0.449179 + 0.536153i
\(364\) 0 0
\(365\) −41.5855 24.0094i −2.17669 1.25671i
\(366\) 0 0
\(367\) −21.3100 12.3034i −1.11237 0.642230i −0.172931 0.984934i \(-0.555324\pi\)
−0.939443 + 0.342704i \(0.888657\pi\)
\(368\) 0 0
\(369\) −5.28830 + 29.7223i −0.275298 + 1.54728i
\(370\) 0 0
\(371\) −6.02601 3.40245i −0.312855 0.176646i
\(372\) 0 0
\(373\) 5.21870 + 9.03905i 0.270214 + 0.468024i 0.968916 0.247388i \(-0.0795723\pi\)
−0.698703 + 0.715412i \(0.746239\pi\)
\(374\) 0 0
\(375\) 15.0074 2.63423i 0.774977 0.136031i
\(376\) 0 0
\(377\) −21.4873 −1.10665
\(378\) 0 0
\(379\) 31.8728 1.63719 0.818597 0.574368i \(-0.194752\pi\)
0.818597 + 0.574368i \(0.194752\pi\)
\(380\) 0 0
\(381\) −30.5789 + 5.36749i −1.56660 + 0.274985i
\(382\) 0 0
\(383\) −12.3860 21.4531i −0.632893 1.09620i −0.986957 0.160982i \(-0.948534\pi\)
0.354065 0.935221i \(-0.384799\pi\)
\(384\) 0 0
\(385\) 14.6411 8.64141i 0.746180 0.440407i
\(386\) 0 0
\(387\) −10.8438 9.12763i −0.551219 0.463983i
\(388\) 0 0
\(389\) 19.9382 + 11.5113i 1.01091 + 0.583647i 0.911457 0.411395i \(-0.134958\pi\)
0.0994499 + 0.995043i \(0.468292\pi\)
\(390\) 0 0
\(391\) −19.5608 11.2934i −0.989232 0.571133i
\(392\) 0 0
\(393\) −18.8767 22.5318i −0.952204 1.13658i
\(394\) 0 0
\(395\) −3.25577 5.63916i −0.163816 0.283737i
\(396\) 0 0
\(397\) 23.9247 + 13.8129i 1.20075 + 0.693251i 0.960721 0.277515i \(-0.0895109\pi\)
0.240025 + 0.970767i \(0.422844\pi\)
\(398\) 0 0
\(399\) 2.01780 + 10.8813i 0.101016 + 0.544747i
\(400\) 0 0
\(401\) 4.61058 2.66192i 0.230241 0.132930i −0.380442 0.924805i \(-0.624228\pi\)
0.610683 + 0.791875i \(0.290895\pi\)
\(402\) 0 0
\(403\) 7.82817 13.5588i 0.389949 0.675412i
\(404\) 0 0
\(405\) 29.8541 + 10.9708i 1.48346 + 0.545143i
\(406\) 0 0
\(407\) 2.85137 1.64624i 0.141337 0.0816009i
\(408\) 0 0
\(409\) 7.06575i 0.349379i −0.984624 0.174690i \(-0.944108\pi\)
0.984624 0.174690i \(-0.0558922\pi\)
\(410\) 0 0
\(411\) −9.04915 3.30155i −0.446362 0.162854i
\(412\) 0 0
\(413\) 11.8505 + 20.0782i 0.583124 + 0.987985i
\(414\) 0 0
\(415\) −19.8522 + 34.3851i −0.974507 + 1.68790i
\(416\) 0 0
\(417\) −23.3923 8.53459i −1.14552 0.417941i
\(418\) 0 0
\(419\) 1.94504 3.36891i 0.0950215 0.164582i −0.814596 0.580029i \(-0.803041\pi\)
0.909618 + 0.415447i \(0.136375\pi\)
\(420\) 0 0
\(421\) −2.18533 3.78510i −0.106506 0.184475i 0.807846 0.589393i \(-0.200633\pi\)
−0.914353 + 0.404919i \(0.867300\pi\)
\(422\) 0 0
\(423\) 22.3439 8.09338i 1.08640 0.393514i
\(424\) 0 0
\(425\) −43.9519 −2.13198
\(426\) 0 0
\(427\) 23.6379 13.9515i 1.14392 0.675158i
\(428\) 0 0
\(429\) −6.49886 7.75722i −0.313768 0.374522i
\(430\) 0 0
\(431\) −10.7505 + 6.20681i −0.517834 + 0.298971i −0.736048 0.676929i \(-0.763310\pi\)
0.218214 + 0.975901i \(0.429977\pi\)
\(432\) 0 0
\(433\) 25.2979i 1.21574i −0.794036 0.607870i \(-0.792024\pi\)
0.794036 0.607870i \(-0.207976\pi\)
\(434\) 0 0
\(435\) −38.4523 14.0292i −1.84365 0.672649i
\(436\) 0 0
\(437\) 9.29455 0.444619
\(438\) 0 0
\(439\) 26.8362i 1.28082i −0.768033 0.640410i \(-0.778764\pi\)
0.768033 0.640410i \(-0.221236\pi\)
\(440\) 0 0
\(441\) −3.28119 + 20.7421i −0.156247 + 0.987718i
\(442\) 0 0
\(443\) 13.0540i 0.620213i 0.950702 + 0.310107i \(0.100365\pi\)
−0.950702 + 0.310107i \(0.899635\pi\)
\(444\) 0 0
