Properties

Label 3420.2.bj.c.2629.5
Level $3420$
Weight $2$
Character 3420.2629
Analytic conductor $27.309$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,2,Mod(1189,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2629.5
Root \(0.392182 - 0.226426i\) of defining polynomial
Character \(\chi\) \(=\) 3420.2629
Dual form 3420.2.bj.c.1189.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.207009 - 2.22647i) q^{5} -2.54366i q^{7} -2.22377 q^{11} +(6.08116 + 3.51096i) q^{13} +(-2.21492 + 1.27878i) q^{17} +(2.70498 + 3.41805i) q^{19} +(6.95328 + 4.01448i) q^{23} +(-4.91429 + 0.921799i) q^{25} +(0.941734 - 1.63113i) q^{29} +5.98111 q^{31} +(-5.66338 + 0.526562i) q^{35} +2.86105i q^{37} +(3.67524 + 6.36571i) q^{41} +(3.19919 - 1.84706i) q^{43} +(-4.09540 - 2.36448i) q^{47} +0.529782 q^{49} +(8.91226 + 5.14549i) q^{53} +(0.460341 + 4.95114i) q^{55} +(3.73666 + 6.47208i) q^{59} +(4.17839 - 7.23719i) q^{61} +(6.55817 - 14.2663i) q^{65} +(-10.7040 - 6.17997i) q^{67} +(4.13931 + 7.16950i) q^{71} +(-10.9489 + 6.32134i) q^{73} +5.65651i q^{77} +(-2.13067 - 3.69043i) q^{79} -14.7613i q^{83} +(3.30568 + 4.66672i) q^{85} +(7.19403 - 12.4604i) q^{89} +(8.93069 - 15.4684i) q^{91} +(7.05022 - 6.73011i) q^{95} +(5.04871 - 2.91488i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.207009 2.22647i −0.0925774 0.995705i
\(6\) 0 0
\(7\) 2.54366i 0.961414i −0.876881 0.480707i \(-0.840380\pi\)
0.876881 0.480707i \(-0.159620\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.22377 −0.670491 −0.335246 0.942131i \(-0.608819\pi\)
−0.335246 + 0.942131i \(0.608819\pi\)
\(12\) 0 0
\(13\) 6.08116 + 3.51096i 1.68661 + 0.973765i 0.957084 + 0.289812i \(0.0935927\pi\)
0.729526 + 0.683953i \(0.239741\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.21492 + 1.27878i −0.537197 + 0.310151i −0.743942 0.668244i \(-0.767046\pi\)
0.206745 + 0.978395i \(0.433713\pi\)
\(18\) 0 0
\(19\) 2.70498 + 3.41805i 0.620565 + 0.784155i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.95328 + 4.01448i 1.44986 + 0.837076i 0.998472 0.0552521i \(-0.0175962\pi\)
0.451387 + 0.892329i \(0.350930\pi\)
\(24\) 0 0
\(25\) −4.91429 + 0.921799i −0.982859 + 0.184360i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.941734 1.63113i 0.174876 0.302893i −0.765243 0.643742i \(-0.777381\pi\)
0.940118 + 0.340849i \(0.110714\pi\)
\(30\) 0 0
\(31\) 5.98111 1.07424 0.537120 0.843506i \(-0.319512\pi\)
0.537120 + 0.843506i \(0.319512\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.66338 + 0.526562i −0.957285 + 0.0890052i
\(36\) 0 0
\(37\) 2.86105i 0.470353i 0.971953 + 0.235177i \(0.0755669\pi\)
−0.971953 + 0.235177i \(0.924433\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.67524 + 6.36571i 0.573977 + 0.994157i 0.996152 + 0.0876426i \(0.0279333\pi\)
−0.422175 + 0.906514i \(0.638733\pi\)
\(42\) 0 0
\(43\) 3.19919 1.84706i 0.487873 0.281673i −0.235819 0.971797i \(-0.575777\pi\)
0.723691 + 0.690124i \(0.242444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.09540 2.36448i −0.597376 0.344895i 0.170633 0.985335i \(-0.445419\pi\)
−0.768008 + 0.640440i \(0.778752\pi\)
\(48\) 0 0
\(49\) 0.529782 0.0756832
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.91226 + 5.14549i 1.22419 + 0.706788i 0.965809 0.259255i \(-0.0834769\pi\)
0.258383 + 0.966042i \(0.416810\pi\)
\(54\) 0 0
\(55\) 0.460341 + 4.95114i 0.0620724 + 0.667612i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.73666 + 6.47208i 0.486472 + 0.842593i 0.999879 0.0155515i \(-0.00495040\pi\)
−0.513408 + 0.858145i \(0.671617\pi\)
\(60\) 0 0
\(61\) 4.17839 7.23719i 0.534988 0.926627i −0.464176 0.885743i \(-0.653649\pi\)
0.999164 0.0408838i \(-0.0130174\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.55817 14.2663i 0.813441 1.76952i
\(66\) 0 0
\(67\) −10.7040 6.17997i −1.30771 0.755004i −0.325993 0.945372i \(-0.605698\pi\)
−0.981713 + 0.190368i \(0.939032\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.13931 + 7.16950i 0.491246 + 0.850863i 0.999949 0.0100790i \(-0.00320829\pi\)
−0.508703 + 0.860942i \(0.669875\pi\)
\(72\) 0 0
\(73\) −10.9489 + 6.32134i −1.28147 + 0.739857i −0.977117 0.212703i \(-0.931773\pi\)
−0.304352 + 0.952559i \(0.598440\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65651i 0.644620i
\(78\) 0 0
\(79\) −2.13067 3.69043i −0.239719 0.415206i 0.720914 0.693024i \(-0.243722\pi\)
−0.960634 + 0.277818i \(0.910389\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.7613i 1.62026i −0.586248 0.810132i \(-0.699396\pi\)
0.586248 0.810132i \(-0.300604\pi\)
\(84\) 0 0
\(85\) 3.30568 + 4.66672i 0.358551 + 0.506177i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.19403 12.4604i 0.762566 1.32080i −0.178958 0.983857i \(-0.557273\pi\)
