Properties

Label 960.2.h.a.191.3
Level $960$
Weight $2$
Character 960.191
Analytic conductor $7.666$
Analytic rank $1$
Dimension $4$
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] = [960,2,Mod(191,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (of order \(2\), degree \(1\), not 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: \(7.66563859404\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 960.191
Dual form 960.2.h.a.191.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.41421i) q^{3} -1.00000i q^{5} +4.82843i q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\) \(q+(-1.00000 + 1.41421i) q^{3} -1.00000i q^{5} +4.82843i q^{7} +(-1.00000 - 2.82843i) q^{9} -4.82843 q^{11} -1.17157 q^{13} +(1.41421 + 1.00000i) q^{15} -0.828427i q^{17} -6.00000i q^{19} +(-6.82843 - 4.82843i) q^{21} +0.828427 q^{23} -1.00000 q^{25} +(5.00000 + 1.41421i) q^{27} +6.00000i q^{29} -2.00000i q^{31} +(4.82843 - 6.82843i) q^{33} +4.82843 q^{35} -6.82843 q^{37} +(1.17157 - 1.65685i) q^{39} -9.65685i q^{41} +1.17157i q^{43} +(-2.82843 + 1.00000i) q^{45} -3.17157 q^{47} -16.3137 q^{49} +(1.17157 + 0.828427i) q^{51} -7.65685i q^{53} +4.82843i q^{55} +(8.48528 + 6.00000i) q^{57} -12.8284 q^{59} +13.3137 q^{61} +(13.6569 - 4.82843i) q^{63} +1.17157i q^{65} -4.48528i q^{67} +(-0.828427 + 1.17157i) q^{69} -11.3137 q^{71} -2.00000 q^{73} +(1.00000 - 1.41421i) q^{75} -23.3137i q^{77} -6.00000i q^{79} +(-7.00000 + 5.65685i) q^{81} +5.31371 q^{83} -0.828427 q^{85} +(-8.48528 - 6.00000i) q^{87} -5.65685i q^{91} +(2.82843 + 2.00000i) q^{93} -6.00000 q^{95} -6.00000 q^{97} +(4.82843 + 13.6569i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{9} - 8 q^{11} - 16 q^{13} - 16 q^{21} - 8 q^{23} - 4 q^{25} + 20 q^{27} + 8 q^{33} + 8 q^{35} - 16 q^{37} + 16 q^{39} - 24 q^{47} - 20 q^{49} + 16 q^{51} - 40 q^{59} + 8 q^{61} + 32 q^{63} + 8 q^{69} - 8 q^{73} + 4 q^{75} - 28 q^{81} - 24 q^{83} + 8 q^{85} - 24 q^{95} - 24 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 + 1.41421i −0.577350 + 0.816497i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 4.82843i 1.82497i 0.409106 + 0.912487i \(0.365841\pi\)
−0.409106 + 0.912487i \(0.634159\pi\)
\(8\) 0 0
\(9\) −1.00000 2.82843i −0.333333 0.942809i
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) −1.17157 −0.324936 −0.162468 0.986714i \(-0.551945\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(14\) 0 0
\(15\) 1.41421 + 1.00000i 0.365148 + 0.258199i
\(16\) 0 0
\(17\) 0.828427i 0.200923i −0.994941 0.100462i \(-0.967968\pi\)
0.994941 0.100462i \(-0.0320319\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) −6.82843 4.82843i −1.49008 1.05365i
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 2.00000i 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 0 0
\(33\) 4.82843 6.82843i 0.840521 1.18868i
\(34\) 0 0
\(35\) 4.82843 0.816153
\(36\) 0 0
\(37\) −6.82843 −1.12259 −0.561293 0.827617i \(-0.689696\pi\)
−0.561293 + 0.827617i \(0.689696\pi\)
\(38\) 0 0
\(39\) 1.17157 1.65685i 0.187602 0.265309i
\(40\) 0 0
\(41\) 9.65685i 1.50815i −0.656790 0.754074i \(-0.728086\pi\)
0.656790 0.754074i \(-0.271914\pi\)
\(42\) 0 0
\(43\) 1.17157i 0.178663i 0.996002 + 0.0893316i \(0.0284731\pi\)
−0.996002 + 0.0893316i \(0.971527\pi\)
\(44\) 0 0
\(45\) −2.82843 + 1.00000i −0.421637 + 0.149071i
\(46\) 0 0
\(47\) −3.17157 −0.462621 −0.231311 0.972880i \(-0.574301\pi\)
−0.231311 + 0.972880i \(0.574301\pi\)
\(48\) 0 0
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) 1.17157 + 0.828427i 0.164053 + 0.116003i
\(52\) 0 0
\(53\) 7.65685i 1.05175i −0.850562 0.525875i \(-0.823738\pi\)
0.850562 0.525875i \(-0.176262\pi\)
\(54\) 0 0
\(55\) 4.82843i 0.651065i
\(56\) 0 0
\(57\) 8.48528 + 6.00000i 1.12390 + 0.794719i
\(58\) 0 0
\(59\) −12.8284 −1.67012 −0.835059 0.550160i \(-0.814567\pi\)
−0.835059 + 0.550160i \(0.814567\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 13.6569 4.82843i 1.72060 0.608325i
\(64\) 0 0
\(65\) 1.17157i 0.145316i
\(66\) 0 0
\(67\) 4.48528i 0.547964i −0.961735 0.273982i \(-0.911659\pi\)
0.961735 0.273982i \(-0.0883409\pi\)
\(68\) 0 0
\(69\) −0.828427 + 1.17157i −0.0997309 + 0.141041i
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 1.00000 1.41421i 0.115470 0.163299i
\(76\) 0 0
\(77\) 23.3137i 2.65684i
\(78\) 0 0
\(79\) 6.00000i 0.675053i −0.941316 0.337526i \(-0.890410\pi\)
0.941316 0.337526i \(-0.109590\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 0 0
\(83\) 5.31371 0.583255 0.291628 0.956532i \(-0.405803\pi\)
0.291628 + 0.956532i \(0.405803\pi\)
\(84\) 0 0