\(445\) 48.7546 2.31119
\(446\) 0 0
\(447\) 8.52526 7.14230i 0.403231 0.337819i
\(448\) 0 0
\(449\) 0.468645i 0.0221167i −0.999939 0.0110584i \(-0.996480\pi\)
0.999939 0.0110584i \(-0.00352006\pi\)
\(450\) 0 0
\(451\) 15.8460 9.14868i 0.746158 0.430795i
\(452\) 0 0
\(453\) 9.49406 26.0220i 0.446070 1.22262i
\(454\) 0 0
\(455\) 25.8741 15.2713i 1.21300 0.715930i
\(456\) 0 0
\(457\) −4.12972 −0.193180 −0.0965901 0.995324i \(-0.530794\pi\)
−0.0965901 + 0.995324i \(0.530794\pi\)
\(458\) 0 0
\(459\) −26.3736 15.3085i −1.23101 0.714542i
\(460\) 0 0
\(461\) −12.9115 22.3633i −0.601347 1.04156i −0.992617 0.121288i \(-0.961298\pi\)
0.391271 0.920276i \(-0.372036\pi\)
\(462\) 0 0
\(463\) −2.56402 + 4.44101i −0.119160 + 0.206391i −0.919435 0.393242i \(-0.871353\pi\)
0.800275 + 0.599633i \(0.204687\pi\)
\(464\) 0 0
\(465\) 22.8615 19.1529i 1.06017 0.888195i
\(466\) 0 0
\(467\) 1.26091 2.18396i 0.0583480 0.101062i −0.835376 0.549679i \(-0.814750\pi\)
0.893724 + 0.448617i \(0.148083\pi\)
\(468\) 0 0
\(469\) −10.5536 17.8809i −0.487319 0.825662i
\(470\) 0 0
\(471\) −4.17044 23.7592i −0.192164 1.09477i
\(472\) 0 0
\(473\) 8.59070i 0.395001i
\(474\) 0 0
\(475\) 15.6632 9.04316i 0.718677 0.414929i
\(476\) 0 0
\(477\) −5.05312 + 6.00318i −0.231366 + 0.274867i
\(478\) 0 0
\(479\) −5.74622 + 9.95275i −0.262552 + 0.454753i −0.966919 0.255083i \(-0.917897\pi\)
0.704368 + 0.709835i \(0.251231\pi\)
\(480\) 0 0
\(481\) 5.03900 2.90927i 0.229759 0.132651i
\(482\) 0 0
\(483\) 16.6260 + 5.88578i 0.756507 + 0.267812i
\(484\) 0 0
\(485\) 48.7629 + 28.1533i 2.21421 + 1.27837i
\(486\) 0 0
\(487\) −10.8463 18.7864i −0.491493 0.851292i 0.508459 0.861086i \(-0.330215\pi\)
−0.999952 + 0.00979483i \(0.996882\pi\)
\(488\) 0 0
\(489\) −9.25506 + 25.3670i −0.418528 + 1.14713i
\(490\) 0 0
\(491\) −19.3592 11.1771i −0.873669 0.504413i −0.00510349 0.999987i \(-0.501624\pi\)
−0.868566 + 0.495574i \(0.834958\pi\)
\(492\) 0 0
\(493\) 33.9861 + 19.6219i 1.53066 + 0.883726i
\(494\) 0 0
\(495\) −6.56521 18.1250i −0.295084 0.814658i
\(496\) 0 0
\(497\) 17.3838 10.2602i 0.779769 0.460232i
\(498\) 0 0
\(499\) −8.76231 15.1768i −0.392255 0.679405i 0.600492 0.799631i \(-0.294972\pi\)
−0.992747 + 0.120226i \(0.961638\pi\)
\(500\) 0 0
\(501\) −7.82115 + 21.4368i −0.349423 + 0.957725i
\(502\) 0 0
\(503\) 7.60926 0.339280 0.169640 0.985506i \(-0.445739\pi\)
0.169640 + 0.985506i \(0.445739\pi\)
\(504\) 0 0
\(505\) 39.3159 1.74953
\(506\) 0 0
\(507\) 2.97514 + 3.55121i 0.132130 + 0.157715i
\(508\) 0 0
\(509\) −3.39373 5.87811i −0.150424 0.260543i 0.780959 0.624582i \(-0.214731\pi\)
−0.931384 + 0.364039i \(0.881397\pi\)
\(510\) 0 0
\(511\) −31.3042 17.6752i −1.38482 0.781905i
\(512\) 0 0
\(513\) 12.5486 + 0.0291328i 0.554032 + 0.00128624i
\(514\) 0 0
\(515\) 1.84559 + 1.06555i 0.0813265 + 0.0469539i
\(516\) 0 0
\(517\) −12.4738 7.20174i −0.548596 0.316732i
\(518\) 0 0
\(519\) 19.1206 3.35622i 0.839300 0.147322i
\(520\) 0 0
\(521\) 13.8435 + 23.9777i 0.606497 + 1.05048i 0.991813 + 0.127698i \(0.0407590\pi\)
−0.385316 + 0.922785i \(0.625908\pi\)
\(522\) 0 0
\(523\) −0.834923 0.482043i −0.0365086 0.0210783i 0.481635 0.876372i \(-0.340043\pi\)
−0.518143 + 0.855294i \(0.673377\pi\)
\(524\) 0 0
\(525\) 33.7447 6.25752i 1.47274 0.273101i