0.941524 0.336946i \(-0.109394\pi\)
\(90\) 0 0
\(91\) 8.93069 15.4684i 0.936191 1.62153i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.05022 6.73011i 0.723337 0.690495i
\(96\) 0 0
\(97\) 5.04871 2.91488i 0.512619 0.295961i −0.221291 0.975208i \(-0.571027\pi\)
0.733910 + 0.679247i \(0.237694\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.47686 + 11.2183i −0.644472 + 1.11626i 0.339951 + 0.940443i \(0.389589\pi\)
−0.984423 + 0.175815i \(0.943744\pi\)
\(102\) 0 0
\(103\) 16.6116i 1.63679i −0.574658 0.818394i \(-0.694865\pi\)
0.574658 0.818394i \(-0.305135\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.59275i 0.443998i −0.975047 0.221999i \(-0.928742\pi\)
0.975047 0.221999i \(-0.0712582\pi\)
\(108\) 0 0
\(109\) −1.30902 2.26728i −0.125381 0.217166i 0.796501 0.604637i \(-0.206682\pi\)
−0.921882 + 0.387471i \(0.873349\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.509357i 0.0479163i 0.999713 + 0.0239581i \(0.00762684\pi\)
−0.999713 + 0.0239581i \(0.992373\pi\)
\(114\) 0 0
\(115\) 7.49870 16.3123i 0.699257 1.52113i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.25279 + 5.63401i 0.298183 + 0.516468i
\(120\) 0 0
\(121\) −6.05486 −0.550441
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.06966 + 10.7507i 0.274559 + 0.961570i
\(126\) 0 0
\(127\) −0.636081 0.367242i −0.0564431 0.0325874i 0.471513 0.881859i \(-0.343708\pi\)
−0.527956 + 0.849272i \(0.677041\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.84274 4.92377i −0.248371 0.430192i 0.714703 0.699428i \(-0.246562\pi\)
−0.963074 + 0.269236i \(0.913229\pi\)
\(132\) 0 0
\(133\) 8.69437 6.88055i 0.753898 0.596620i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.2812 9.39993i −1.39099 0.803091i −0.397568 0.917573i \(-0.630146\pi\)
−0.993425 + 0.114482i \(0.963479\pi\)
\(138\) 0 0
\(139\) 2.05362 3.55697i 0.174186 0.301698i −0.765694 0.643206i \(-0.777604\pi\)
0.939879 + 0.341507i \(0.110937\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.5231 7.80756i −1.13086 0.652901i
\(144\) 0 0
\(145\) −3.82660 1.75908i −0.317782 0.146084i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.08175 + 3.60570i 0.170544 + 0.295391i 0.938610 0.344980i \(-0.112114\pi\)
−0.768066 + 0.640370i \(0.778781\pi\)
\(150\) 0 0
\(151\) 17.7955 1.44818 0.724090 0.689706i \(-0.242260\pi\)
0.724090 + 0.689706i \(0.242260\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.23815 13.3167i −0.0994504 1.06963i
\(156\) 0 0
\(157\) −8.09693 + 4.67476i −0.646205 + 0.373087i −0.787001 0.616952i \(-0.788367\pi\)
0.140796 + 0.990039i \(0.455034\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.2115 17.6868i 0.804777 1.39391i
\(162\) 0 0
\(163\) 1.55215i 0.121574i 0.998151 + 0.0607869i \(0.0193610\pi\)
−0.998151 + 0.0607869i \(0.980639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.03177 + 4.05980i 0.544135 + 0.314157i 0.746753 0.665101i \(-0.231612\pi\)
−0.202618 + 0.979258i \(0.564945\pi\)
\(168\) 0 0
\(169\) 18.1537 + 31.4431i 1.39644 + 2.41870i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.633386 0.365686i 0.0481555 0.0278026i −0.475729 0.879592i \(-0.657816\pi\)
0.523884 + 0.851789i \(0.324482\pi\)
\(174\) 0 0
\(175\) 2.34474 + 12.5003i 0.177246 + 0.944934i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.68858 0.200954 0.100477 0.994939i \(-0.467963\pi\)
0.100477 + 0.994939i \(0.467963\pi\)
\(180\) 0 0
\(181\) 2.75090 4.76469i 0.204473 0.354157i −0.745492 0.666515i \(-0.767785\pi\)
0.949965 + 0.312358i \(0.101119\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.37002 0.592264i 0.468333 0.0435441i
\(186\) 0 0
\(187\) 4.92547 2.84372i 0.360186 0.207953i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.39845 −0.680048 −0.340024 0.940417i \(-0.610435\pi\)
−0.340024 + 0.940417i \(0.610435\pi\)
\(192\) 0 0
\(193\) 10.0089 5.77864i 0.720457 0.415956i −0.0944641 0.995528i \(-0.530114\pi\)
0.814921 + 0.579572i \(0.196780\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.11825i 0.364660i 0.983237 + 0.182330i \(0.0583639\pi\)
−0.983237 + 0.182330i \(0.941636\pi\)
\(198\) 0 0
\(199\) 6.71897 11.6376i 0.476295 0.824968i −0.523336 0.852127i \(-0.675313\pi\)
0.999631 + 0.0271588i \(0.00864598\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.14905 2.39545i −0.291206 0.168128i
\(204\) 0 0
\(205\) 13.4122 9.50056i 0.936750 0.663548i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.01524 7.60096i −0.416083 0.525769i
\(210\) 0 0
\(211\) 2.54063 + 4.40050i 0.174904 + 0.302943i 0.940128 0.340821i \(-0.110705\pi\)
−0.765224 + 0.643764i \(0.777372\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.77467 6.74054i −0.325630 0.459701i