\(85\) −0.828427 −0.0898555
\(86\) 0 0
\(87\) −8.48528 6.00000i −0.909718 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 5.65685i 0.592999i
\(92\) 0 0
\(93\) 2.82843 + 2.00000i 0.293294 + 0.207390i
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 4.82843 + 13.6569i 0.485275 + 1.37257i
\(100\) 0 0
\(101\) 7.65685i 0.761885i 0.924599 + 0.380943i \(0.124401\pi\)
−0.924599 + 0.380943i \(0.875599\pi\)
\(102\) 0 0
\(103\) 0.828427i 0.0816274i −0.999167 0.0408137i \(-0.987005\pi\)
0.999167 0.0408137i \(-0.0129950\pi\)
\(104\) 0 0
\(105\) −4.82843 + 6.82843i −0.471206 + 0.666386i
\(106\) 0 0
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 0 0
\(109\) −7.65685 −0.733394 −0.366697 0.930341i \(-0.619511\pi\)
−0.366697 + 0.930341i \(0.619511\pi\)
\(110\) 0 0
\(111\) 6.82843 9.65685i 0.648126 0.916588i
\(112\) 0 0
\(113\) 7.17157i 0.674645i −0.941389 0.337322i \(-0.890479\pi\)
0.941389 0.337322i \(-0.109521\pi\)
\(114\) 0 0
\(115\) 0.828427i 0.0772512i
\(116\) 0 0
\(117\) 1.17157 + 3.31371i 0.108312 + 0.306352i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 13.6569 + 9.65685i 1.23140 + 0.870729i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 6.48528i 0.575476i 0.957709 + 0.287738i \(0.0929032\pi\)
−0.957709 + 0.287738i \(0.907097\pi\)
\(128\) 0 0
\(129\) −1.65685 1.17157i −0.145878 0.103151i
\(130\) 0 0
\(131\) 2.48528 0.217140 0.108570 0.994089i \(-0.465373\pi\)
0.108570 + 0.994089i \(0.465373\pi\)
\(132\) 0 0
\(133\) 28.9706 2.51207
\(134\) 0 0
\(135\) 1.41421 5.00000i 0.121716 0.430331i
\(136\) 0 0
\(137\) 16.1421i 1.37912i 0.724231 + 0.689558i \(0.242195\pi\)
−0.724231 + 0.689558i \(0.757805\pi\)
\(138\) 0 0
\(139\) 21.3137i 1.80781i 0.427738 + 0.903903i \(0.359310\pi\)
−0.427738 + 0.903903i \(0.640690\pi\)
\(140\) 0 0
\(141\) 3.17157 4.48528i 0.267095 0.377729i
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 16.3137 23.0711i 1.34553 1.90287i
\(148\) 0 0
\(149\) 12.3431i 1.01119i 0.862771 + 0.505595i \(0.168727\pi\)
−0.862771 + 0.505595i \(0.831273\pi\)
\(150\) 0 0
\(151\) 7.65685i 0.623106i 0.950229 + 0.311553i \(0.100849\pi\)
−0.950229 + 0.311553i \(0.899151\pi\)
\(152\) 0 0
\(153\) −2.34315 + 0.828427i −0.189432 + 0.0669744i
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −0.485281 −0.0387297 −0.0193648 0.999812i \(-0.506164\pi\)
−0.0193648 + 0.999812i \(0.506164\pi\)
\(158\) 0 0
\(159\) 10.8284 + 7.65685i 0.858750 + 0.607228i
\(160\) 0 0
\(161\) 4.00000i 0.315244i
\(162\) 0 0
\(163\) 0.485281i 0.0380102i 0.999819 + 0.0190051i \(0.00604987\pi\)
−0.999819 + 0.0190051i \(0.993950\pi\)
\(164\) 0 0
\(165\) −6.82843 4.82843i −0.531592 0.375893i
\(166\) 0 0
\(167\) −24.8284 −1.92128 −0.960641 0.277794i \(-0.910397\pi\)
−0.960641 + 0.277794i \(0.910397\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 0 0
\(171\) −16.9706 + 6.00000i −1.29777 + 0.458831i
\(172\) 0 0
\(173\) 10.9706i 0.834076i 0.908889 + 0.417038i \(0.136932\pi\)
−0.908889 + 0.417038i \(0.863068\pi\)
\(174\) 0 0
\(175\) 4.82843i 0.364995i
\(176\) 0 0
\(177\) 12.8284 18.1421i 0.964244 1.36365i
\(178\) 0 0
\(179\) −3.17157 −0.237054 −0.118527 0.992951i \(-0.537817\pi\)
−0.118527 + 0.992951i \(0.537817\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −13.3137 + 18.8284i −0.984178 + 1.39184i
\(184\) 0 0
\(185\) 6.82843i 0.502036i
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) −6.82843 + 24.1421i −0.496695 + 1.75608i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 18.9706 1.36553 0.682765 0.730638i \(-0.260777\pi\)
0.682765 + 0.730638i \(0.260777\pi\)
\(194\) 0 0
\(195\) −1.65685 1.17157i −0.118650 0.0838981i
\(196\) 0 0
\(197\) 13.3137i 0.948562i 0.880373 + 0.474281i \(0.157292\pi\)
−0.880373 + 0.474281i \(0.842708\pi\)
\(198\) 0 0
\(199\) 7.65685i 0.542780i −0.962469 0.271390i \(-0.912517\pi\)
0.962469 0.271390i \(-0.0874833\pi\)
\(200\) 0 0
\(201\) 6.34315 + 4.48528i 0.447411 + 0.316367i
\(202\) 0 0
\(203\) −28.9706 −2.03333
\(204\) 0 0
\(205\) −9.65685 −0.674464
\(206\) 0 0
\(207\) −0.828427 2.34315i −0.0575797 0.162860i
\(208\) 0 0
\(209\) 28.9706i 2.00394i
\(210\) 0 0
\(211\) 6.97056i 0.479873i 0.970789 + 0.239937i \(0.0771267\pi\)
−0.970789 + 0.239937i \(0.922873\pi\)
\(212\) 0 0
\(213\) 11.3137 16.0000i 0.775203 1.09630i
\(214\) 0 0
\(215\) 1.17157 0.0799006
\(216\) 0 0
\(217\) 9.65685 0.655550
\(218\) 0 0
\(219\) 2.00000 2.82843i 0.135147 0.191127i
\(220\) 0 0
\(221\) 0.970563i 0.0652871i
\(222\) 0 0
\(223\) 6.48528i 0.434287i 0.976140 + 0.217143i \(0.0696739\pi\)
−0.976140 + 0.217143i \(0.930326\pi\)