\(526\) 0 0
\(527\) −24.7634 + 14.2972i −1.07871 + 0.622794i
\(528\) 0 0
\(529\) −4.09370 + 7.09049i −0.177987 + 0.308282i
\(530\) 0 0
\(531\) 24.8559 9.00327i 1.07865 0.390709i
\(532\) 0 0
\(533\) 28.0034 16.1678i 1.21296 0.700304i
\(534\) 0 0
\(535\) 46.2518i 1.99964i
\(536\) 0 0
\(537\) 14.1470 11.8521i 0.610490 0.511457i
\(538\) 0 0
\(539\) 10.8985 6.57430i 0.469433 0.283175i
\(540\) 0 0
\(541\) 2.36867 4.10266i 0.101837 0.176387i −0.810604 0.585594i \(-0.800861\pi\)
0.912442 + 0.409207i \(0.134195\pi\)
\(542\) 0 0
\(543\) 3.56034 + 20.2835i 0.152789 + 0.870447i
\(544\) 0 0
\(545\) 4.84374 8.38960i 0.207483 0.359371i
\(546\) 0 0
\(547\) −0.840875 1.45644i −0.0359532 0.0622728i 0.847489 0.530813i \(-0.178113\pi\)
−0.883442 + 0.468540i \(0.844780\pi\)
\(548\) 0 0
\(549\) −10.5994 29.2626i −0.452373 1.24890i
\(550\) 0 0
\(551\) −16.1489 −0.687967
\(552\) 0 0
\(553\) −2.47784 4.19820i −0.105369 0.178525i
\(554\) 0 0
\(555\) 10.9170 1.91625i 0.463400 0.0813402i
\(556\) 0 0
\(557\) −8.14983 + 4.70530i −0.345319 + 0.199370i −0.662622 0.748954i \(-0.730556\pi\)
0.317303 + 0.948324i \(0.397223\pi\)
\(558\) 0 0
\(559\) 15.1817i 0.642117i
\(560\) 0 0
\(561\) 3.19536 + 18.2042i 0.134908 + 0.768580i
\(562\) 0 0
\(563\) −31.0638 −1.30918 −0.654591 0.755983i \(-0.727159\pi\)
−0.654591 + 0.755983i \(0.727159\pi\)
\(564\) 0 0
\(565\) 27.1419i 1.14187i
\(566\) 0 0
\(567\) 22.4282 + 7.99849i 0.941896 + 0.335905i
\(568\) 0 0
\(569\) 20.1032i 0.842769i −0.906882 0.421384i \(-0.861544\pi\)
0.906882 0.421384i \(-0.138456\pi\)
\(570\) 0 0
\(571\) −7.87413 −0.329522 −0.164761 0.986333i \(-0.552685\pi\)
−0.164761 + 0.986333i \(0.552685\pi\)
\(572\) 0 0
\(573\) 0.0670662 + 0.382080i 0.00280173 + 0.0159616i
\(574\) 0 0
\(575\) 28.8239i 1.20204i
\(576\) 0 0
\(577\) −2.77842 + 1.60412i −0.115667 + 0.0667805i −0.556717 0.830702i \(-0.687939\pi\)
0.441050 + 0.897482i \(0.354606\pi\)
\(578\) 0 0
\(579\) −44.4823 + 7.80794i −1.84862 + 0.324487i
\(580\) 0 0
\(581\) −14.6148 + 25.8839i −0.606322 + 1.07385i
\(582\) 0 0
\(583\) 4.75587 0.196968
\(584\) 0 0
\(585\) −11.6022 32.0309i −0.479692 1.32432i
\(586\) 0 0
\(587\) 2.37708 + 4.11722i 0.0981125 + 0.169936i 0.910903 0.412620i \(-0.135386\pi\)
−0.812791 + 0.582556i \(0.802053\pi\)
\(588\) 0 0
\(589\) 5.88332 10.1902i 0.242418 0.419880i
\(590\) 0 0
\(591\) −0.468004 2.66625i −0.0192511 0.109675i
\(592\) 0 0
\(593\) −13.0354 + 22.5780i −0.535300 + 0.927166i 0.463849 + 0.885914i \(0.346468\pi\)
−0.999149 + 0.0412519i \(0.986865\pi\)
\(594\) 0 0
\(595\) −54.8703 + 0.526519i −2.24946 + 0.0215852i
\(596\) 0 0
\(597\) −20.6544 + 17.3039i −0.845328 + 0.708200i
\(598\) 0 0
\(599\) 27.5024i 1.12372i −0.827233 0.561859i \(-0.810086\pi\)
0.827233 0.561859i \(-0.189914\pi\)
\(600\) 0 0
\(601\) 26.8647 15.5104i 1.09584 0.632681i 0.160712 0.987001i \(-0.448621\pi\)
0.935124 + 0.354320i \(0.115288\pi\)
\(602\) 0 0
\(603\) −22.1357 + 8.01795i −0.901435 + 0.326516i
\(604\) 0 0
\(605\) 13.5951 23.5475i 0.552721 0.957340i
\(606\) 0 0
\(607\) −33.9745 + 19.6152i −1.37898 + 0.796156i −0.992037 0.125947i \(-0.959803\pi\)
−0.386945 + 0.922103i \(0.626470\pi\)
\(608\) 0 0
\(609\) −28.8869 10.2263i −1.17056 0.414391i
\(610\) 0 0
\(611\) −22.0440 12.7271i −0.891803 0.514883i