\(216\) 0 0
\(217\) 15.2139i 1.03279i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −17.9590 −1.20806
\(222\) 0 0
\(223\) −20.5186 + 11.8464i −1.37403 + 0.793295i −0.991432 0.130621i \(-0.958303\pi\)
−0.382595 + 0.923916i \(0.624970\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.0306i 1.26310i 0.775334 + 0.631551i \(0.217582\pi\)
−0.775334 + 0.631551i \(0.782418\pi\)
\(228\) 0 0
\(229\) 27.9233 1.84523 0.922613 0.385727i \(-0.126049\pi\)
0.922613 + 0.385727i \(0.126049\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.2680 + 7.08296i −0.803706 + 0.464020i −0.844765 0.535137i \(-0.820260\pi\)
0.0410592 + 0.999157i \(0.486927\pi\)
\(234\) 0 0
\(235\) −4.41665 + 9.60774i −0.288110 + 0.626740i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.0237 1.10117 0.550585 0.834779i \(-0.314405\pi\)
0.550585 + 0.834779i \(0.314405\pi\)
\(240\) 0 0
\(241\) −8.59549 + 14.8878i −0.553684 + 0.959009i 0.444321 + 0.895868i \(0.353445\pi\)
−0.998005 + 0.0631409i \(0.979888\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.109670 1.17954i −0.00700655 0.0753581i
\(246\) 0 0
\(247\) 4.44876 + 30.2828i 0.283068 + 1.92685i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.8853 20.5860i 0.750195 1.29938i −0.197534 0.980296i \(-0.563293\pi\)
0.947728 0.319079i \(-0.103374\pi\)
\(252\) 0 0
\(253\) −15.4625 8.92727i −0.972118 0.561252i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.24086 0.716409i −0.0774026 0.0446884i 0.460799 0.887504i \(-0.347563\pi\)
−0.538202 + 0.842816i \(0.680896\pi\)
\(258\) 0 0
\(259\) 7.27754 0.452204
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.17484 + 3.56505i −0.380757 + 0.219830i −0.678148 0.734926i \(-0.737217\pi\)
0.297390 + 0.954756i \(0.403884\pi\)
\(264\) 0 0
\(265\) 9.61134 20.9080i 0.590420 1.28437i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.13452 + 15.8214i 0.556941 + 0.964651i 0.997750 + 0.0670494i \(0.0213585\pi\)
−0.440808 + 0.897601i \(0.645308\pi\)
\(270\) 0 0
\(271\) 2.90265 + 5.02754i 0.176324 + 0.305401i 0.940619 0.339465i \(-0.110246\pi\)
−0.764295 + 0.644867i \(0.776913\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.9282 2.04987i 0.658998 0.123612i
\(276\) 0 0
\(277\) 5.72209i 0.343807i 0.985114 + 0.171904i \(0.0549918\pi\)
−0.985114 + 0.171904i \(0.945008\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.07155 + 5.32009i −0.183233 + 0.317370i −0.942980 0.332850i \(-0.891990\pi\)
0.759746 + 0.650220i \(0.225323\pi\)
\(282\) 0 0
\(283\) −14.2484 + 8.22632i −0.846980 + 0.489004i −0.859631 0.510916i \(-0.829306\pi\)
0.0126510 + 0.999920i \(0.495973\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.1922 9.34858i 0.955796 0.551829i
\(288\) 0 0
\(289\) −5.22942 + 9.05763i −0.307613 + 0.532801i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 19.5684i 1.14320i −0.820533 0.571599i \(-0.806323\pi\)
0.820533 0.571599i \(-0.193677\pi\)
\(294\) 0 0
\(295\) 13.6363 9.65932i 0.793939 0.562388i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 28.1893 + 48.8253i 1.63023 + 2.82364i
\(300\) 0 0
\(301\) −4.69829 8.13767i −0.270805 0.469047i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.9783 7.80488i −0.972175 0.446906i
\(306\) 0 0
\(307\) 24.9405 14.3994i 1.42343 0.821817i 0.426839 0.904327i \(-0.359627\pi\)
0.996590 + 0.0825100i \(0.0262937\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.4309 1.10182 0.550911 0.834564i \(-0.314280\pi\)
0.550911 + 0.834564i \(0.314280\pi\)
\(312\) 0 0
\(313\) 8.34649 + 4.81885i 0.471772 + 0.272377i 0.716981 0.697093i \(-0.245523\pi\)
−0.245209 + 0.969470i \(0.578857\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.199789 0.115348i −0.0112213 0.00647860i 0.494379 0.869247i \(-0.335396\pi\)
−0.505600 + 0.862768i \(0.668729\pi\)
\(318\) 0 0
\(319\) −2.09420 + 3.62726i −0.117253 + 0.203087i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.3623 4.11163i −0.576572 0.228777i
\(324\) 0 0
\(325\) −33.1210 11.6483i −1.83722 0.646130i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.01444 + 10.4173i −0.331587 + 0.574325i
\(330\) 0 0
\(331\) −12.1135 −0.665820 −0.332910 0.942959i \(-0.608031\pi\)
−0.332910 + 0.942959i \(0.608031\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.5437 + 25.1115i −0.630698 + 1.37199i
\(336\) 0 0
\(337\) −9.64909 + 5.57090i −0.525619 + 0.303466i −0.739231 0.673452i \(-0.764811\pi\)
0.213611 + 0.976919i \(0.431477\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.3006 −0.720268
\(342\) 0 0
\(343\) 19.1532i 1.03418i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.92730 2.84478i 0.264511 0.152716i −0.361880 0.932225i \(-0.617865\pi\)
0.626391 + 0.779509i \(0.284532\pi\)