\(224\) 0 0
\(225\) 1.00000 + 2.82843i 0.0666667 + 0.188562i
\(226\) 0 0
\(227\) −11.6569 −0.773693 −0.386846 0.922144i \(-0.626436\pi\)
−0.386846 + 0.922144i \(0.626436\pi\)
\(228\) 0 0
\(229\) −18.9706 −1.25361 −0.626805 0.779176i \(-0.715638\pi\)
−0.626805 + 0.779176i \(0.715638\pi\)
\(230\) 0 0
\(231\) 32.9706 + 23.3137i 2.16930 + 1.53393i
\(232\) 0 0
\(233\) 20.8284i 1.36452i −0.731112 0.682258i \(-0.760998\pi\)
0.731112 0.682258i \(-0.239002\pi\)
\(234\) 0 0
\(235\) 3.17157i 0.206891i
\(236\) 0 0
\(237\) 8.48528 + 6.00000i 0.551178 + 0.389742i
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) −1.00000 15.5563i −0.0641500 0.997940i
\(244\) 0 0
\(245\) 16.3137i 1.04224i
\(246\) 0 0
\(247\) 7.02944i 0.447272i
\(248\) 0 0
\(249\) −5.31371 + 7.51472i −0.336743 + 0.476226i
\(250\) 0 0
\(251\) 17.7990 1.12346 0.561731 0.827320i \(-0.310136\pi\)
0.561731 + 0.827320i \(0.310136\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0.828427 1.17157i 0.0518781 0.0733667i
\(256\) 0 0
\(257\) 26.4853i 1.65211i 0.563592 + 0.826053i \(0.309419\pi\)
−0.563592 + 0.826053i \(0.690581\pi\)
\(258\) 0 0
\(259\) 32.9706i 2.04869i
\(260\) 0 0
\(261\) 16.9706 6.00000i 1.05045 0.371391i
\(262\) 0 0
\(263\) −2.48528 −0.153249 −0.0766245 0.997060i \(-0.524414\pi\)
−0.0766245 + 0.997060i \(0.524414\pi\)
\(264\) 0 0
\(265\) −7.65685 −0.470357
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.65685i 0.222962i 0.993767 + 0.111481i \(0.0355595\pi\)
−0.993767 + 0.111481i \(0.964441\pi\)
\(270\) 0 0
\(271\) 14.9706i 0.909397i −0.890645 0.454698i \(-0.849747\pi\)
0.890645 0.454698i \(-0.150253\pi\)
\(272\) 0 0
\(273\) 8.00000 + 5.65685i 0.484182 + 0.342368i
\(274\) 0 0
\(275\) 4.82843 0.291165
\(276\) 0 0
\(277\) −3.51472 −0.211179 −0.105589 0.994410i \(-0.533673\pi\)
−0.105589 + 0.994410i \(0.533673\pi\)
\(278\) 0 0
\(279\) −5.65685 + 2.00000i −0.338667 + 0.119737i
\(280\) 0 0
\(281\) 23.3137i 1.39078i −0.718633 0.695390i \(-0.755232\pi\)
0.718633 0.695390i \(-0.244768\pi\)
\(282\) 0 0
\(283\) 17.1716i 1.02074i −0.859954 0.510372i \(-0.829508\pi\)
0.859954 0.510372i \(-0.170492\pi\)
\(284\) 0 0
\(285\) 6.00000 8.48528i 0.355409 0.502625i
\(286\) 0 0
\(287\) 46.6274 2.75233
\(288\) 0 0
\(289\) 16.3137 0.959630
\(290\) 0 0
\(291\) 6.00000 8.48528i 0.351726 0.497416i
\(292\) 0 0
\(293\) 0.343146i 0.0200468i 0.999950 + 0.0100234i \(0.00319060\pi\)
−0.999950 + 0.0100234i \(0.996809\pi\)
\(294\) 0 0
\(295\) 12.8284i 0.746900i
\(296\) 0 0
\(297\) −24.1421 6.82843i −1.40087 0.396226i
\(298\) 0 0
\(299\) −0.970563 −0.0561291
\(300\) 0 0
\(301\) −5.65685 −0.326056
\(302\) 0 0
\(303\) −10.8284 7.65685i −0.622077 0.439875i
\(304\) 0 0
\(305\) 13.3137i 0.762341i
\(306\) 0 0
\(307\) 20.4853i 1.16916i 0.811337 + 0.584578i \(0.198740\pi\)
−0.811337 + 0.584578i \(0.801260\pi\)
\(308\) 0 0
\(309\) 1.17157 + 0.828427i 0.0666485 + 0.0471276i
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) 25.3137 1.43082 0.715408 0.698707i \(-0.246241\pi\)
0.715408 + 0.698707i \(0.246241\pi\)
\(314\) 0 0
\(315\) −4.82843 13.6569i −0.272051 0.769477i
\(316\) 0 0
\(317\) 5.31371i 0.298448i −0.988803 0.149224i \(-0.952323\pi\)
0.988803 0.149224i \(-0.0476775\pi\)
\(318\) 0 0
\(319\) 28.9706i 1.62204i
\(320\) 0 0
\(321\) 5.31371 7.51472i 0.296582 0.419431i
\(322\) 0 0
\(323\) −4.97056 −0.276570
\(324\) 0 0
\(325\) 1.17157 0.0649872
\(326\) 0 0
\(327\) 7.65685 10.8284i 0.423425 0.598813i
\(328\) 0 0
\(329\) 15.3137i 0.844272i
\(330\) 0 0
\(331\) 10.9706i 0.602997i −0.953467 0.301498i \(-0.902513\pi\)
0.953467 0.301498i \(-0.0974868\pi\)
\(332\) 0 0
\(333\) 6.82843 + 19.3137i 0.374196 + 1.05838i
\(334\) 0 0
\(335\) −4.48528 −0.245057
\(336\) 0 0
\(337\) 34.9706 1.90497 0.952484 0.304589i \(-0.0985190\pi\)
0.952484 + 0.304589i \(0.0985190\pi\)
\(338\) 0 0
\(339\) 10.1421 + 7.17157i 0.550845 + 0.389506i
\(340\) 0 0
\(341\) 9.65685i 0.522948i
\(342\) 0 0
\(343\) 44.9706i 2.42818i
\(344\) 0 0
\(345\) 1.17157 + 0.828427i 0.0630754 + 0.0446010i
\(346\) 0 0
\(347\) 13.3137 0.714717 0.357359 0.933967i \(-0.383677\pi\)
0.357359 + 0.933967i \(0.383677\pi\)
\(348\) 0 0
\(349\) −10.9706 −0.587241 −0.293620 0.955922i \(-0.594860\pi\)
−0.293620 + 0.955922i \(0.594860\pi\)
\(350\) 0 0
\(351\) −5.85786 1.65685i −0.312670 0.0884363i
\(352\) 0 0
\(353\) 16.8284i 0.895687i 0.894112 + 0.447843i \(0.147808\pi\)
−0.894112 + 0.447843i \(0.852192\pi\)
\(354\) 0 0
\(355\) 11.3137i 0.600469i
\(356\) 0 0
\(357\) −4.00000 + 5.65685i −0.211702 + 0.299392i
\(358\) 0 0
\(359\) −31.3137 −1.65267 −0.826337 0.563176i \(-0.809579\pi\)
