Properties

Label 960.2.b.a.671.4
Level $960$
Weight $2$
Character 960.671
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,2,Mod(671,960)]
 
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, 1, 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("960.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 671.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 960.671
Dual form 960.2.b.a.671.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 + 1.00000i) q^{3} -1.00000 q^{5} +2.00000i q^{7} +(1.00000 + 2.82843i) q^{9} +O(q^{10})\) \(q+(1.41421 + 1.00000i) q^{3} -1.00000 q^{5} +2.00000i q^{7} +(1.00000 + 2.82843i) q^{9} -2.00000i q^{11} +2.82843i q^{13} +(-1.41421 - 1.00000i) q^{15} +2.82843i q^{17} +5.65685 q^{19} +(-2.00000 + 2.82843i) q^{21} -8.48528 q^{23} +1.00000 q^{25} +(-1.41421 + 5.00000i) q^{27} +2.00000 q^{29} +6.00000i q^{31} +(2.00000 - 2.82843i) q^{33} -2.00000i q^{35} -2.82843i q^{37} +(-2.82843 + 4.00000i) q^{39} +11.3137i q^{41} -8.48528 q^{43} +(-1.00000 - 2.82843i) q^{45} -2.82843 q^{47} +3.00000 q^{49} +(-2.82843 + 4.00000i) q^{51} +10.0000 q^{53} +2.00000i q^{55} +(8.00000 + 5.65685i) q^{57} -10.0000i q^{59} +(-5.65685 + 2.00000i) q^{63} -2.82843i q^{65} -2.82843 q^{67} +(-12.0000 - 8.48528i) q^{69} +5.65685 q^{71} +10.0000 q^{73} +(1.41421 + 1.00000i) q^{75} +4.00000 q^{77} -2.00000i q^{79} +(-7.00000 + 5.65685i) q^{81} +2.00000i q^{83} -2.82843i q^{85} +(2.82843 + 2.00000i) q^{87} +5.65685i q^{89} -5.65685 q^{91} +(-6.00000 + 8.48528i) q^{93} -5.65685 q^{95} -14.0000 q^{97} +(5.65685 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{9} - 8 q^{21} + 4 q^{25} + 8 q^{29} + 8 q^{33} - 4 q^{45} + 12 q^{49} + 40 q^{53} + 32 q^{57} - 48 q^{69} + 40 q^{73} + 16 q^{77} - 28 q^{81} - 24 q^{93} - 56 q^{97}+O(q^{100}) \) 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
\(3\) 1.41421 + 1.00000i 0.816497 + 0.577350i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 1.00000 + 2.82843i 0.333333 + 0.942809i
\(10\) 0 0
\(11\) 2.00000i 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) −1.41421 1.00000i −0.365148 0.258199i
\(16\) 0 0
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) −2.00000 + 2.82843i −0.436436 + 0.617213i
\(22\) 0 0
\(23\) −8.48528 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.41421 + 5.00000i −0.272166 + 0.962250i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 2.00000 2.82843i 0.348155 0.492366i
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) 2.82843i 0.464991i −0.972598 0.232495i \(-0.925311\pi\)
0.972598 0.232495i \(-0.0746890\pi\)
\(38\) 0 0
\(39\) −2.82843 + 4.00000i −0.452911 + 0.640513i
\(40\) 0 0
\(41\) 11.3137i 1.76690i 0.468521 + 0.883452i \(0.344787\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) −1.00000 2.82843i −0.149071 0.421637i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.82843 + 4.00000i −0.396059 + 0.560112i
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 2.00000i 0.269680i
\(56\) 0 0
\(57\) 8.00000 + 5.65685i 1.05963 + 0.749269i
\(58\) 0 0
\(59\) 10.0000i 1.30189i −0.759125 0.650945i \(-0.774373\pi\)
0.759125 0.650945i \(-0.225627\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −5.65685 + 2.00000i −0.712697 + 0.251976i
\(64\) 0 0
\(65\) 2.82843i 0.350823i
\(66\) 0 0
\(67\) −2.82843 −0.345547 −0.172774 0.984962i \(-0.555273\pi\)