\(348\) 0 0
\(349\) −0.369374 −0.0197722 −0.00988608 0.999951i \(-0.503147\pi\)
−0.00988608 + 0.999951i \(0.503147\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.87124i 0.0995962i 0.998759 + 0.0497981i \(0.0158578\pi\)
−0.998759 + 0.0497981i \(0.984142\pi\)
\(354\) 0 0
\(355\) 15.1058 10.7002i 0.801731 0.567907i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.47802 + 14.6844i 0.447453 + 0.775011i 0.998219 0.0596485i \(-0.0189980\pi\)
−0.550767 + 0.834659i \(0.685665\pi\)
\(360\) 0 0
\(361\) −4.36618 + 18.4915i −0.229799 + 0.973238i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.3408 + 23.0687i 0.855315 + 1.20747i
\(366\) 0 0
\(367\) −3.27769 1.89237i −0.171094 0.0987812i 0.412008 0.911180i \(-0.364828\pi\)
−0.583102 + 0.812399i \(0.698161\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.0884 22.6698i 0.679516 1.17696i
\(372\) 0 0
\(373\) 4.51203i 0.233624i 0.993154 + 0.116812i \(0.0372675\pi\)
−0.993154 + 0.116812i \(0.962732\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.4537 6.61278i 0.589894 0.340575i
\(378\) 0 0
\(379\) −4.64242 −0.238465 −0.119233 0.992866i \(-0.538043\pi\)
−0.119233 + 0.992866i \(0.538043\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.1086 6.99088i 0.618719 0.357218i −0.157651 0.987495i \(-0.550392\pi\)
0.776370 + 0.630277i \(0.217059\pi\)
\(384\) 0 0
\(385\) 12.5940 1.17095i 0.641851 0.0596772i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.93460 + 6.81492i −0.199492 + 0.345530i −0.948364 0.317185i \(-0.897263\pi\)
0.748872 + 0.662715i \(0.230596\pi\)
\(390\) 0 0
\(391\) −20.5346 −1.03848
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.77555 + 5.50782i −0.391231 + 0.277129i
\(396\) 0 0
\(397\) 14.6969 8.48528i 0.737618 0.425864i −0.0835846 0.996501i \(-0.526637\pi\)
0.821203 + 0.570637i \(0.193304\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.16651 + 12.4128i 0.357878 + 0.619864i 0.987606 0.156952i \(-0.0501669\pi\)
−0.629728 + 0.776816i \(0.716834\pi\)
\(402\) 0 0
\(403\) 36.3721 + 20.9994i 1.81182 + 1.04606i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.36231i 0.315368i
\(408\) 0 0
\(409\) −0.122530 + 0.212228i −0.00605872 + 0.0104940i −0.869039 0.494744i \(-0.835262\pi\)
0.862980 + 0.505238i \(0.168595\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.4628 9.50480i 0.810081 0.467701i
\(414\) 0 0
\(415\) −32.8655 + 3.05573i −1.61330 + 0.150000i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −25.4690 −1.24424 −0.622122 0.782920i \(-0.713729\pi\)
−0.622122 + 0.782920i \(0.713729\pi\)
\(420\) 0 0
\(421\) −3.02703 5.24297i −0.147529 0.255527i 0.782785 0.622292i \(-0.213798\pi\)
−0.930313 + 0.366765i \(0.880465\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.70598 8.32603i 0.470809 0.403872i
\(426\) 0 0
\(427\) −18.4090 10.6284i −0.890872 0.514345i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.1005 19.2267i 0.534694 0.926117i −0.464484 0.885581i \(-0.653760\pi\)
0.999178 0.0405356i \(-0.0129064\pi\)
\(432\) 0 0
\(433\) 21.0094 + 12.1298i 1.00965 + 0.582920i 0.911089 0.412211i \(-0.135243\pi\)
0.0985593 + 0.995131i \(0.468577\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.08677 + 34.6258i 0.243333 + 1.65637i
\(438\) 0 0
\(439\) −11.9487 20.6957i −0.570279 0.987753i −0.996537 0.0831509i \(-0.973502\pi\)
0.426258 0.904602i \(-0.359832\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.9017 + 10.3356i 0.850536 + 0.491057i 0.860832 0.508890i \(-0.169944\pi\)
−0.0102958 + 0.999947i \(0.503277\pi\)
\(444\) 0 0
\(445\) −29.2319 13.4378i −1.38573 0.637015i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.64143 −0.313429 −0.156714 0.987644i \(-0.550090\pi\)
−0.156714 + 0.987644i \(0.550090\pi\)
\(450\) 0 0
\(451\) −8.17289 14.1559i −0.384846 0.666573i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −36.2886 16.6818i −1.70124 0.782053i
\(456\) 0 0
\(457\) 14.3433i 0.670952i −0.942049 0.335476i \(-0.891103\pi\)
0.942049 0.335476i \(-0.108897\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −13.3682 23.1544i −0.622618 1.07841i −0.988996 0.147940i \(-0.952736\pi\)
0.366378 0.930466i \(-0.380598\pi\)
\(462\) 0 0
\(463\) 12.0950i 0.562101i −0.959693 0.281051i \(-0.909317\pi\)
0.959693 0.281051i \(-0.0906829\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.3169i 1.26408i −0.774938 0.632038i \(-0.782219\pi\)
0.774938 0.632038i \(-0.217781\pi\)
\(468\) 0 0
\(469\) −15.7198 + 27.2274i −0.725871 + 1.25725i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.11427 + 4.10742i −0.327114 + 0.188860i
\(474\) 0 0
\(475\) −16.4438 14.3039i −0.754494 0.656307i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.69624 + 6.40208i −0.168886 + 0.292519i −0.938028 0.346559i \(-0.887350\pi\)