−0.826337 + 0.563176i \(0.809579\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) −12.3137 + 17.4142i −0.646302 + 0.914009i
\(364\) 0 0
\(365\) 2.00000i 0.104685i
\(366\) 0 0
\(367\) 28.1421i 1.46901i 0.678605 + 0.734504i \(0.262585\pi\)
−0.678605 + 0.734504i \(0.737415\pi\)
\(368\) 0 0
\(369\) −27.3137 + 9.65685i −1.42189 + 0.502716i
\(370\) 0 0
\(371\) 36.9706 1.91942
\(372\) 0 0
\(373\) −16.4853 −0.853576 −0.426788 0.904352i \(-0.640355\pi\)
−0.426788 + 0.904352i \(0.640355\pi\)
\(374\) 0 0
\(375\) −1.41421 1.00000i −0.0730297 0.0516398i
\(376\) 0 0
\(377\) 7.02944i 0.362034i
\(378\) 0 0
\(379\) 10.6863i 0.548918i −0.961599 0.274459i \(-0.911501\pi\)
0.961599 0.274459i \(-0.0884987\pi\)
\(380\) 0 0
\(381\) −9.17157 6.48528i −0.469874 0.332251i
\(382\) 0 0
\(383\) −24.1421 −1.23361 −0.616803 0.787118i \(-0.711572\pi\)
−0.616803 + 0.787118i \(0.711572\pi\)
\(384\) 0 0
\(385\) −23.3137 −1.18818
\(386\) 0 0
\(387\) 3.31371 1.17157i 0.168445 0.0595544i
\(388\) 0 0
\(389\) 28.6274i 1.45147i −0.687976 0.725734i \(-0.741500\pi\)
0.687976 0.725734i \(-0.258500\pi\)
\(390\) 0 0
\(391\) 0.686292i 0.0347073i
\(392\) 0 0
\(393\) −2.48528 + 3.51472i −0.125366 + 0.177294i
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 24.4853 1.22888 0.614441 0.788963i \(-0.289382\pi\)
0.614441 + 0.788963i \(0.289382\pi\)
\(398\) 0 0
\(399\) −28.9706 + 40.9706i −1.45034 + 2.05109i
\(400\) 0 0
\(401\) 16.9706i 0.847469i −0.905786 0.423735i \(-0.860719\pi\)
0.905786 0.423735i \(-0.139281\pi\)
\(402\) 0 0
\(403\) 2.34315i 0.116720i
\(404\) 0 0
\(405\) 5.65685 + 7.00000i 0.281091 + 0.347833i
\(406\) 0 0
\(407\) 32.9706 1.63429
\(408\) 0 0
\(409\) −10.6863 −0.528403 −0.264202 0.964467i \(-0.585108\pi\)
−0.264202 + 0.964467i \(0.585108\pi\)
\(410\) 0 0
\(411\) −22.8284 16.1421i −1.12604 0.796233i
\(412\) 0 0
\(413\) 61.9411i 3.04792i
\(414\) 0 0
\(415\) 5.31371i 0.260840i
\(416\) 0 0
\(417\) −30.1421 21.3137i −1.47607 1.04374i
\(418\) 0 0
\(419\) −16.8284 −0.822122 −0.411061 0.911608i \(-0.634842\pi\)
−0.411061 + 0.911608i \(0.634842\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 3.17157 + 8.97056i 0.154207 + 0.436164i
\(424\) 0 0
\(425\) 0.828427i 0.0401846i
\(426\) 0 0
\(427\) 64.2843i 3.11093i
\(428\) 0 0
\(429\) −5.65685 + 8.00000i −0.273115 + 0.386244i
\(430\) 0 0
\(431\) 17.6569 0.850501 0.425250 0.905076i \(-0.360186\pi\)
0.425250 + 0.905076i \(0.360186\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) −6.00000 + 8.48528i −0.287678 + 0.406838i
\(436\) 0 0
\(437\) 4.97056i 0.237774i
\(438\) 0 0
\(439\) 26.2843i 1.25448i 0.778826 + 0.627240i \(0.215815\pi\)
−0.778826 + 0.627240i \(0.784185\pi\)
\(440\) 0 0
\(441\) 16.3137 + 46.1421i 0.776843 + 2.19724i
\(442\) 0 0
\(443\) 19.6569 0.933925 0.466963 0.884277i \(-0.345348\pi\)
0.466963 + 0.884277i \(0.345348\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −17.4558 12.3431i −0.825633 0.583811i
\(448\) 0 0
\(449\) 22.3431i 1.05444i 0.849729 + 0.527219i \(0.176765\pi\)
−0.849729 + 0.527219i \(0.823235\pi\)
\(450\) 0 0
\(451\) 46.6274i 2.19560i
\(452\) 0 0
\(453\) −10.8284 7.65685i −0.508764 0.359750i
\(454\) 0 0
\(455\) −5.65685 −0.265197
\(456\) 0 0
\(457\) −34.9706 −1.63585 −0.817927 0.575322i \(-0.804877\pi\)
−0.817927 + 0.575322i \(0.804877\pi\)
\(458\) 0 0
\(459\) 1.17157 4.14214i 0.0546843 0.193338i
\(460\) 0 0
\(461\) 13.3137i 0.620081i −0.950723 0.310041i \(-0.899657\pi\)
0.950723 0.310041i \(-0.100343\pi\)
\(462\) 0 0
\(463\) 7.17157i 0.333291i −0.986017 0.166646i \(-0.946706\pi\)
0.986017 0.166646i \(-0.0532936\pi\)
\(464\) 0 0
\(465\) 2.00000 2.82843i 0.0927478 0.131165i
\(466\) 0 0
\(467\) −23.6569 −1.09471 −0.547354 0.836901i \(-0.684365\pi\)
−0.547354 + 0.836901i \(0.684365\pi\)
\(468\) 0 0
\(469\) 21.6569 1.00002
\(470\) 0 0
\(471\) 0.485281 0.686292i 0.0223606 0.0316226i
\(472\) 0 0
\(473\) 5.65685i 0.260102i
\(474\) 0 0
\(475\) 6.00000i 0.275299i
\(476\) 0 0
\(477\) −21.6569 + 7.65685i −0.991599 + 0.350583i
\(478\) 0 0
\(479\) 2.34315 0.107061 0.0535305 0.998566i \(-0.482953\pi\)
0.0535305 + 0.998566i \(0.482953\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) −5.65685 4.00000i −0.257396 0.182006i
\(484\) 0 0
\(485\) 6.00000i 0.272446i
\(486\) 0 0
\(487\) 23.4558i 1.06289i −0.847094 0.531443i \(-0.821650\pi\)
0.847094 0.531443i \(-0.178350\pi\)
\(488\) 0 0
\(489\) −0.686292 0.485281i −0.0310352 0.0219452i
\(490\) 0 0
\(491\) −15.1716 −0.684683 −0.342342 0.939576i \(-0.611220\pi\)
−0.342342 + 0.939576i \(0.611220\pi\)
\(492\) 0 0
