Properties

Label 3420.2.bj.c.1189.5
Level $3420$
Weight $2$
Character 3420.1189
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 1189.5
Root \(0.392182 + 0.226426i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.5

$q$-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 \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 \(0.159620\pi\)
−0.876881 + 0.480707i \(0.840380\pi\)
\(8\) 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 0.729526 0.683953i \(-0.239741\pi\)
0.957084 0.289812i \(-0.0935927\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.21492 1.27878i −0.537197 0.310151i 0.206745 0.978395i \(-0.433713\pi\)
−0.743942 + 0.668244i \(0.767046\pi\)
\(18\) 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 0.451387 0.892329i \(-0.350930\pi\)
0.998472 + 0.0552521i \(0.0175962\pi\)
\(24\) 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 0.940118 0.340849i \(-0.110714\pi\)
−0.765243 + 0.643742i \(0.777381\pi\)
\(30\) 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 \(-0.924433\pi\)
0.971953 0.235177i \(-0.0755669\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.67524 6.36571i 0.573977 0.994157i −0.422175 0.906514i \(-0.638733\pi\)
0.996152 0.0876426i \(-0.0279333\pi\)
\(42\) 0 0
\(43\) 3.19919 + 1.84706i 0.487873 + 0.281673i 0.723691 0.690124i \(-0.242444\pi\)
−0.235819 + 0.971797i \(0.575777\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.09540 + 2.36448i −0.597376 + 0.344895i −0.768008 0.640440i \(-0.778752\pi\)
0.170633 + 0.985335i \(0.445419\pi\)
\(48\) 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 0.258383 0.966042i \(-0.416810\pi\)
0.965809 + 0.259255i \(0.0834769\pi\)
\(54\) 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 −0.513408 0.858145i \(-0.671617\pi\)
0.999879 + 0.0155515i \(0.00495040\pi\)
\(60\) 0 0
\(61\) 4.17839 + 7.23719i 0.534988 + 0.926627i 0.999164 + 0.0408838i \(0.0130174\pi\)
−0.464176 + 0.885743i \(0.653649\pi\)
\(62\) 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 −0.981713 0.190368i \(-0.939032\pi\)
−0.325993 + 0.945372i \(0.605698\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.13931 7.16950i 0.491246 0.850863i −0.508703 0.860942i \(-0.669875\pi\)
0.999949 + 0.0100790i \(0.00320829\pi\)
\(72\) 0 0
\(73\) −10.9489 6.32134i −1.28147 0.739857i −0.304352 0.952559i \(-0.598440\pi\)
−0.977117 + 0.212703i \(0.931773\pi\)
\(74\) 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 −0.960634 0.277818i \(-0.910389\pi\)
0.720914 + 0.693024i \(0.243722\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.7613i 1.62026i 0.586248 + 0.810132i \(0.300604\pi\)
−0.586248 + 0.810132i \(0.699396\pi\)
\(84\) 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 0.941524 + 0.336946i \(0.109394\pi\)
−0.178958 + 0.983857i \(0.557273\pi\)
\(90\) 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 0.733910 0.679247i \(-0.237694\pi\)
−0.221291 + 0.975208i \(0.571027\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.47686 11.2183i −0.644472 1.11626i −0.984423 0.175815i \(-0.943744\pi\)
0.339951 0.940443i \(-0.389589\pi\)
\(102\) 0 0
\(103\) 16.6116i 1.63679i 0.574658 + 0.818394i \(0.305135\pi\)
−0.574658 + 0.818394i \(0.694865\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.59275i 0.443998i 0.975047 + 0.221999i \(0.0712582\pi\)
−0.975047 + 0.221999i \(0.928742\pi\)
\(108\) 0 0
\(109\) −1.30902 + 2.26728i −0.125381 + 0.217166i −0.921882 0.387471i \(-0.873349\pi\)
0.796501 + 0.604637i \(0.206682\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.509357i 0.0479163i −0.999713 0.0239581i \(-0.992373\pi\)
0.999713 0.0239581i \(-0.00762684\pi\)
\(114\) 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 −0.527956 0.849272i \(-0.677041\pi\)
0.471513 + 0.881859i \(0.343708\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.84274 + 4.92377i −0.248371 + 0.430192i −0.963074 0.269236i \(-0.913229\pi\)
0.714703 + 0.699428i \(0.246562\pi\)
\(132\) 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 −0.993425 0.114482i \(-0.963479\pi\)
−0.397568 + 0.917573i \(0.630146\pi\)
\(138\) 0 0