0.769143 + 0.639077i \(0.220684\pi\)
\(480\) 0 0
\(481\) −10.0450 + 17.3985i −0.458013 + 0.793302i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.53500 10.6374i −0.342147 0.483018i
\(486\) 0 0
\(487\) 12.9424i 0.586477i −0.956039 0.293239i \(-0.905267\pi\)
0.956039 0.293239i \(-0.0947330\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.59974 9.69903i −0.252713 0.437711i 0.711559 0.702626i \(-0.247989\pi\)
−0.964272 + 0.264915i \(0.914656\pi\)
\(492\) 0 0
\(493\) 4.81710i 0.216951i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.2368 10.5290i 0.818032 0.472291i
\(498\) 0 0
\(499\) −12.9699 22.4645i −0.580611 1.00565i −0.995407 0.0957335i \(-0.969480\pi\)
0.414796 0.909914i \(-0.363853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.6079 13.6300i −1.05262 0.607733i −0.129241 0.991613i \(-0.541254\pi\)
−0.923383 + 0.383881i \(0.874587\pi\)
\(504\) 0 0
\(505\) 26.3178 + 12.0982i 1.17113 + 0.538364i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.6203 + 21.8591i −0.559386 + 0.968885i 0.438161 + 0.898896i \(0.355630\pi\)
−0.997548 + 0.0699892i \(0.977704\pi\)
\(510\) 0 0
\(511\) 16.0794 + 27.8503i 0.711309 + 1.23202i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −36.9851 + 3.43875i −1.62976 + 0.151530i
\(516\) 0 0
\(517\) 9.10722 + 5.25806i 0.400535 + 0.231249i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.8142 1.08713 0.543565 0.839367i \(-0.317074\pi\)
0.543565 + 0.839367i \(0.317074\pi\)
\(522\) 0 0
\(523\) −5.33581 3.08063i −0.233318 0.134706i 0.378784 0.925485i \(-0.376342\pi\)
−0.612102 + 0.790779i \(0.709676\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.2477 + 7.64855i −0.577078 + 0.333176i
\(528\) 0 0
\(529\) 20.7321 + 35.9090i 0.901394 + 1.56126i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 51.6145i 2.23567i
\(534\) 0 0
\(535\) −10.2256 + 0.950743i −0.442092 + 0.0411042i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.17811 −0.0507449
\(540\) 0 0
\(541\) −3.27394 + 5.67063i −0.140758 + 0.243799i −0.927782 0.373122i \(-0.878287\pi\)
0.787025 + 0.616922i \(0.211620\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.77704 + 3.38383i −0.204626 + 0.144947i
\(546\) 0 0
\(547\) −19.5323 11.2770i −0.835141 0.482169i 0.0204683 0.999791i \(-0.493484\pi\)
−0.855610 + 0.517621i \(0.826818\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.12266 1.19328i 0.346037 0.0508353i
\(552\) 0 0
\(553\) −9.38722 + 5.41971i −0.399185 + 0.230470i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −14.0691 8.12281i −0.596128 0.344175i 0.171389 0.985203i \(-0.445175\pi\)
−0.767517 + 0.641029i \(0.778508\pi\)
\(558\) 0 0
\(559\) 25.9397 1.09713
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.4581i 1.36795i 0.729506 + 0.683974i \(0.239750\pi\)
−0.729506 + 0.683974i \(0.760250\pi\)
\(564\) 0 0
\(565\) 1.13407 0.105442i 0.0477105 0.00443597i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.5422 1.07078 0.535392 0.844604i \(-0.320164\pi\)
0.535392 + 0.844604i \(0.320164\pi\)
\(570\) 0 0
\(571\) 25.9974 1.08796 0.543979 0.839099i \(-0.316917\pi\)
0.543979 + 0.839099i \(0.316917\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −37.8710 13.3188i −1.57933 0.555432i
\(576\) 0 0
\(577\) 44.5844i 1.85607i 0.372489 + 0.928037i \(0.378504\pi\)
−0.372489 + 0.928037i \(0.621496\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −37.5478 −1.55774
\(582\) 0 0
\(583\) −19.8188 11.4424i −0.820810 0.473895i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.6246 7.28884i 0.521074 0.300842i −0.216300 0.976327i \(-0.569399\pi\)
0.737374 + 0.675485i \(0.236065\pi\)
\(588\) 0 0
\(589\) 16.1788 + 20.4438i 0.666635 + 0.842371i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.5520 + 11.8657i 0.843968 + 0.487265i 0.858611 0.512628i \(-0.171328\pi\)
−0.0146433 + 0.999893i \(0.504661\pi\)
\(594\) 0 0
\(595\) 11.8706 8.40853i 0.486645 0.344716i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.86173 17.0810i 0.402939 0.697911i −0.591140 0.806569i \(-0.701322\pi\)
0.994079 + 0.108658i \(0.0346552\pi\)
\(600\) 0 0
\(601\) −45.0351 −1.83702 −0.918509 0.395400i \(-0.870606\pi\)
−0.918509 + 0.395400i \(0.870606\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.25341 + 13.4809i 0.0509585 + 0.548078i
\(606\) 0 0
\(607\) 7.24160i 0.293928i 0.989142 + 0.146964i \(0.0469501\pi\)
−0.989142 + 0.146964i \(0.953050\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.6032 28.7576i −0.671693 1.16341i
\(612\) 0 0
\(613\) −31.8788 + 18.4052i −1.28757 + 0.743381i −0.978221 0.207567i \(-0.933446\pi\)