\(612\) 0 0
\(613\) −11.5364 19.9817i −0.465951 0.807051i 0.533293 0.845931i \(-0.320954\pi\)
−0.999244 + 0.0388795i \(0.987621\pi\)
\(614\) 0 0
\(615\) 60.6692 10.6492i 2.44642 0.429418i
\(616\) 0 0
\(617\) 5.15715 + 2.97748i 0.207619 + 0.119869i 0.600204 0.799847i \(-0.295086\pi\)
−0.392585 + 0.919716i \(0.628419\pi\)
\(618\) 0 0
\(619\) −28.0289 16.1825i −1.12658 0.650429i −0.183504 0.983019i \(-0.558744\pi\)
−0.943072 + 0.332590i \(0.892078\pi\)
\(620\) 0 0
\(621\) 10.0394 17.2960i 0.402869 0.694063i
\(622\) 0 0
\(623\) 36.4987 0.350230i 1.46229 0.0140317i
\(624\) 0 0
\(625\) 3.17879 + 5.50582i 0.127152 + 0.220233i
\(626\) 0 0
\(627\) −4.88426 5.83000i −0.195059 0.232828i
\(628\) 0 0
\(629\) −10.6268 −0.423719
\(630\) 0 0
\(631\) −35.6689 −1.41996 −0.709978 0.704224i \(-0.751295\pi\)
−0.709978 + 0.704224i \(0.751295\pi\)
\(632\) 0 0
\(633\) −8.29000 + 22.7219i −0.329498 + 0.903113i
\(634\) 0 0
\(635\) 31.6729 + 54.8591i 1.25690 + 2.17702i
\(636\) 0 0
\(637\) 19.2602 11.6183i 0.763116 0.460333i
\(638\) 0 0
\(639\) −7.79505 21.5203i −0.308367 0.851330i
\(640\) 0 0
\(641\) 21.5239 + 12.4268i 0.850141 + 0.490829i 0.860699 0.509115i \(-0.170027\pi\)
−0.0105572 + 0.999944i \(0.503361\pi\)
\(642\) 0 0
\(643\) −14.8270 8.56038i −0.584720 0.337588i 0.178287 0.983979i \(-0.442944\pi\)
−0.763007 + 0.646390i \(0.776278\pi\)
\(644\) 0 0
\(645\) −9.91224 + 27.1682i −0.390294 + 1.06975i
\(646\) 0 0
\(647\) −16.7208 28.9613i −0.657363 1.13859i −0.981296 0.192506i \(-0.938339\pi\)
0.323933 0.946080i \(-0.394995\pi\)
\(648\) 0 0
\(649\) −13.8761 8.01138i −0.544686 0.314474i
\(650\) 0 0
\(651\) 16.9770 14.5025i 0.665379 0.568397i
\(652\) 0 0
\(653\) 22.8837 13.2119i 0.895508 0.517022i 0.0197678 0.999805i \(-0.493707\pi\)
0.875740 + 0.482783i \(0.160374\pi\)
\(654\) 0 0
\(655\) −29.9873 + 51.9395i −1.17170 + 2.02944i
\(656\) 0 0
\(657\) −26.2502 + 31.1856i −1.02412 + 1.21667i
\(658\) 0 0
\(659\) 25.8862 14.9454i 1.00838 0.582191i 0.0976658 0.995219i \(-0.468862\pi\)
0.910718 + 0.413029i \(0.135529\pi\)
\(660\) 0 0
\(661\) 4.53368i 0.176340i 0.996105 + 0.0881699i \(0.0281018\pi\)
−0.996105 + 0.0881699i \(0.971898\pi\)
\(662\) 0 0
\(663\) 5.64692 + 32.1708i 0.219308 + 1.24941i
\(664\) 0 0
\(665\) 19.4459 11.4773i 0.754079 0.445069i
\(666\) 0 0
\(667\) −12.8682 + 22.2883i −0.498257 + 0.863006i
\(668\) 0 0
\(669\) −32.8451 + 27.5170i −1.26986 + 1.06387i
\(670\) 0 0
\(671\) −9.43172 + 16.3362i −0.364107 + 0.630652i
\(672\) 0 0
\(673\) 4.12239 + 7.14019i 0.158907 + 0.275234i 0.934475 0.356030i \(-0.115870\pi\)
−0.775568 + 0.631264i \(0.782536\pi\)
\(674\) 0 0
\(675\) 0.0903455 38.9151i 0.00347740 1.49784i
\(676\) 0 0
\(677\) 5.51263 0.211867 0.105934 0.994373i \(-0.466217\pi\)
0.105934 + 0.994373i \(0.466217\pi\)
\(678\) 0 0
\(679\) 36.7071 + 20.7258i 1.40869 + 0.795383i
\(680\) 0 0
\(681\) 3.70532 10.1558i 0.141988 0.389172i
\(682\) 0 0
\(683\) −24.0097 + 13.8620i −0.918706 + 0.530415i −0.883222 0.468955i \(-0.844631\pi\)
−0.0354840 + 0.999370i \(0.511297\pi\)
\(684\) 0 0
\(685\) 19.6540i 0.750942i
\(686\) 0 0
\(687\) 21.0537 17.6384i 0.803248 0.672947i
\(688\) 0 0
\(689\) 8.40469 0.320193
\(690\) 0 0
\(691\) 42.1826i 1.60470i 0.596853 + 0.802351i \(0.296418\pi\)