\(493\) 4.97056 0.223863
\(494\) 0 0
\(495\) 13.6569 4.82843i 0.613830 0.217022i
\(496\) 0 0
\(497\) 54.6274i 2.45037i
\(498\) 0 0
\(499\) 25.3137i 1.13320i −0.823994 0.566599i \(-0.808259\pi\)
0.823994 0.566599i \(-0.191741\pi\)
\(500\) 0 0
\(501\) 24.8284 35.1127i 1.10925 1.56872i
\(502\) 0 0
\(503\) 0.828427 0.0369377 0.0184689 0.999829i \(-0.494121\pi\)
0.0184689 + 0.999829i \(0.494121\pi\)
\(504\) 0 0
\(505\) 7.65685 0.340726
\(506\) 0 0
\(507\) 11.6274 16.4437i 0.516392 0.730288i
\(508\) 0 0
\(509\) 12.3431i 0.547100i −0.961858 0.273550i \(-0.911802\pi\)
0.961858 0.273550i \(-0.0881980\pi\)
\(510\) 0 0
\(511\) 9.65685i 0.427194i
\(512\) 0 0
\(513\) 8.48528 30.0000i 0.374634 1.32453i
\(514\) 0 0
\(515\) −0.828427 −0.0365049
\(516\) 0 0
\(517\) 15.3137 0.673496
\(518\) 0 0
\(519\) −15.5147 10.9706i −0.681021 0.481554i
\(520\) 0 0
\(521\) 12.0000i 0.525730i 0.964833 + 0.262865i \(0.0846673\pi\)
−0.964833 + 0.262865i \(0.915333\pi\)
\(522\) 0 0
\(523\) 4.48528i 0.196128i −0.995180 0.0980638i \(-0.968735\pi\)
0.995180 0.0980638i \(-0.0312649\pi\)
\(524\) 0 0
\(525\) 6.82843 + 4.82843i 0.298017 + 0.210730i
\(526\) 0 0
\(527\) −1.65685 −0.0721737
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) 12.8284 + 36.2843i 0.556706 + 1.57460i
\(532\) 0 0
\(533\) 11.3137i 0.490051i
\(534\) 0 0
\(535\) 5.31371i 0.229732i
\(536\) 0 0
\(537\) 3.17157 4.48528i 0.136863 0.193554i
\(538\) 0 0
\(539\) 78.7696 3.39284
\(540\) 0 0
\(541\) −36.6274 −1.57474 −0.787368 0.616483i \(-0.788557\pi\)
−0.787368 + 0.616483i \(0.788557\pi\)
\(542\) 0 0
\(543\) −2.00000 + 2.82843i −0.0858282 + 0.121379i
\(544\) 0 0
\(545\) 7.65685i 0.327984i
\(546\) 0 0
\(547\) 21.4558i 0.917386i −0.888595 0.458693i \(-0.848318\pi\)
0.888595 0.458693i \(-0.151682\pi\)
\(548\) 0 0
\(549\) −13.3137 37.6569i −0.568215 1.60716i
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) 28.9706 1.23195
\(554\) 0 0
\(555\) −9.65685 6.82843i −0.409911 0.289851i
\(556\) 0 0
\(557\) 1.02944i 0.0436187i 0.999762 + 0.0218093i \(0.00694268\pi\)
−0.999762 + 0.0218093i \(0.993057\pi\)
\(558\) 0 0
\(559\) 1.37258i 0.0580541i
\(560\) 0 0
\(561\) −5.65685 4.00000i −0.238833 0.168880i
\(562\) 0 0
\(563\) 17.3137 0.729686 0.364843 0.931069i \(-0.381123\pi\)
0.364843 + 0.931069i \(0.381123\pi\)
\(564\) 0 0
\(565\) −7.17157 −0.301710
\(566\) 0 0
\(567\) −27.3137 33.7990i −1.14707 1.41942i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 34.9706i 1.46347i −0.681588 0.731736i \(-0.738710\pi\)
0.681588 0.731736i \(-0.261290\pi\)
\(572\) 0 0
\(573\) 12.0000 16.9706i 0.501307 0.708955i
\(574\) 0 0
\(575\) −0.828427 −0.0345478
\(576\) 0 0
\(577\) −14.9706 −0.623233 −0.311616 0.950208i \(-0.600870\pi\)
−0.311616 + 0.950208i \(0.600870\pi\)
\(578\) 0 0
\(579\) −18.9706 + 26.8284i −0.788390 + 1.11495i
\(580\) 0 0
\(581\) 25.6569i 1.06443i
\(582\) 0 0
\(583\) 36.9706i 1.53116i
\(584\) 0 0
\(585\) 3.31371 1.17157i 0.137005 0.0484386i
\(586\) 0 0
\(587\) −16.6274 −0.686287 −0.343143 0.939283i \(-0.611492\pi\)
−0.343143 + 0.939283i \(0.611492\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −18.8284 13.3137i −0.774498 0.547653i
\(592\) 0 0
\(593\) 36.1421i 1.48418i 0.670300 + 0.742090i \(0.266165\pi\)
−0.670300 + 0.742090i \(0.733835\pi\)
\(594\) 0 0
\(595\) 4.00000i 0.163984i
\(596\) 0 0
\(597\) 10.8284 + 7.65685i 0.443178 + 0.313374i
\(598\) 0 0
\(599\) 18.6274 0.761096 0.380548 0.924761i \(-0.375735\pi\)
0.380548 + 0.924761i \(0.375735\pi\)
\(600\) 0 0
\(601\) −36.6274 −1.49406 −0.747032 0.664788i \(-0.768522\pi\)
−0.747032 + 0.664788i \(0.768522\pi\)
\(602\) 0 0
\(603\) −12.6863 + 4.48528i −0.516626 + 0.182655i
\(604\) 0 0
\(605\) 12.3137i 0.500623i
\(606\) 0 0
\(607\) 21.5147i 0.873255i 0.899642 + 0.436628i \(0.143827\pi\)
−0.899642 + 0.436628i \(0.856173\pi\)
\(608\) 0 0
\(609\) 28.9706 40.9706i 1.17395 1.66021i
\(610\) 0 0
\(611\) 3.71573 0.150322
\(612\) 0 0
\(613\) −25.4558 −1.02815 −0.514076 0.857745i \(-0.671865\pi\)
−0.514076 + 0.857745i \(0.671865\pi\)
\(614\) 0 0
\(615\) 9.65685 13.6569i 0.389402 0.550698i
\(616\) 0 0
\(617\) 3.17157i 0.127683i −0.997960 0.0638414i \(-0.979665\pi\)
0.997960 0.0638414i \(-0.0203352\pi\)
\(618\) 0 0
\(619\) 6.00000i 0.241160i −0.992704 0.120580i \(-0.961525\pi\)
0.992704 0.120580i \(-0.0384755\pi\)
\(620\) 0 0
\(621\) 4.14214 + 1.17157i 0.166218 + 0.0470136i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −40.9706 28.9706i −1.63621 1.15697i
\(628\) 0 0
\(629\) 5.65685i 0.225554i
\(630\) 0 0
\(631\) 33.3137i 1.32620i 0.748532 + 0.663099i \(0.230759\pi\)