−0.309352 + 0.950947i \(0.600112\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.7741 18.3448i −1.27918 0.738533i −0.302479 0.953156i \(-0.597814\pi\)
−0.976697 + 0.214623i \(0.931148\pi\)
\(618\) 0 0
\(619\) −24.1569 −0.970948 −0.485474 0.874251i \(-0.661353\pi\)
−0.485474 + 0.874251i \(0.661353\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −31.6951 18.2992i −1.26984 0.733142i
\(624\) 0 0
\(625\) 23.3006 9.05998i 0.932023 0.362399i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.65866 6.33699i −0.145880 0.252672i
\(630\) 0 0
\(631\) −7.11258 + 12.3193i −0.283147 + 0.490426i −0.972158 0.234325i \(-0.924712\pi\)
0.689011 + 0.724751i \(0.258045\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.685976 + 1.49223i −0.0272221 + 0.0592175i
\(636\) 0 0
\(637\) 3.22169 + 1.86004i 0.127648 + 0.0736976i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.1834 17.6382i −0.402222 0.696668i 0.591772 0.806105i \(-0.298429\pi\)
−0.993994 + 0.109437i \(0.965095\pi\)
\(642\) 0 0
\(643\) 10.0041 5.77586i 0.394523 0.227778i −0.289595 0.957149i \(-0.593521\pi\)
0.684118 + 0.729371i \(0.260187\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.1233i 0.515932i −0.966154 0.257966i \(-0.916948\pi\)
0.966154 0.257966i \(-0.0830522\pi\)
\(648\) 0 0
\(649\) −8.30946 14.3924i −0.326175 0.564952i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.4406i 0.760769i −0.924828 0.380385i \(-0.875792\pi\)
0.924828 0.380385i \(-0.124208\pi\)
\(654\) 0 0
\(655\) −10.3741 + 7.34853i −0.405351 + 0.287131i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.7848 23.8759i 0.536979 0.930074i −0.462086 0.886835i \(-0.652899\pi\)
0.999065 0.0432391i \(-0.0137677\pi\)
\(660\) 0 0
\(661\) 3.88006 6.72046i 0.150917 0.261396i −0.780648 0.624971i \(-0.785111\pi\)
0.931565 + 0.363575i \(0.118444\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −17.1191 17.9334i −0.663851 0.695427i
\(666\) 0 0
\(667\) 13.0963 7.56114i 0.507090 0.292769i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.29178 + 16.0938i −0.358705 + 0.621295i
\(672\) 0 0
\(673\) 24.2617i 0.935220i 0.883935 + 0.467610i \(0.154885\pi\)
−0.883935 + 0.467610i \(0.845115\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.88851i 0.187881i −0.995578 0.0939403i \(-0.970054\pi\)
0.995578 0.0939403i \(-0.0299463\pi\)
\(678\) 0 0
\(679\) −7.41446 12.8422i −0.284541 0.492839i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.560031i 0.0214290i 0.999943 + 0.0107145i \(0.00341060\pi\)
−0.999943 + 0.0107145i \(0.996589\pi\)
\(684\) 0 0
\(685\) −17.5583 + 38.1953i −0.670867 + 1.45937i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 36.1312 + 62.5811i 1.37649 + 2.38415i
\(690\) 0 0
\(691\) −14.5255 −0.552576 −0.276288 0.961075i \(-0.589104\pi\)
−0.276288 + 0.961075i \(0.589104\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.34459 3.83598i −0.316528 0.145507i
\(696\) 0 0
\(697\) −16.2807 9.39969i −0.616677 0.356039i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.23926 2.14647i −0.0468064 0.0810710i 0.841673 0.539987i \(-0.181571\pi\)
−0.888479 + 0.458916i \(0.848238\pi\)
\(702\) 0 0
\(703\) −9.77921 + 7.73907i −0.368830 + 0.291885i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.5354 + 16.4749i 1.07319 + 0.619604i
\(708\) 0 0
\(709\) −16.1966 + 28.0533i −0.608276 + 1.05356i 0.383249 + 0.923645i \(0.374805\pi\)
−0.991525 + 0.129919i \(0.958528\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 41.5884 + 24.0110i 1.55750 + 0.899221i
\(714\) 0 0
\(715\) −14.5838 + 31.7249i −0.545405 + 1.18644i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.64373 14.9714i −0.322357 0.558338i 0.658617 0.752478i \(-0.271142\pi\)
−0.980974 + 0.194140i \(0.937808\pi\)
\(720\) 0 0
\(721\) −42.2542 −1.57363
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.12438 + 8.88395i −0.116037 + 0.329942i
\(726\) 0 0
\(727\) 16.4031 9.47034i 0.608358 0.351236i −0.163965 0.986466i \(-0.552428\pi\)
0.772323 + 0.635231i \(0.219095\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.72397 + 8.18216i −0.174722 + 0.302628i
\(732\) 0 0
\(733\) 0.542118i 0.0200236i −0.999950 0.0100118i \(-0.996813\pi\)
0.999950 0.0100118i \(-0.00318690\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.8033 + 13.7428i 0.876805 + 0.506224i
\(738\) 0 0
\(739\) −2.87033 4.97156i −0.105587 0.182882i 0.808391 0.588646i \(-0.200339\pi\)
−0.913978 + 0.405764i \(0.867005\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.9290 11.5060i 0.731125 0.422115i −0.0877087 0.996146i \(-0.527954\pi\)
0.818834 + 0.574031i \(0.194621\pi\)
\(744\) 0 0
\(745\) 7.59703 5.38137i 0.278334 0.197158i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.6824 −0.426866
\(750\) 0 0