\(139\) 2.05362 + 3.55697i 0.174186 + 0.301698i 0.939879 0.341507i \(-0.110937\pi\)
−0.765694 + 0.643206i \(0.777604\pi\)
\(140\) 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 −0.768066 0.640370i \(-0.778781\pi\)
0.938610 + 0.344980i \(0.112114\pi\)
\(150\) 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 0.140796 0.990039i \(-0.455034\pi\)
−0.787001 + 0.616952i \(0.788367\pi\)
\(158\) 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 \(-0.980639\pi\)
0.998151 0.0607869i \(-0.0193610\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.03177 4.05980i 0.544135 0.314157i −0.202618 0.979258i \(-0.564945\pi\)
0.746753 + 0.665101i \(0.231612\pi\)
\(168\) 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 0.523884 0.851789i \(-0.324482\pi\)
−0.475729 + 0.879592i \(0.657816\pi\)
\(174\) 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 0.949965 0.312358i \(-0.101119\pi\)
−0.745492 + 0.666515i \(0.767785\pi\)
\(182\) 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 0.814921 0.579572i \(-0.196780\pi\)
−0.0944641 + 0.995528i \(0.530114\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.11825i 0.364660i −0.983237 0.182330i \(-0.941636\pi\)
0.983237 0.182330i \(-0.0583639\pi\)
\(198\) 0 0
\(199\) 6.71897 + 11.6376i 0.476295 + 0.824968i 0.999631 0.0271588i \(-0.00864598\pi\)
−0.523336 + 0.852127i \(0.675313\pi\)
\(200\) 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 −0.765224 0.643764i \(-0.777372\pi\)
0.940128 + 0.340821i \(0.110705\pi\)
\(212\) 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 −0.382595 0.923916i \(-0.624970\pi\)
−0.991432 + 0.130621i \(0.958303\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.0306i 1.26310i −0.775334 0.631551i \(-0.782418\pi\)
0.775334 0.631551i \(-0.217582\pi\)
\(228\) 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 0.0410592 0.999157i \(-0.486927\pi\)
−0.844765 + 0.535137i \(0.820260\pi\)
\(234\) 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 −0.998005 0.0631409i \(-0.979888\pi\)
0.444321 0.895868i \(-0.353445\pi\)
\(242\) 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 0.947728 + 0.319079i \(0.103374\pi\)
−0.197534 + 0.980296i \(0.563293\pi\)
\(252\) 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 −0.538202 0.842816i \(-0.680896\pi\)
0.460799 + 0.887504i \(0.347563\pi\)
\(258\) 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 0.297390 0.954756i \(-0.403884\pi\)
−0.678148 + 0.734926i \(0.737217\pi\)
\(264\) 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 −0.440808 0.897601i \(-0.645308\pi\)
0.997750 0.0670494i \(-0.0213585\pi\)
\(270\) 0 0
\(271\) 2.90265 5.02754i 0.176324 0.305401i −0.764295 0.644867i \(-0.776913\pi\)
0.940619 + 0.339465i \(0.110246\pi\)
\(272\) 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 \(-0.945008\pi\)
0.985114 0.171904i \(-0.0549918\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.07155 5.32009i −0.183233 0.317370i 0.759746 0.650220i \(-0.225323\pi\)
−0.942980 + 0.332850i \(0.891990\pi\)
\(282\) 0 0
\(283\) −14.2484 8.22632i −0.846980 0.489004i 0.0126510 0.999920i \(-0.495973\pi\)
−0.859631 + 0.510916i \(0.829306\pi\)
\(284\) 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 \(0.193677\pi\)
−0.820533 + 0.571599i \(0.806323\pi\)
\(294\) 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 0.996590 0.0825100i \(-0.0262937\pi\)
0.426839 + 0.904327i \(0.359627\pi\)
\(308\) 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 −0.245209 0.969470i \(-0.578857\pi\)
0.716981 + 0.697093i \(0.245523\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.199789 + 0.115348i −0.0112213 + 0.00647860i −0.505600 0.862768i \(-0.668729\pi\)
0.494379 + 0.869247i \(0.335396\pi\)
\(318\) 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 0.213611 0.976919i \(-0.431477\pi\)
−0.739231 + 0.673452i \(0.764811\pi\)
\(338\) 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 0.626391 0.779509i \(-0.284532\pi\)
−0.361880 + 0.932225i \(0.617865\pi\)
\(348\) 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 \(-0.984142\pi\)
0.998759 0.0497981i \(-0.0158578\pi\)