−0.748532 + 0.663099i \(0.769241\pi\)
\(632\) 0 0
\(633\) −9.85786 6.97056i −0.391815 0.277055i
\(634\) 0 0
\(635\) 6.48528 0.257361
\(636\) 0 0
\(637\) 19.1127 0.757273
\(638\) 0 0
\(639\) 11.3137 + 32.0000i 0.447563 + 1.26590i
\(640\) 0 0
\(641\) 14.6274i 0.577748i −0.957367 0.288874i \(-0.906719\pi\)
0.957367 0.288874i \(-0.0932809\pi\)
\(642\) 0 0
\(643\) 5.17157i 0.203947i 0.994787 + 0.101973i \(0.0325157\pi\)
−0.994787 + 0.101973i \(0.967484\pi\)
\(644\) 0 0
\(645\) −1.17157 + 1.65685i −0.0461306 + 0.0652386i
\(646\) 0 0
\(647\) 15.1716 0.596456 0.298228 0.954495i \(-0.403604\pi\)
0.298228 + 0.954495i \(0.403604\pi\)
\(648\) 0 0
\(649\) 61.9411 2.43140
\(650\) 0 0
\(651\) −9.65685 + 13.6569i −0.378482 + 0.535254i
\(652\) 0 0
\(653\) 30.2843i 1.18512i 0.805528 + 0.592558i \(0.201882\pi\)
−0.805528 + 0.592558i \(0.798118\pi\)
\(654\) 0 0
\(655\) 2.48528i 0.0971080i
\(656\) 0 0
\(657\) 2.00000 + 5.65685i 0.0780274 + 0.220695i
\(658\) 0 0
\(659\) 6.20101 0.241557 0.120779 0.992679i \(-0.461461\pi\)
0.120779 + 0.992679i \(0.461461\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) −1.37258 0.970563i −0.0533067 0.0376935i
\(664\) 0 0
\(665\) 28.9706i 1.12343i
\(666\) 0 0
\(667\) 4.97056i 0.192461i
\(668\) 0 0
\(669\) −9.17157 6.48528i −0.354593 0.250735i
\(670\) 0 0
\(671\) −64.2843 −2.48167
\(672\) 0 0
\(673\) −31.9411 −1.23124 −0.615620 0.788043i \(-0.711094\pi\)
−0.615620 + 0.788043i \(0.711094\pi\)
\(674\) 0 0
\(675\) −5.00000 1.41421i −0.192450 0.0544331i
\(676\) 0 0
\(677\) 11.6569i 0.448009i 0.974588 + 0.224005i \(0.0719131\pi\)
−0.974588 + 0.224005i \(0.928087\pi\)
\(678\) 0 0
\(679\) 28.9706i 1.11179i
\(680\) 0 0
\(681\) 11.6569 16.4853i 0.446692 0.631717i
\(682\) 0 0
\(683\) 31.6569 1.21132 0.605658 0.795725i \(-0.292910\pi\)
0.605658 + 0.795725i \(0.292910\pi\)
\(684\) 0 0
\(685\) 16.1421 0.616759
\(686\) 0 0
\(687\) 18.9706 26.8284i 0.723772 1.02357i
\(688\) 0 0
\(689\) 8.97056i 0.341751i
\(690\) 0 0
\(691\) 42.9706i 1.63468i −0.576158 0.817339i \(-0.695449\pi\)
0.576158 0.817339i \(-0.304551\pi\)
\(692\) 0 0
\(693\) −65.9411 + 23.3137i −2.50490 + 0.885615i
\(694\) 0 0
\(695\) 21.3137 0.808475
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) 29.4558 + 20.8284i 1.11412 + 0.787803i
\(700\) 0 0
\(701\) 39.9411i 1.50856i 0.656555 + 0.754278i \(0.272013\pi\)
−0.656555 + 0.754278i \(0.727987\pi\)
\(702\) 0 0
\(703\) 40.9706i 1.54523i
\(704\) 0 0
\(705\) −4.48528 3.17157i −0.168925 0.119448i
\(706\) 0 0
\(707\) −36.9706 −1.39042
\(708\) 0 0
\(709\) −9.02944 −0.339108 −0.169554 0.985521i \(-0.554233\pi\)
−0.169554 + 0.985521i \(0.554233\pi\)
\(710\) 0 0
\(711\) −16.9706 + 6.00000i −0.636446 + 0.225018i
\(712\) 0 0
\(713\) 1.65685i 0.0620497i
\(714\) 0 0
\(715\) 5.65685i 0.211554i
\(716\) 0 0
\(717\) −16.9706 + 24.0000i −0.633777 + 0.896296i
\(718\) 0 0
\(719\) 20.6863 0.771468 0.385734 0.922610i \(-0.373948\pi\)
0.385734 + 0.922610i \(0.373948\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) −6.00000 + 8.48528i −0.223142 + 0.315571i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 24.8284i 0.920835i −0.887702 0.460418i \(-0.847700\pi\)
0.887702 0.460418i \(-0.152300\pi\)
\(728\) 0 0
\(729\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(730\) 0 0
\(731\) 0.970563 0.0358976
\(732\) 0 0
\(733\) 25.1716 0.929733 0.464867 0.885381i \(-0.346102\pi\)
0.464867 + 0.885381i \(0.346102\pi\)
\(734\) 0 0
\(735\) −23.0711 16.3137i −0.850989 0.601740i
\(736\) 0 0
\(737\) 21.6569i 0.797740i
\(738\) 0 0
\(739\) 1.31371i 0.0483255i −0.999708 0.0241628i \(-0.992308\pi\)
0.999708 0.0241628i \(-0.00769200\pi\)
\(740\) 0 0
\(741\) −9.94113 7.02944i −0.365196 0.258233i
\(742\) 0 0
\(743\) 31.4558 1.15400 0.577001 0.816743i \(-0.304223\pi\)
0.577001 + 0.816743i \(0.304223\pi\)
\(744\) 0 0
\(745\) 12.3431 0.452218
\(746\) 0 0
\(747\) −5.31371 15.0294i −0.194418 0.549898i
\(748\) 0 0
\(749\) 25.6569i 0.937481i
\(750\) 0 0
\(751\) 2.68629i 0.0980242i 0.998798 + 0.0490121i \(0.0156073\pi\)
−0.998798 + 0.0490121i \(0.984393\pi\)
\(752\) 0 0
\(753\) −17.7990 + 25.1716i −0.648631 + 0.917303i
\(754\) 0 0
\(755\) 7.65685 0.278661
\(756\) 0 0
\(757\) −18.1421 −0.659387 −0.329694 0.944088i \(-0.606945\pi\)
−0.329694 + 0.944088i \(0.606945\pi\)
\(758\) 0 0
\(759\) 4.00000 5.65685i 0.145191 0.205331i
\(760\) 0 0
\(761\) 8.68629i 0.314878i −0.987529 0.157439i \(-0.949676\pi\)
0.987529 0.157439i \(-0.0503237\pi\)
\(762\) 0 0
\(763\) 36.9706i 1.33842i
\(764\) 0 0
\(765\) 0.828427 + 2.34315i 0.0299518 + 0.0847166i
\(766\) 0 0