\(751\) 25.9704 44.9821i 0.947674 1.64142i 0.197368 0.980330i \(-0.436761\pi\)
0.750306 0.661090i \(-0.229906\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.68384 39.6211i −0.134069 1.44196i
\(756\) 0 0
\(757\) 11.0229 6.36406i 0.400633 0.231306i −0.286124 0.958193i \(-0.592367\pi\)
0.686757 + 0.726887i \(0.259034\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.6089 0.384573 0.192287 0.981339i \(-0.438410\pi\)
0.192287 + 0.981339i \(0.438410\pi\)
\(762\) 0 0
\(763\) −5.76720 + 3.32969i −0.208787 + 0.120543i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.4770i 1.89484i
\(768\) 0 0
\(769\) 10.8089 18.7215i 0.389777 0.675114i −0.602642 0.798012i \(-0.705885\pi\)
0.992419 + 0.122898i \(0.0392187\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −37.5227 21.6637i −1.34960 0.779190i −0.361404 0.932409i \(-0.617702\pi\)
−0.988192 + 0.153220i \(0.951036\pi\)
\(774\) 0 0
\(775\) −29.3930 + 5.51338i −1.05583 + 0.198047i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.8169 + 29.7813i −0.423384 + 1.06703i
\(780\) 0 0
\(781\) −9.20487 15.9433i −0.329376 0.570496i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.0843 + 17.0598i 0.431309 + 0.608891i
\(786\) 0 0
\(787\) 29.0046i 1.03390i 0.856015 + 0.516950i \(0.172933\pi\)
−0.856015 + 0.516950i \(0.827067\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.29563 0.0460674
\(792\) 0 0
\(793\) 50.8189 29.3403i 1.80463 1.04191i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.2045i 0.503150i −0.967838 0.251575i \(-0.919051\pi\)
0.967838 0.251575i \(-0.0809485\pi\)
\(798\) 0 0
\(799\) 12.0946 0.427878
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 24.3478 14.0572i 0.859214 0.496067i
\(804\) 0 0
\(805\) −41.4929 19.0742i −1.46243 0.672276i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.9781 0.526603 0.263301 0.964714i \(-0.415189\pi\)
0.263301 + 0.964714i \(0.415189\pi\)
\(810\) 0 0
\(811\) 20.5388 35.5742i 0.721214 1.24918i −0.239300 0.970946i \(-0.576918\pi\)
0.960513 0.278233i \(-0.0897489\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.45581 0.321310i 0.121052 0.0112550i
\(816\) 0 0
\(817\) 14.9671 + 5.93877i 0.523632 + 0.207771i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.5219 + 37.2771i −0.751120 + 1.30098i 0.196160 + 0.980572i \(0.437153\pi\)
−0.947280 + 0.320407i \(0.896180\pi\)
\(822\) 0 0
\(823\) 12.4379 + 7.18105i 0.433559 + 0.250316i 0.700862 0.713297i \(-0.252799\pi\)
−0.267302 + 0.963613i \(0.586132\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.4326 22.1891i −1.33643 0.771590i −0.350157 0.936691i \(-0.613872\pi\)
−0.986277 + 0.165101i \(0.947205\pi\)
\(828\) 0 0
\(829\) −35.6112 −1.23683 −0.618413 0.785853i \(-0.712224\pi\)
−0.618413 + 0.785853i \(0.712224\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.17342 + 0.677477i −0.0406567 + 0.0234732i
\(834\) 0 0
\(835\) 7.58335 16.4964i 0.262433 0.570882i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.1147 + 36.5718i 0.728962 + 1.26260i 0.957322 + 0.289022i \(0.0933301\pi\)
−0.228361 + 0.973577i \(0.573337\pi\)
\(840\) 0 0
\(841\) 12.7263 + 22.0426i 0.438837 + 0.760088i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 66.2489 46.9275i 2.27903 1.61435i
\(846\) 0 0
\(847\) 15.4015i 0.529202i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11.4856 + 19.8937i −0.393722 + 0.681946i
\(852\) 0 0
\(853\) 17.1632 9.90917i 0.587656 0.339284i −0.176514 0.984298i \(-0.556482\pi\)
0.764170 + 0.645015i \(0.223149\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.5279 25.7082i 1.52104 0.878175i 0.521352 0.853342i \(-0.325428\pi\)
0.999692 0.0248328i \(-0.00790535\pi\)
\(858\) 0 0
\(859\) −10.5243 + 18.2286i −0.359084 + 0.621951i −0.987808 0.155677i \(-0.950244\pi\)
0.628724 + 0.777628i \(0.283577\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.8358i 1.32199i 0.750391 + 0.660994i \(0.229865\pi\)
−0.750391 + 0.660994i \(0.770135\pi\)
\(864\) 0 0
\(865\) −0.945303 1.33451i −0.0321413 0.0453748i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.73812 + 8.20667i 0.160730 + 0.278392i
\(870\) 0 0
\(871\) −43.3953 75.1628i −1.47039 2.54679i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 27.3461 7.80817i 0.924467 0.263964i
\(876\) 0 0
\(877\) −22.5823 + 13.0379i −0.762551 + 0.440259i −0.830211 0.557450i \(-0.811780\pi\)
0.0676600 + 0.997708i \(0.478447\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8.23096 −0.277308 −0.138654 0.990341i \(-0.544278\pi\)
−0.138654 + 0.990341i \(0.544278\pi\)
\(882\) 0 0
\(883\) −28.4622 16.4326i −0.957828 0.553003i −0.0623244 0.998056i \(-0.519851\pi\)
−0.895504 + 0.445053i \(0.853185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.3177 7.11165i −0.413589 0.238786i 0.278741 0.960366i \(-0.410083\pi\)