\(354\) 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 −0.550767 0.834659i \(-0.685665\pi\)
0.998219 + 0.0596485i \(0.0189980\pi\)
\(360\) 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 −0.583102 0.812399i \(-0.698161\pi\)
0.412008 + 0.911180i \(0.364828\pi\)
\(368\) 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 \(-0.962732\pi\)
0.993154 0.116812i \(-0.0372675\pi\)
\(374\) 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 0.776370 0.630277i \(-0.217059\pi\)
−0.157651 + 0.987495i \(0.550392\pi\)
\(384\) 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 0.748872 0.662715i \(-0.230596\pi\)
−0.948364 + 0.317185i \(0.897263\pi\)
\(390\) 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 0.821203 0.570637i \(-0.193304\pi\)
−0.0835846 + 0.996501i \(0.526637\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.16651 12.4128i 0.357878 0.619864i −0.629728 0.776816i \(-0.716834\pi\)
0.987606 + 0.156952i \(0.0501669\pi\)
\(402\) 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 0.862980 0.505238i \(-0.168595\pi\)
−0.869039 + 0.494744i \(0.835262\pi\)
\(410\) 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 −0.930313 0.366765i \(-0.880465\pi\)
0.782785 + 0.622292i \(0.213798\pi\)
\(422\) 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 0.999178 + 0.0405356i \(0.0129064\pi\)
−0.464484 + 0.885581i \(0.653760\pi\)
\(432\) 0 0
\(433\) 21.0094 12.1298i 1.00965 0.582920i 0.0985593 0.995131i \(-0.468577\pi\)
0.911089 + 0.412211i \(0.135243\pi\)
\(434\) 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 0.426258 + 0.904602i \(0.359832\pi\)
−0.996537 + 0.0831509i \(0.973502\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.9017 10.3356i 0.850536 0.491057i −0.0102958 0.999947i \(-0.503277\pi\)
0.860832 + 0.508890i \(0.169944\pi\)
\(444\) 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 \(0.108897\pi\)
−0.942049 + 0.335476i \(0.891103\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −13.3682 + 23.1544i −0.622618 + 1.07841i 0.366378 + 0.930466i \(0.380598\pi\)
−0.988996 + 0.147940i \(0.952736\pi\)
\(462\) 0 0
\(463\) 12.0950i 0.562101i 0.959693 + 0.281051i \(0.0906829\pi\)
−0.959693 + 0.281051i \(0.909317\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.3169i 1.26408i 0.774938 + 0.632038i \(0.217781\pi\)
−0.774938 + 0.632038i \(0.782219\pi\)
\(468\) 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 0.769143 0.639077i \(-0.220684\pi\)
−0.938028 + 0.346559i \(0.887350\pi\)
\(480\) 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 \(0.0947330\pi\)
−0.956039 + 0.293239i \(0.905267\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.59974 + 9.69903i −0.252713 + 0.437711i −0.964272 0.264915i \(-0.914656\pi\)
0.711559 + 0.702626i \(0.247989\pi\)
\(492\) 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 0.414796 + 0.909914i \(0.363853\pi\)
−0.995407 + 0.0957335i \(0.969480\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.6079 + 13.6300i −1.05262 + 0.607733i −0.923383 0.383881i \(-0.874587\pi\)
−0.129241 + 0.991613i \(0.541254\pi\)
\(504\) 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 −0.997548 0.0699892i \(-0.977704\pi\)
0.438161 0.898896i \(-0.355630\pi\)
\(510\) 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 −0.612102 0.790779i \(-0.709676\pi\)
0.378784 + 0.925485i \(0.376342\pi\)
\(524\) 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 0.787025 0.616922i \(-0.211620\pi\)
−0.927782 + 0.373122i \(0.878287\pi\)
\(542\) 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 −0.855610 0.517621i \(-0.826818\pi\)
0.0204683 + 0.999791i \(0.493484\pi\)
\(548\) 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 −0.767517 0.641029i \(-0.778508\pi\)
0.171389 + 0.985203i \(0.445175\pi\)
\(558\) 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 \(-0.760250\pi\)
0.729506 0.683974i \(-0.239750\pi\)
\(564\) 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 \(-0.621496\pi\)
0.372489 0.928037i \(-0.378504\pi\)
\(578\) 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 0.737374 0.675485i \(-0.236065\pi\)