−0.172774 + 0.984962i \(0.555273\pi\)
\(68\) 0 0
\(69\) −12.0000 8.48528i −1.44463 1.02151i
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 1.41421 + 1.00000i 0.163299 + 0.115470i
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 2.00000i 0.225018i −0.993651 0.112509i \(-0.964111\pi\)
0.993651 0.112509i \(-0.0358886\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 2.82843i 0.306786i
\(86\) 0 0
\(87\) 2.82843 + 2.00000i 0.303239 + 0.214423i
\(88\) 0 0
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) 0 0
\(93\) −6.00000 + 8.48528i −0.622171 + 0.879883i
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 5.65685 2.00000i 0.568535 0.201008i
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 2.00000 2.82843i 0.195180 0.276026i
\(106\) 0 0
\(107\) 14.0000i 1.35343i −0.736245 0.676716i \(-0.763403\pi\)
0.736245 0.676716i \(-0.236597\pi\)
\(108\) 0 0
\(109\) 16.9706i 1.62549i −0.582623 0.812743i \(-0.697974\pi\)
0.582623 0.812743i \(-0.302026\pi\)
\(110\) 0 0
\(111\) 2.82843 4.00000i 0.268462 0.379663i
\(112\) 0 0
\(113\) 14.1421i 1.33038i −0.746674 0.665190i \(-0.768350\pi\)
0.746674 0.665190i \(-0.231650\pi\)
\(114\) 0 0
\(115\) 8.48528 0.791257
\(116\) 0 0
\(117\) −8.00000 + 2.82843i −0.739600 + 0.261488i
\(118\) 0 0
\(119\) −5.65685 −0.518563
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) −11.3137 + 16.0000i −1.02012 + 1.44267i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) −12.0000 8.48528i −1.05654 0.747087i
\(130\) 0 0
\(131\) 6.00000i 0.524222i 0.965038 + 0.262111i \(0.0844187\pi\)
−0.965038 + 0.262111i \(0.915581\pi\)
\(132\) 0 0
\(133\) 11.3137i 0.981023i
\(134\) 0 0
\(135\) 1.41421 5.00000i 0.121716 0.430331i
\(136\) 0 0
\(137\) 2.82843i 0.241649i 0.992674 + 0.120824i \(0.0385538\pi\)
−0.992674 + 0.120824i \(0.961446\pi\)
\(138\) 0 0
\(139\) 16.9706 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(140\) 0 0
\(141\) −4.00000 2.82843i −0.336861 0.238197i
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 4.24264 + 3.00000i 0.349927 + 0.247436i
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 2.00000i 0.162758i −0.996683 0.0813788i \(-0.974068\pi\)
0.996683 0.0813788i \(-0.0259324\pi\)
\(152\) 0 0
\(153\) −8.00000 + 2.82843i −0.646762 + 0.228665i
\(154\) 0 0
\(155\) 6.00000i 0.481932i
\(156\) 0 0
\(157\) 19.7990i 1.58013i 0.613022 + 0.790066i \(0.289954\pi\)
−0.613022 + 0.790066i \(0.710046\pi\)
\(158\) 0 0
\(159\) 14.1421 + 10.0000i 1.12154 + 0.793052i
\(160\) 0 0
\(161\) 16.9706i 1.33747i
\(162\) 0 0
\(163\) −8.48528 −0.664619 −0.332309 0.943170i \(-0.607828\pi\)
−0.332309 + 0.943170i \(0.607828\pi\)
\(164\) 0 0
\(165\) −2.00000 + 2.82843i −0.155700 + 0.220193i
\(166\) 0 0
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 5.65685 + 16.0000i 0.432590 + 1.22355i
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 10.0000 14.1421i 0.751646 1.06299i
\(178\) 0 0
\(179\) 18.0000i 1.34538i −0.739923 0.672692i \(-0.765138\pi\)
0.739923 0.672692i \(-0.234862\pi\)
\(180\) 0 0
\(181\) 22.6274i 1.68188i 0.541126 + 0.840941i \(0.317998\pi\)
−0.541126 + 0.840941i \(0.682002\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.82843i 0.207950i
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) −10.0000 2.82843i −0.727393 0.205738i
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 2.82843 4.00000i 0.202548 0.286446i