−0.596853 + 0.802351i \(0.703582\pi\)
\(692\) 0 0
\(693\) −5.04505 13.5216i −0.191645 0.513642i
\(694\) 0 0
\(695\) 50.8061i 1.92719i
\(696\) 0 0
\(697\) −59.0568 −2.23693
\(698\) 0 0
\(699\) 5.53461 + 2.01929i 0.209338 + 0.0763764i
\(700\) 0 0
\(701\) 1.07738i 0.0406922i 0.999793 + 0.0203461i \(0.00647681\pi\)
−0.999793 + 0.0203461i \(0.993523\pi\)
\(702\) 0 0
\(703\) 3.78710 2.18648i 0.142833 0.0824648i
\(704\) 0 0
\(705\) −31.1389 37.1683i −1.17276 1.39984i
\(706\) 0 0
\(707\) 29.4326 0.282426i 1.10693 0.0106217i
\(708\) 0 0
\(709\) −21.9612 −0.824769 −0.412385 0.911010i \(-0.635304\pi\)
−0.412385 + 0.911010i \(0.635304\pi\)
\(710\) 0 0
\(711\) −5.19717 + 1.88251i −0.194909 + 0.0705997i
\(712\) 0 0
\(713\) −9.37617 16.2400i −0.351140 0.608193i
\(714\) 0 0
\(715\) −10.3240 + 17.8817i −0.386095 + 0.668737i
\(716\) 0 0
\(717\) 30.7159 + 11.2066i 1.14711 + 0.418518i
\(718\) 0 0
\(719\) 5.89267 10.2064i 0.219760 0.380635i −0.734975 0.678094i \(-0.762806\pi\)
0.954734 + 0.297460i \(0.0961394\pi\)
\(720\) 0 0
\(721\) 1.38930 + 0.784436i 0.0517403 + 0.0292139i
\(722\) 0 0
\(723\) 18.1988 + 6.63977i 0.676820 + 0.246936i
\(724\) 0 0
\(725\) 50.0804i 1.85994i
\(726\) 0 0
\(727\) 10.1963 5.88682i 0.378159 0.218330i −0.298858 0.954298i \(-0.596606\pi\)
0.677017 + 0.735967i \(0.263272\pi\)
\(728\) 0 0
\(729\) 13.6084 23.3198i 0.504016 0.863694i
\(730\) 0 0
\(731\) 13.8637 24.0127i 0.512768 0.888140i
\(732\) 0 0
\(733\) −37.9020 + 21.8828i −1.39994 + 0.808258i −0.994386 0.105812i \(-0.966256\pi\)
−0.405558 + 0.914069i \(0.632923\pi\)
\(734\) 0 0
\(735\) 42.0525 8.21624i 1.55113 0.303060i
\(736\) 0 0
\(737\) 12.3575 + 7.13462i 0.455195 + 0.262807i
\(738\) 0 0
\(739\) 13.9728 + 24.2016i 0.513998 + 0.890270i 0.999868 + 0.0162393i \(0.00516935\pi\)
−0.485870 + 0.874031i \(0.661497\pi\)
\(740\) 0 0
\(741\) −8.63159 10.3029i −0.317089 0.378487i
\(742\) 0 0
\(743\) 31.0951 + 17.9528i 1.14077 + 0.658623i 0.946621 0.322350i \(-0.104473\pi\)
0.194147 + 0.980972i \(0.437806\pi\)
\(744\) 0 0
\(745\) −19.6521 11.3462i −0.719998 0.415691i
\(746\) 0 0
\(747\) 25.7859 + 21.7050i 0.943455 + 0.794144i
\(748\) 0 0
\(749\) 0.332251 + 34.6250i 0.0121402 + 1.26517i
\(750\) 0 0
\(751\) 1.17516 + 2.03543i 0.0428821 + 0.0742740i 0.886670 0.462403i \(-0.153013\pi\)
−0.843788 + 0.536677i \(0.819679\pi\)
\(752\) 0 0
\(753\) 26.3636 4.62758i 0.960743 0.168638i
\(754\) 0 0
\(755\) −56.5178 −2.05689
\(756\) 0 0
\(757\) 8.88061 0.322771 0.161386 0.986891i \(-0.448404\pi\)
0.161386 + 0.986891i \(0.448404\pi\)
\(758\) 0 0
\(759\) −11.9384 + 2.09554i −0.433336 + 0.0760632i
\(760\) 0 0
\(761\) 13.4883 + 23.3624i 0.488951 + 0.846888i 0.999919 0.0127118i \(-0.00404640\pi\)
−0.510968 + 0.859600i \(0.670713\pi\)
\(762\) 0 0
\(763\) 3.56585 6.31541i 0.129092 0.228633i
\(764\) 0 0
\(765\) −10.8992 + 61.2579i −0.394061 + 2.21478i
\(766\) 0 0
\(767\) −24.5222 14.1579i −0.885446 0.511213i
\(768\) 0 0
\(769\) −7.88232 4.55086i −0.284244 0.164108i 0.351099 0.936338i \(-0.385808\pi\)
−0.635343 + 0.772230i \(0.719141\pi\)
\(770\) 0 0
\(771\) 11.8403 + 14.1329i 0.426419 + 0.508985i
\(772\) 0 0
\(773\) 6.46864 + 11.2040i 0.232661 + 0.402980i 0.958590 0.284789i \(-0.0919235\pi\)
−0.725929 + 0.687769i \(0.758590\pi\)
\(774\) 0 0