\(767\) 15.0294 0.542682
\(768\) 0 0
\(769\) 33.3137 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) 0 0
\(771\) −37.4558 26.4853i −1.34894 0.953844i
\(772\) 0 0
\(773\) 6.68629i 0.240489i 0.992744 + 0.120245i \(0.0383679\pi\)
−0.992744 + 0.120245i \(0.961632\pi\)
\(774\) 0 0
\(775\) 2.00000i 0.0718421i
\(776\) 0 0
\(777\) 46.6274 + 32.9706i 1.67275 + 1.18281i
\(778\) 0 0
\(779\) −57.9411 −2.07596
\(780\) 0 0
\(781\) 54.6274 1.95472
\(782\) 0 0
\(783\) −8.48528 + 30.0000i −0.303239 + 1.07211i
\(784\) 0 0
\(785\) 0.485281i 0.0173204i
\(786\) 0 0
\(787\) 0.201010i 0.00716524i 0.999994 + 0.00358262i \(0.00114039\pi\)
−0.999994 + 0.00358262i \(0.998860\pi\)
\(788\) 0 0
\(789\) 2.48528 3.51472i 0.0884784 0.125127i
\(790\) 0 0
\(791\) 34.6274 1.23121
\(792\) 0 0
\(793\) −15.5980 −0.553901
\(794\) 0 0
\(795\) 7.65685 10.8284i 0.271561 0.384045i
\(796\) 0 0
\(797\) 3.37258i 0.119463i −0.998214 0.0597315i \(-0.980976\pi\)
0.998214 0.0597315i \(-0.0190245\pi\)
\(798\) 0 0
\(799\) 2.62742i 0.0929513i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.65685 0.340783
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) −5.17157 3.65685i −0.182048 0.128727i
\(808\) 0 0
\(809\) 5.65685i 0.198884i 0.995043 + 0.0994422i \(0.0317058\pi\)
−0.995043 + 0.0994422i \(0.968294\pi\)
\(810\) 0 0
\(811\) 38.2843i 1.34434i −0.740396 0.672171i \(-0.765362\pi\)
0.740396 0.672171i \(-0.234638\pi\)
\(812\) 0 0
\(813\) 21.1716 + 14.9706i 0.742519 + 0.525041i
\(814\) 0 0
\(815\) 0.485281 0.0169987
\(816\) 0 0
\(817\) 7.02944 0.245929
\(818\) 0 0
\(819\) −16.0000 + 5.65685i −0.559085 + 0.197666i
\(820\) 0 0
\(821\) 35.6569i 1.24443i −0.782845 0.622216i \(-0.786232\pi\)
0.782845 0.622216i \(-0.213768\pi\)
\(822\) 0 0
\(823\) 39.4558i 1.37534i −0.726021 0.687672i \(-0.758633\pi\)
0.726021 0.687672i \(-0.241367\pi\)
\(824\) 0 0
\(825\) −4.82843 + 6.82843i −0.168104 + 0.237735i
\(826\) 0 0
\(827\) 37.3137 1.29752 0.648762 0.760991i \(-0.275287\pi\)
0.648762 + 0.760991i \(0.275287\pi\)
\(828\) 0 0
\(829\) 56.9117 1.97662 0.988312 0.152443i \(-0.0487139\pi\)
0.988312 + 0.152443i \(0.0487139\pi\)
\(830\) 0 0
\(831\) 3.51472 4.97056i 0.121924 0.172427i
\(832\) 0 0
\(833\) 13.5147i 0.468257i
\(834\) 0 0
\(835\) 24.8284i 0.859223i
\(836\) 0 0
\(837\) 2.82843 10.0000i 0.0977647 0.345651i
\(838\) 0 0
\(839\) 21.9411 0.757492 0.378746 0.925501i \(-0.376355\pi\)
0.378746 + 0.925501i \(0.376355\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 32.9706 + 23.3137i 1.13557 + 0.802967i
\(844\) 0 0
\(845\) 11.6274i 0.399995i
\(846\) 0 0
\(847\) 59.4558i 2.04293i
\(848\) 0 0
\(849\) 24.2843 + 17.1716i 0.833434 + 0.589327i
\(850\) 0 0
\(851\) −5.65685 −0.193914
\(852\) 0 0
\(853\) 48.4853 1.66010 0.830052 0.557686i \(-0.188311\pi\)
0.830052 + 0.557686i \(0.188311\pi\)
\(854\) 0 0
\(855\) 6.00000 + 16.9706i 0.205196 + 0.580381i
\(856\) 0 0
\(857\) 9.79899i 0.334727i −0.985895 0.167364i \(-0.946475\pi\)
0.985895 0.167364i \(-0.0535254\pi\)
\(858\) 0 0
\(859\) 22.0000i 0.750630i −0.926897 0.375315i \(-0.877534\pi\)
0.926897 0.375315i \(-0.122466\pi\)
\(860\) 0 0
\(861\) −46.6274 + 65.9411i −1.58906 + 2.24727i
\(862\) 0 0
\(863\) 57.7990 1.96750 0.983750 0.179542i \(-0.0574617\pi\)
0.983750 + 0.179542i \(0.0574617\pi\)
\(864\) 0 0
\(865\) 10.9706 0.373010
\(866\) 0 0
\(867\) −16.3137 + 23.0711i −0.554043 + 0.783535i
\(868\) 0 0
\(869\) 28.9706i 0.982759i
\(870\) 0 0
\(871\) 5.25483i 0.178053i
\(872\) 0 0
\(873\) 6.00000 + 16.9706i 0.203069 + 0.574367i
\(874\) 0 0
\(875\) −4.82843 −0.163231
\(876\) 0 0
\(877\) −27.7990 −0.938705 −0.469353 0.883011i \(-0.655513\pi\)
−0.469353 + 0.883011i \(0.655513\pi\)
\(878\) 0 0
\(879\) −0.485281 0.343146i −0.0163681 0.0115740i
\(880\) 0 0
\(881\) 39.5980i 1.33409i 0.745018 + 0.667045i \(0.232441\pi\)
−0.745018 + 0.667045i \(0.767559\pi\)
\(882\) 0 0
\(883\) 15.5147i 0.522112i −0.965324 0.261056i \(-0.915929\pi\)
0.965324 0.261056i \(-0.0840707\pi\)
\(884\) 0 0
\(885\) −18.1421 12.8284i −0.609841 0.431223i
\(886\) 0 0
\(887\) −40.8284 −1.37088 −0.685442 0.728127i \(-0.740391\pi\)
−0.685442 + 0.728127i \(0.740391\pi\)
\(888\) 0 0
\(889\) −31.3137 −1.05023
\(890\) 0 0
\(891\) 33.7990 27.3137i 1.13231 0.915044i
\(892\) 0 0
\(893\) 19.0294i 0.636796i
\(894\) 0 0
\(895\) 3.17157i 0.106014i
\(896\) 0 0
\(897\) 0.970563 1.37258i 0.0324061 0.0458292i
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −6.34315 −0.211321
\(902\) 0 0
\(903\) 5.65685 8.00000i 0.188248 0.266223i
\(904\) 0 0
\(905\) 2.00000i 0.0664822i
\(906\) 0 0