−0.692331 + 0.721580i \(0.743416\pi\)
\(888\) 0 0
\(889\) −0.934139 + 1.61798i −0.0313300 + 0.0542652i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.99605 20.3942i −0.100259 0.682465i
\(894\) 0 0
\(895\) −0.556562 5.98603i −0.0186038 0.200091i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.63262 9.75598i 0.187858 0.325380i
\(900\) 0 0
\(901\) −26.3199 −0.876843
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.1779 5.13844i −0.371565 0.170808i
\(906\) 0 0
\(907\) −38.9008 + 22.4594i −1.29168 + 0.745752i −0.978952 0.204090i \(-0.934576\pi\)
−0.312729 + 0.949842i \(0.601243\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −38.3516 −1.27064 −0.635322 0.772247i \(-0.719133\pi\)
−0.635322 + 0.772247i \(0.719133\pi\)
\(912\) 0 0
\(913\) 32.8257i 1.08637i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.5244 + 7.23097i −0.413593 + 0.238788i
\(918\) 0 0
\(919\) 36.3727 1.19983 0.599913 0.800065i \(-0.295202\pi\)
0.599913 + 0.800065i \(0.295202\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 58.1318i 1.91343i
\(924\) 0 0
\(925\) −2.63731 14.0600i −0.0867142 0.462291i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.96542 3.40421i −0.0644834 0.111689i 0.831981 0.554804i \(-0.187207\pi\)
−0.896465 + 0.443115i \(0.853873\pi\)
\(930\) 0 0
\(931\) 1.43305 + 1.81082i 0.0469663 + 0.0593473i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.35106 10.3777i −0.240405 0.339387i
\(936\) 0 0
\(937\) 46.8937 + 27.0741i 1.53195 + 0.884473i 0.999272 + 0.0381558i \(0.0121483\pi\)
0.532680 + 0.846317i \(0.321185\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.9969 + 25.9755i −0.488886 + 0.846776i −0.999918 0.0127858i \(-0.995930\pi\)
0.511032 + 0.859562i \(0.329263\pi\)
\(942\) 0 0
\(943\) 59.0167i 1.92185i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.1378 + 19.7095i −1.10933 + 0.640472i −0.938656 0.344856i \(-0.887928\pi\)
−0.170674 + 0.985328i \(0.554594\pi\)
\(948\) 0 0
\(949\) −88.7758 −2.88179
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.14860 0.663142i 0.0372067 0.0214813i −0.481281 0.876566i \(-0.659828\pi\)
0.518488 + 0.855085i \(0.326495\pi\)
\(954\) 0 0
\(955\) 1.94557 + 20.9253i 0.0629571 + 0.677127i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −23.9103 + 41.4138i −0.772102 + 1.33732i
\(960\) 0 0
\(961\) 4.77372 0.153991
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.9379 21.0882i −0.480867 0.678854i
\(966\) 0 0
\(967\) −6.18431 + 3.57052i −0.198874 + 0.114820i −0.596130 0.802888i \(-0.703296\pi\)
0.397256 + 0.917708i \(0.369962\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.5252 + 21.6943i 0.401953 + 0.696202i 0.993962 0.109729i \(-0.0349983\pi\)
−0.592009 + 0.805931i \(0.701665\pi\)
\(972\) 0 0
\(973\) −9.04773 5.22371i −0.290057 0.167464i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.00409135i 0.000130894i 1.00000 6.54469e-5i \(2.08324e-5\pi\)
−1.00000 6.54469e-5i \(0.999979\pi\)
\(978\) 0 0
\(979\) −15.9979 + 27.7091i −0.511294 + 0.885587i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 38.4495 22.1988i 1.22635 0.708033i 0.260084 0.965586i \(-0.416250\pi\)
0.966264 + 0.257553i \(0.0829163\pi\)
\(984\) 0 0
\(985\) 11.3956 1.05953i 0.363094 0.0337593i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.6599 0.943128
\(990\) 0 0
\(991\) 23.7636 + 41.1598i 0.754877 + 1.30749i 0.945435 + 0.325810i \(0.105637\pi\)
−0.190558 + 0.981676i \(0.561030\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −27.3016 12.5505i −0.865519 0.397877i
\(996\) 0 0
\(997\) −25.6381 14.8021i −0.811965 0.468788i 0.0356725 0.999364i \(-0.488643\pi\)
−0.847638 + 0.530575i \(0.821976\pi\)
\(998\) 0 0
\(999\) 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 3420.2.bj.c.2629.5 20
3.2 odd 2 380.2.r.a.349.6 yes 20
5.4 even 2 inner 3420.2.bj.c.2629.3 20
15.2 even 4 1900.2.i.g.501.5 20
15.8 even 4 1900.2.i.g.501.6 20
15.14 odd 2 380.2.r.a.349.5 yes 20
19.11 even 3 inner 3420.2.bj.c.1189.3 20
57.11 odd 6 380.2.r.a.49.5 20
95.49 even 6 inner 3420.2.bj.c.1189.5 20
285.68 even 12 1900.2.i.g.201.6 20
285.182 even 12 1900.2.i.g.201.5 20
285.239 odd 6 380.2.r.a.49.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 20 57.11 odd 6
380.2.r.a.49.6 yes 20 285.239 odd 6
380.2.r.a.349.5 yes 20 15.14 odd 2
380.2.r.a.349.6 yes 20 3.2 odd 2
1900.2.i.g.201.5 20 285.182 even 12
1900.2.i.g.201.6 20 285.68 even 12
1900.2.i.g.501.5 20 15.2 even 4
1900.2.i.g.501.6 20 15.8 even 4
3420.2.bj.c.1189.3 20 19.11 even 3 inner
3420.2.bj.c.1189.5 20 95.49 even 6 inner
3420.2.bj.c.2629.3 20 5.4 even 2 inner
3420.2.bj.c.2629.5 20 1.1 even 1 trivial