\(775\) −31.6015 18.2451i −1.13516 0.655384i
\(776\) 0 0
\(777\) 8.15890 1.51296i 0.292699 0.0542772i
\(778\) 0 0
\(779\) 21.0462 12.1510i 0.754057 0.435355i
\(780\) 0 0
\(781\) −6.93628 + 12.0140i −0.248199 + 0.429894i
\(782\) 0 0
\(783\) −17.4431 + 30.0510i −0.623366 + 1.07394i
\(784\) 0 0
\(785\) −42.6245 + 24.6093i −1.52134 + 0.878343i
\(786\) 0 0
\(787\) 24.8566i 0.886043i −0.896511 0.443022i \(-0.853906\pi\)
0.896511 0.443022i \(-0.146094\pi\)
\(788\) 0 0
\(789\) −18.0409 6.58216i −0.642272 0.234331i
\(790\) 0 0
\(791\) 0.194975 + 20.3190i 0.00693250 + 0.722460i
\(792\) 0 0
\(793\) −16.6680 + 28.8697i −0.591896 + 1.02519i
\(794\) 0 0
\(795\) 15.0405 + 5.48749i 0.533432 + 0.194621i
\(796\) 0 0
\(797\) −6.71671 + 11.6337i −0.237918 + 0.412086i −0.960117 0.279600i \(-0.909798\pi\)
0.722199 + 0.691686i \(0.243132\pi\)
\(798\) 0 0
\(799\) 23.2444 + 40.2605i 0.822328 + 1.42431i
\(800\) 0 0
\(801\) 7.24994 40.7476i 0.256164 1.43974i
\(802\) 0 0
\(803\) 24.7060 0.871857
\(804\) 0 0
\(805\) −0.345294 35.9843i −0.0121700 1.26828i
\(806\) 0 0
\(807\) −6.98860 8.34180i −0.246011 0.293645i
\(808\) 0 0
\(809\) 9.38524 5.41857i 0.329967 0.190507i −0.325859 0.945418i \(-0.605654\pi\)
0.655827 + 0.754912i \(0.272320\pi\)
\(810\) 0 0
\(811\) 35.3334i 1.24072i −0.784316 0.620361i \(-0.786986\pi\)
0.784316 0.620361i \(-0.213014\pi\)
\(812\) 0 0
\(813\) −1.79343 0.654327i −0.0628983 0.0229483i
\(814\) 0 0
\(815\) 55.0951 1.92990
\(816\) 0 0
\(817\) 11.4099i 0.399182i
\(818\) 0 0
\(819\) −8.91573 23.8956i −0.311541 0.834981i
\(820\) 0 0
\(821\) 19.2156i 0.670631i 0.942106 + 0.335315i \(0.108843\pi\)
−0.942106 + 0.335315i \(0.891157\pi\)
\(822\) 0 0
\(823\) 29.8148 1.03928 0.519639 0.854386i \(-0.326066\pi\)
0.519639 + 0.854386i \(0.326066\pi\)
\(824\) 0 0
\(825\) −18.0798 + 15.1469i −0.629456 + 0.527347i
\(826\) 0 0
\(827\) 0.356137i 0.0123841i −0.999981 0.00619205i \(-0.998029\pi\)
0.999981 0.00619205i \(-0.00197100\pi\)
\(828\) 0 0
\(829\) −17.5571 + 10.1366i −0.609783 + 0.352058i −0.772880 0.634552i \(-0.781185\pi\)
0.163098 + 0.986610i \(0.447851\pi\)
\(830\) 0 0
\(831\) −9.11951 + 24.9955i −0.316352 + 0.867083i
\(832\) 0 0
\(833\) −41.0732 + 0.788324i −1.42310 + 0.0273138i
\(834\) 0 0
\(835\) 46.5590 1.61124
\(836\) 0 0
\(837\) −12.6078 21.9550i −0.435791 0.758875i
\(838\) 0 0
\(839\) 0.662663 + 1.14777i 0.0228777 + 0.0396253i 0.877238 0.480056i \(-0.159384\pi\)
−0.854360 + 0.519682i \(0.826051\pi\)
\(840\) 0 0
\(841\) 7.85791 13.6103i 0.270962 0.469321i
\(842\) 0 0
\(843\) −28.9956 + 24.2920i −0.998662 + 0.836661i
\(844\) 0 0
\(845\) 4.72626 8.18612i 0.162588 0.281611i
\(846\) 0 0
\(847\) 10.0084 17.7258i 0.343894 0.609064i
\(848\) 0 0
\(849\) −3.31743 18.8996i −0.113854 0.648632i
\(850\) 0 0
\(851\) 6.96914i 0.238899i
\(852\) 0 0
\(853\) 41.7558 24.1077i 1.42969 0.825432i 0.432594 0.901589i \(-0.357598\pi\)
0.997096 + 0.0761571i \(0.0242650\pi\)
\(854\) 0 0
\(855\) −8.71972 24.0731i −0.298208 0.823282i
\(856\) 0 0
\(857\) −10.4387 + 18.0803i −0.356578 + 0.617611i −0.987387 0.158328i \(-0.949390\pi\)
0.630809 + 0.775938i \(0.282723\pi\)
\(858\) 0 0
\(859\) 11.8727 6.85469i 0.405090 0.233879i −0.283588 0.958946i \(-0.591525\pi\)
0.688678 + 0.725067i \(0.258191\pi\)
\(860\) 0 0