\(907\) 10.8284i 0.359552i 0.983708 + 0.179776i \(0.0575373\pi\)
−0.983708 + 0.179776i \(0.942463\pi\)
\(908\) 0 0
\(909\) 21.6569 7.65685i 0.718313 0.253962i
\(910\) 0 0
\(911\) −51.5980 −1.70952 −0.854759 0.519026i \(-0.826295\pi\)
−0.854759 + 0.519026i \(0.826295\pi\)
\(912\) 0 0
\(913\) −25.6569 −0.849118
\(914\) 0 0
\(915\) 18.8284 + 13.3137i 0.622449 + 0.440138i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) 25.3137i 0.835022i −0.908672 0.417511i \(-0.862902\pi\)
0.908672 0.417511i \(-0.137098\pi\)
\(920\) 0 0
\(921\) −28.9706 20.4853i −0.954612 0.675013i
\(922\) 0 0
\(923\) 13.2548 0.436288
\(924\) 0 0
\(925\) 6.82843 0.224517
\(926\) 0 0
\(927\) −2.34315 + 0.828427i −0.0769590 + 0.0272091i
\(928\) 0 0
\(929\) 30.3431i 0.995526i −0.867313 0.497763i \(-0.834155\pi\)
0.867313 0.497763i \(-0.165845\pi\)
\(930\) 0 0
\(931\) 97.8823i 3.20796i
\(932\) 0 0
\(933\) 11.3137 16.0000i 0.370394 0.523816i
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 6.97056 0.227718 0.113859 0.993497i \(-0.463679\pi\)
0.113859 + 0.993497i \(0.463679\pi\)
\(938\) 0 0
\(939\) −25.3137 + 35.7990i −0.826082 + 1.16826i
\(940\) 0 0
\(941\) 41.5980i 1.35606i −0.735036 0.678028i \(-0.762835\pi\)
0.735036 0.678028i \(-0.237165\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) 24.1421 + 6.82843i 0.785344 + 0.222129i
\(946\) 0 0
\(947\) 46.2843 1.50404 0.752018 0.659142i \(-0.229080\pi\)
0.752018 + 0.659142i \(0.229080\pi\)
\(948\) 0 0
\(949\) 2.34315 0.0760617
\(950\) 0 0
\(951\) 7.51472 + 5.31371i 0.243681 + 0.172309i
\(952\) 0 0
\(953\) 1.79899i 0.0582750i 0.999575 + 0.0291375i \(0.00927607\pi\)
−0.999575 + 0.0291375i \(0.990724\pi\)
\(954\) 0 0
\(955\) 12.0000i 0.388311i
\(956\) 0 0
\(957\) 40.9706 + 28.9706i 1.32439 + 0.936485i
\(958\) 0 0
\(959\) −77.9411 −2.51685
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 5.31371 + 15.0294i 0.171232 + 0.484317i
\(964\) 0 0
\(965\) 18.9706i 0.610684i
\(966\) 0 0
\(967\) 18.4853i 0.594447i 0.954808 + 0.297223i \(0.0960606\pi\)
−0.954808 + 0.297223i \(0.903939\pi\)
\(968\) 0 0
\(969\) 4.97056 7.02944i 0.159677 0.225818i
\(970\) 0 0
\(971\) −22.7696 −0.730710 −0.365355 0.930868i \(-0.619052\pi\)
−0.365355 + 0.930868i \(0.619052\pi\)
\(972\) 0 0
\(973\) −102.912 −3.29920
\(974\) 0 0
\(975\) −1.17157 + 1.65685i −0.0375204 + 0.0530618i
\(976\) 0 0
\(977\) 29.7990i 0.953354i 0.879078 + 0.476677i \(0.158159\pi\)
−0.879078 + 0.476677i \(0.841841\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.65685 + 21.6569i 0.244465 + 0.691450i
\(982\) 0 0
\(983\) −15.1716 −0.483898 −0.241949 0.970289i \(-0.577787\pi\)
−0.241949 + 0.970289i \(0.577787\pi\)
\(984\) 0 0
\(985\) 13.3137 0.424210
\(986\) 0 0
\(987\) 21.6569 + 15.3137i 0.689345 + 0.487441i
\(988\) 0 0
\(989\) 0.970563i 0.0308621i
\(990\) 0 0
\(991\) 42.2843i 1.34320i −0.740912 0.671602i \(-0.765606\pi\)
0.740912 0.671602i \(-0.234394\pi\)
\(992\) 0 0
\(993\) 15.5147 + 10.9706i 0.492345 + 0.348140i
\(994\) 0 0
\(995\) −7.65685 −0.242739
\(996\) 0 0
\(997\) 27.5147 0.871400 0.435700 0.900092i \(-0.356501\pi\)
0.435700 + 0.900092i \(0.356501\pi\)
\(998\) 0 0
\(999\) −34.1421 9.65685i −1.08021 0.305529i
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 960.2.h.a.191.3 4
3.2 odd 2 960.2.h.e.191.4 4
4.3 odd 2 960.2.h.e.191.1 4
8.3 odd 2 480.2.h.a.191.4 yes 4
8.5 even 2 480.2.h.c.191.2 yes 4
12.11 even 2 inner 960.2.h.a.191.2 4
24.5 odd 2 480.2.h.a.191.1 4
24.11 even 2 480.2.h.c.191.3 yes 4
40.3 even 4 2400.2.o.d.2399.1 4
40.13 odd 4 2400.2.o.g.2399.4 4
40.19 odd 2 2400.2.h.d.1151.2 4
40.27 even 4 2400.2.o.f.2399.4 4
40.29 even 2 2400.2.h.a.1151.3 4
40.37 odd 4 2400.2.o.e.2399.1 4
120.29 odd 2 2400.2.h.d.1151.3 4
120.53 even 4 2400.2.o.f.2399.3 4
120.59 even 2 2400.2.h.a.1151.2 4
120.77 even 4 2400.2.o.d.2399.2 4
120.83 odd 4 2400.2.o.e.2399.2 4
120.107 odd 4 2400.2.o.g.2399.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.h.a.191.1 4 24.5 odd 2
480.2.h.a.191.4 yes 4 8.3 odd 2
480.2.h.c.191.2 yes 4 8.5 even 2
480.2.h.c.191.3 yes 4 24.11 even 2
960.2.h.a.191.2 4 12.11 even 2 inner
960.2.h.a.191.3 4 1.1 even 1 trivial
960.2.h.e.191.1 4 4.3 odd 2
960.2.h.e.191.4 4 3.2 odd 2
2400.2.h.a.1151.2 4 120.59 even 2
2400.2.h.a.1151.3 4 40.29 even 2
2400.2.h.d.1151.2 4 40.19 odd 2
2400.2.h.d.1151.3 4 120.29 odd 2
2400.2.o.d.2399.1 4 40.3 even 4
2400.2.o.d.2399.2 4 120.77 even 4
2400.2.o.e.2399.1 4 40.37 odd 4
2400.2.o.e.2399.2 4 120.83 odd 4
2400.2.o.f.2399.3 4 120.53 even 4
2400.2.o.f.2399.4 4 40.27 even 4
2400.2.o.g.2399.3 4 120.107 odd 4
2400.2.o.g.2399.4 4 40.13 odd 4