−0.216300 + 0.976327i \(0.569399\pi\)
\(588\) 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 −0.0146433 0.999893i \(-0.504661\pi\)
0.858611 + 0.512628i \(0.171328\pi\)
\(594\) 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 0.994079 0.108658i \(-0.0346552\pi\)
−0.591140 + 0.806569i \(0.701322\pi\)
\(600\) 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 \(-0.953050\pi\)
0.989142 0.146964i \(-0.0469501\pi\)
\(608\) 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 −0.309352 0.950947i \(-0.600112\pi\)
−0.978221 + 0.207567i \(0.933446\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.7741 + 18.3448i −1.27918 + 0.738533i −0.976697 0.214623i \(-0.931148\pi\)
−0.302479 + 0.953156i \(0.597814\pi\)
\(618\) 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 0.689011 0.724751i \(-0.258045\pi\)
−0.972158 + 0.234325i \(0.924712\pi\)
\(632\) 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 −0.993994 0.109437i \(-0.965095\pi\)
0.591772 + 0.806105i \(0.298429\pi\)
\(642\) 0 0
\(643\) 10.0041 + 5.77586i 0.394523 + 0.227778i 0.684118 0.729371i \(-0.260187\pi\)
−0.289595 + 0.957149i \(0.593521\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.1233i 0.515932i 0.966154 + 0.257966i \(0.0830522\pi\)
−0.966154 + 0.257966i \(0.916948\pi\)
\(648\) 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 \(0.124208\pi\)
−0.924828 + 0.380385i \(0.875792\pi\)
\(654\) 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 0.999065 + 0.0432391i \(0.0137677\pi\)
−0.462086 + 0.886835i \(0.652899\pi\)
\(660\) 0 0
\(661\) 3.88006 + 6.72046i 0.150917 + 0.261396i 0.931565 0.363575i \(-0.118444\pi\)
−0.780648 + 0.624971i \(0.785111\pi\)
\(662\) 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 \(-0.845115\pi\)
0.883935 0.467610i \(-0.154885\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.88851i 0.187881i 0.995578 + 0.0939403i \(0.0299463\pi\)
−0.995578 + 0.0939403i \(0.970054\pi\)
\(678\) 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 \(-0.996589\pi\)
0.999943 0.0107145i \(-0.00341060\pi\)
\(684\) 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 −0.888479 0.458916i \(-0.848238\pi\)
0.841673 + 0.539987i \(0.181571\pi\)
\(702\) 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 −0.991525 0.129919i \(-0.958528\pi\)
0.383249 0.923645i \(-0.374805\pi\)
\(710\) 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 −0.980974 0.194140i \(-0.937808\pi\)
0.658617 + 0.752478i \(0.271142\pi\)
\(720\) 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 0.772323 0.635231i \(-0.219095\pi\)
−0.163965 + 0.986466i \(0.552428\pi\)
\(728\) 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 \(0.00318690\pi\)
−0.999950 + 0.0100118i \(0.996813\pi\)
\(734\) 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 −0.913978 0.405764i \(-0.867005\pi\)
0.808391 + 0.588646i \(0.200339\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.9290 + 11.5060i 0.731125 + 0.422115i 0.818834 0.574031i \(-0.194621\pi\)
−0.0877087 + 0.996146i \(0.527954\pi\)
\(744\) 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 0.750306 + 0.661090i \(0.229906\pi\)
0.197368 + 0.980330i \(0.436761\pi\)
\(752\) 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 0.686757 0.726887i \(-0.259034\pi\)
−0.286124 + 0.958193i \(0.592367\pi\)
\(758\) 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 0.992419 0.122898i \(-0.0392187\pi\)
−0.602642 + 0.798012i \(0.705885\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −37.5227 + 21.6637i −1.34960 + 0.779190i −0.988192 0.153220i \(-0.951036\pi\)
−0.361404 + 0.932409i \(0.617702\pi\)
\(774\) 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 \(-0.827067\pi\)
0.856015 0.516950i \(-0.172933\pi\)
\(788\) 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 \(0.0809485\pi\)
−0.967838 + 0.251575i \(0.919051\pi\)
\(798\) 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 0.960513 + 0.278233i \(0.0897489\pi\)
−0.239300 + 0.970946i \(0.576918\pi\)
\(812\) 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 −0.947280 0.320407i \(-0.896180\pi\)