\(861\) 45.3417 8.40803i 1.54524 0.286545i
\(862\) 0 0
\(863\) −28.1125 16.2307i −0.956959 0.552501i −0.0617233 0.998093i \(-0.519660\pi\)
−0.895236 + 0.445593i \(0.852993\pi\)
\(864\) 0 0
\(865\) −19.8047 34.3027i −0.673379 1.16633i
\(866\) 0 0
\(867\) 10.3542 28.3795i 0.351646 0.963819i
\(868\) 0 0
\(869\) 2.90139 + 1.67512i 0.0984228 + 0.0568244i
\(870\) 0 0
\(871\) 21.8385 + 12.6085i 0.739970 + 0.427222i
\(872\) 0 0
\(873\) 30.7808 36.5680i 1.04177 1.23764i
\(874\) 0 0
\(875\) −11.8302 20.0438i −0.399933 0.677604i
\(876\) 0 0
\(877\) −10.9669 18.9953i −0.370326 0.641424i 0.619289 0.785163i \(-0.287421\pi\)
−0.989616 + 0.143739i \(0.954087\pi\)
\(878\) 0 0
\(879\) 14.9993 41.1112i 0.505913 1.38665i
\(880\) 0 0
\(881\) −21.8019 −0.734526 −0.367263 0.930117i \(-0.619705\pi\)
−0.367263 + 0.930117i \(0.619705\pi\)
\(882\) 0 0
\(883\) −24.4085 −0.821411 −0.410705 0.911768i \(-0.634717\pi\)
−0.410705 + 0.911768i \(0.634717\pi\)
\(884\) 0 0
\(885\) −34.6397 41.3469i −1.16440 1.38986i
\(886\) 0 0
\(887\) −16.8722 29.2235i −0.566513 0.981230i −0.996907 0.0785887i \(-0.974959\pi\)
0.430394 0.902641i \(-0.358375\pi\)
\(888\) 0 0
\(889\) 24.1051 + 40.8411i 0.808458 + 1.36977i
\(890\) 0 0
\(891\) −16.1246 + 2.79176i −0.540193 + 0.0935274i
\(892\) 0 0
\(893\) −16.5673 9.56513i −0.554403 0.320085i
\(894\) 0 0
\(895\) −32.6113 18.8281i −1.09007 0.629355i
\(896\) 0 0
\(897\) −21.0978 + 3.70328i −0.704436 + 0.123649i
\(898\) 0 0
\(899\) 16.2907 + 28.2164i 0.543326 + 0.941068i
\(900\) 0 0
\(901\) −13.2936 7.67505i −0.442873 0.255693i
\(902\) 0 0
\(903\) −7.22533 + 20.4099i −0.240444 + 0.679198i
\(904\) 0 0
\(905\) 36.3889 21.0092i 1.20961 0.698368i
\(906\) 0 0
\(907\) −7.52832 + 13.0394i −0.249974 + 0.432967i −0.963518 0.267643i \(-0.913755\pi\)
0.713544 + 0.700610i \(0.247089\pi\)
\(908\) 0 0
\(909\) 5.84637 32.8589i 0.193912 1.08986i
\(910\) 0 0
\(911\) −21.5047 + 12.4157i −0.712481 + 0.411351i −0.811979 0.583686i \(-0.801610\pi\)
0.0994977 + 0.995038i \(0.468276\pi\)
\(912\) 0 0
\(913\) 20.4282i 0.676075i
\(914\) 0 0
\(915\) −48.6772 + 40.7809i −1.60922 + 1.34817i
\(916\) 0 0
\(917\) −22.0759 + 39.0983i −0.729012 + 1.29114i
\(918\) 0 0
\(919\) −12.0689 + 20.9039i −0.398115 + 0.689555i −0.993493 0.113890i \(-0.963669\pi\)
0.595378 + 0.803445i \(0.297002\pi\)
\(920\) 0 0
\(921\) −3.65813 20.8406i −0.120539 0.686720i
\(922\) 0 0
\(923\) −12.2580 + 21.2314i −0.403476 + 0.698840i
\(924\) 0 0
\(925\) −6.78064 11.7444i −0.222946 0.386154i
\(926\) 0 0
\(927\) 1.16500 1.38404i 0.0382636 0.0454577i
\(928\) 0 0
\(929\) −26.6620 −0.874752 −0.437376 0.899279i \(-0.644092\pi\)
−0.437376 + 0.899279i \(0.644092\pi\)
\(930\) 0 0
\(931\) 14.4751 8.73180i 0.474403 0.286173i
\(932\) 0 0
\(933\) 16.1891 2.84166i 0.530007 0.0930317i
\(934\) 0 0
\(935\) 32.6586 18.8555i 1.06805 0.616640i
\(936\) 0 0
\(937\) 8.17165i 0.266956i −0.991052 0.133478i \(-0.957385\pi\)
0.991052 0.133478i \(-0.0426146\pi\)
\(938\) 0 0
\(939\) −4.05790 23.1181i −0.132424 0.754430i
\(940\) 0 0
\(941\) 32.0272 1.04406 0.522028 0.852928i \(-0.325176\pi\)
0.522028 + 0.852928i \(0.325176\pi\)
\(942\) 0 0
\(943\) 38.7298i 1.26122i
\(944\) 0 0
\(945\) −0.353392 48.5833i −0.0114958 1.58042i
\(946\) 0 0
\(947\) 17.9002i 0.581678i 0.956772 + 0.290839i \(0.0939344\pi\)
−0.956772 + 0.290839i \(0.906066\pi\)
\(948\) 0 0
\(949\) 43.6611 1.41730
\(950\) 0 0
\(951\) 4.75640 + 27.0975i 0.154237 + 0.878697i
\(952\) 0 0
\(953\) 11.2737i 0.365191i −0.983188 0.182596i \(-0.941550\pi\)
0.983188 0.182596i \(-0.0584499\pi\)
\(954\) 0 0
\(955\) 0.685459 0.395750i 0.0221809 0.0128062i
\(956\) 0 0
\(957\) 20.7425 3.64091i 0.670510 0.117694i
\(958\) 0 0
\(959\) 0.141185 + 14.7134i 0.00455911 + 0.475120i
\(960\) 0 0
\(961\) 7.26007 0.234196
\(962\) 0 0
\(963\) 38.6558 + 6.87776i 1.24567 + 0.221633i
\(964\) 0 0
\(965\) 46.0738 + 79.8022i 1.48317 + 2.56892i
\(966\) 0 0
\(967\) 15.3313 26.5546i 0.493022 0.853939i −0.506946 0.861978i \(-0.669226\pi\)
0.999968 + 0.00803913i \(0.00255896\pi\)
\(968\) 0 0
\(969\) 4.24398 + 24.1782i 0.136336 + 0.776716i
\(970\) 0 0
\(971\) −19.0807 + 33.0488i −0.612330 + 1.06059i 0.378517 + 0.925595i \(0.376434\pi\)
−0.990847 + 0.134992i \(0.956899\pi\)
\(972\) 0 0
\(973\) 0.364967 + 38.0345i 0.0117003 + 1.21933i
\(974\) 0 0
\(975\) −31.9510 + 26.7680i −1.02325 + 0.857261i
\(976\) 0 0
\(977\) 30.6991i 0.982151i 0.871117 + 0.491076i \(0.163396\pi\)
−0.871117 + 0.491076i \(0.836604\pi\)
\(978\) 0 0
\(979\) −21.7239 + 12.5423i −0.694299 + 0.400854i
\(980\) 0 0
\(981\) −6.29148 5.29580i −0.200872 0.169082i
\(982\) 0 0
\(983\) 12.9028 22.3484i 0.411536 0.712802i −0.583522 0.812098i \(-0.698326\pi\)
0.995058 + 0.0992957i \(0.0316590\pi\)
\(984\) 0 0
\(985\) −4.78330 + 2.76164i −0.152409 + 0.0879932i
\(986\) 0 0
\(987\) −23.5782 27.6012i −0.750502 0.878556i
\(988\) 0 0
\(989\) 15.7476 + 9.09191i 0.500746 + 0.289106i
\(990\) 0 0
\(991\) −11.4388 19.8126i −0.363366 0.629369i 0.625146 0.780508i \(-0.285039\pi\)
−0.988513 + 0.151139i \(0.951706\pi\)
\(992\) 0 0
\(993\) −18.7212 + 3.28612i −0.594100 + 0.104282i
\(994\) 0 0
\(995\) 47.6118 + 27.4887i 1.50940 + 0.871450i
\(996\) 0 0
\(997\) −35.9753 20.7704i −1.13935 0.657804i −0.193080 0.981183i \(-0.561848\pi\)
−0.946269 + 0.323379i \(0.895181\pi\)
\(998\) 0 0
\(999\) 0.0218440 9.40901i 0.000691115 0.297688i
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 504.2.bs.a.257.1 48
3.2 odd 2 1512.2.bs.a.1097.3 48
4.3 odd 2 1008.2.ca.e.257.24 48
7.3 odd 6 504.2.cx.a.185.7 yes 48
9.2 odd 6 504.2.cx.a.425.7 yes 48
9.7 even 3 1512.2.cx.a.89.3 48
12.11 even 2 3024.2.ca.e.2609.3 48
21.17 even 6 1512.2.cx.a.17.3 48
28.3 even 6 1008.2.df.e.689.18 48
36.7 odd 6 3024.2.df.e.1601.3 48
36.11 even 6 1008.2.df.e.929.18 48
63.38 even 6 inner 504.2.bs.a.353.1 yes 48
63.52 odd 6 1512.2.bs.a.521.3 48
84.59 odd 6 3024.2.df.e.17.3 48
252.115 even 6 3024.2.ca.e.2033.3 48
252.227 odd 6 1008.2.ca.e.353.24 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.1 48 1.1 even 1 trivial
504.2.bs.a.353.1 yes 48 63.38 even 6 inner
504.2.cx.a.185.7 yes 48 7.3 odd 6
504.2.cx.a.425.7 yes 48 9.2 odd 6
1008.2.ca.e.257.24 48 4.3 odd 2
1008.2.ca.e.353.24 48 252.227 odd 6
1008.2.df.e.689.18 48 28.3 even 6
1008.2.df.e.929.18 48 36.11 even 6
1512.2.bs.a.521.3 48 63.52 odd 6
1512.2.bs.a.1097.3 48 3.2 odd 2
1512.2.cx.a.17.3 48 21.17 even 6
1512.2.cx.a.89.3 48 9.7 even 3
3024.2.ca.e.2033.3 48 252.115 even 6
3024.2.ca.e.2609.3 48 12.11 even 2
3024.2.df.e.17.3 48 84.59 odd 6
3024.2.df.e.1601.3 48 36.7 odd 6