0.196160 0.980572i \(-0.437153\pi\)
\(822\) 0 0
\(823\) 12.4379 7.18105i 0.433559 0.250316i −0.267302 0.963613i \(-0.586132\pi\)
0.700862 + 0.713297i \(0.252799\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.4326 + 22.1891i −1.33643 + 0.771590i −0.986277 0.165101i \(-0.947205\pi\)
−0.350157 + 0.936691i \(0.613872\pi\)
\(828\) 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 −0.228361 0.973577i \(-0.573337\pi\)
0.957322 0.289022i \(-0.0933301\pi\)
\(840\) 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 0.764170 0.645015i \(-0.223149\pi\)
−0.176514 + 0.984298i \(0.556482\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.5279 + 25.7082i 1.52104 + 0.878175i 0.999692 + 0.0248328i \(0.00790535\pi\)
0.521352 + 0.853342i \(0.325428\pi\)
\(858\) 0 0
\(859\) −10.5243 18.2286i −0.359084 0.621951i 0.628724 0.777628i \(-0.283577\pi\)
−0.987808 + 0.155677i \(0.950244\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.8358i 1.32199i −0.750391 0.660994i \(-0.770135\pi\)
0.750391 0.660994i \(-0.229865\pi\)
\(864\) 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 0.0676600 0.997708i \(-0.478447\pi\)
−0.830211 + 0.557450i \(0.811780\pi\)
\(878\) 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 −0.895504 0.445053i \(-0.853185\pi\)
−0.0623244 + 0.998056i \(0.519851\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.3177 + 7.11165i −0.413589 + 0.238786i −0.692331 0.721580i \(-0.743416\pi\)
0.278741 + 0.960366i \(0.410083\pi\)
\(888\) 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 −0.312729 0.949842i \(-0.601243\pi\)
−0.978952 + 0.204090i \(0.934576\pi\)
\(908\) 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 −0.896465 0.443115i \(-0.853873\pi\)
0.831981 + 0.554804i \(0.187207\pi\)
\(930\) 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 0.532680 0.846317i \(-0.321185\pi\)
0.999272 0.0381558i \(-0.0121483\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −14.9969 25.9755i −0.488886 0.846776i 0.511032 0.859562i \(-0.329263\pi\)
−0.999918 + 0.0127858i \(0.995930\pi\)
\(942\) 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 −0.170674 0.985328i \(-0.554594\pi\)
−0.938656 + 0.344856i \(0.887928\pi\)
\(948\) 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 0.518488 0.855085i \(-0.326495\pi\)
−0.481281 + 0.876566i \(0.659828\pi\)
\(954\) 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 0.397256 0.917708i \(-0.369962\pi\)
−0.596130 + 0.802888i \(0.703296\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.5252 21.6943i 0.401953 0.696202i −0.592009 0.805931i \(-0.701665\pi\)
0.993962 + 0.109729i \(0.0349983\pi\)
\(972\) 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 \(-0.999979\pi\)
1.00000 6.54469e-5i \(-2.08324e-5\pi\)
\(978\) 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 0.966264 0.257553i \(-0.0829163\pi\)
0.260084 + 0.965586i \(0.416250\pi\)
\(984\) 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 −0.190558 0.981676i \(-0.561030\pi\)
0.945435 0.325810i \(-0.105637\pi\)
\(992\) 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 −0.847638 0.530575i \(-0.821976\pi\)
0.0356725 + 0.999364i \(0.488643\pi\)
\(998\) 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.1189.5 20
3.2 odd 2 380.2.r.a.49.6 yes 20
5.4 even 2 inner 3420.2.bj.c.1189.3 20
15.2 even 4 1900.2.i.g.201.6 20
15.8 even 4 1900.2.i.g.201.5 20
15.14 odd 2 380.2.r.a.49.5 20
19.7 even 3 inner 3420.2.bj.c.2629.3 20
57.26 odd 6 380.2.r.a.349.5 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.5 20
285.83 even 12 1900.2.i.g.501.5 20
285.197 even 12 1900.2.i.g.501.6 20
285.254 odd 6 380.2.r.a.349.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 20 15.14 odd 2
380.2.r.a.49.6 yes 20 3.2 odd 2
380.2.r.a.349.5 yes 20 57.26 odd 6
380.2.r.a.349.6 yes 20 285.254 odd 6
1900.2.i.g.201.5 20 15.8 even 4
1900.2.i.g.201.6 20 15.2 even 4
1900.2.i.g.501.5 20 285.83 even 12
1900.2.i.g.501.6 20 285.197 even 12
3420.2.bj.c.1189.3 20 5.4 even 2 inner
3420.2.bj.c.1189.5 20 1.1 even 1 trivial
3420.2.bj.c.2629.3 20 19.7 even 3 inner
3420.2.bj.c.2629.5 20 95.64 even 6 inner