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 26.0000i 1.84309i −0.388270 0.921546i \(-0.626927\pi\)
0.388270 0.921546i \(-0.373073\pi\)
\(200\) 0 0
\(201\) −4.00000 2.82843i −0.282138 0.199502i
\(202\) 0 0
\(203\) 4.00000i 0.280745i
\(204\) 0 0
\(205\) 11.3137i 0.790184i
\(206\) 0 0
\(207\) −8.48528 24.0000i −0.589768 1.66812i
\(208\) 0 0
\(209\) 11.3137i 0.782586i
\(210\) 0 0
\(211\) −11.3137 −0.778868 −0.389434 0.921054i \(-0.627329\pi\)
−0.389434 + 0.921054i \(0.627329\pi\)
\(212\) 0 0
\(213\) 8.00000 + 5.65685i 0.548151 + 0.387601i
\(214\) 0 0
\(215\) 8.48528 0.578691
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 14.1421 + 10.0000i 0.955637 + 0.675737i
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 14.0000i 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 1.00000 + 2.82843i 0.0666667 + 0.188562i
\(226\) 0 0
\(227\) 10.0000i 0.663723i 0.943328 + 0.331862i \(0.107677\pi\)
−0.943328 + 0.331862i \(0.892323\pi\)
\(228\) 0 0
\(229\) 28.2843i 1.86908i −0.355862 0.934539i \(-0.615813\pi\)
0.355862 0.934539i \(-0.384187\pi\)
\(230\) 0 0
\(231\) 5.65685 + 4.00000i 0.372194 + 0.263181i
\(232\) 0 0
\(233\) 8.48528i 0.555889i 0.960597 + 0.277945i \(0.0896532\pi\)
−0.960597 + 0.277945i \(0.910347\pi\)
\(234\) 0 0
\(235\) 2.82843 0.184506
\(236\) 0 0
\(237\) 2.00000 2.82843i 0.129914 0.183726i
\(238\) 0 0
\(239\) 22.6274 1.46365 0.731823 0.681495i \(-0.238670\pi\)
0.731823 + 0.681495i \(0.238670\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) −15.5563 + 1.00000i −0.997940 + 0.0641500i
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) −2.00000 + 2.82843i −0.126745 + 0.179244i
\(250\) 0 0
\(251\) 18.0000i 1.13615i −0.822977 0.568075i \(-0.807688\pi\)
0.822977 0.568075i \(-0.192312\pi\)
\(252\) 0 0
\(253\) 16.9706i 1.06693i
\(254\) 0 0
\(255\) 2.82843 4.00000i 0.177123 0.250490i
\(256\) 0 0
\(257\) 19.7990i 1.23503i −0.786560 0.617514i \(-0.788140\pi\)
0.786560 0.617514i \(-0.211860\pi\)
\(258\) 0 0
\(259\) 5.65685 0.351500
\(260\) 0 0
\(261\) 2.00000 + 5.65685i 0.123797 + 0.350150i
\(262\) 0 0
\(263\) 14.1421 0.872041 0.436021 0.899937i \(-0.356387\pi\)
0.436021 + 0.899937i \(0.356387\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −5.65685 + 8.00000i −0.346194 + 0.489592i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 10.0000i 0.607457i −0.952759 0.303728i \(-0.901768\pi\)
0.952759 0.303728i \(-0.0982315\pi\)
\(272\) 0 0
\(273\) −8.00000 5.65685i −0.484182 0.342368i
\(274\) 0 0
\(275\) 2.00000i 0.120605i
\(276\) 0 0
\(277\) 25.4558i 1.52949i −0.644331 0.764747i \(-0.722864\pi\)
0.644331 0.764747i \(-0.277136\pi\)
\(278\) 0 0
\(279\) −16.9706 + 6.00000i −1.01600 + 0.359211i
\(280\) 0 0
\(281\) 11.3137i 0.674919i −0.941340 0.337460i \(-0.890432\pi\)
0.941340 0.337460i \(-0.109568\pi\)
\(282\) 0 0
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) 0 0
\(285\) −8.00000 5.65685i −0.473879 0.335083i
\(286\) 0 0
\(287\) −22.6274 −1.33565
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) −19.7990 14.0000i −1.16064 0.820695i
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 10.0000i 0.582223i
\(296\) 0 0
\(297\) 10.0000 + 2.82843i 0.580259 + 0.164122i
\(298\) 0 0
\(299\) 24.0000i 1.38796i
\(300\) 0 0
\(301\) 16.9706i 0.978167i
\(302\) 0 0
\(303\) 25.4558 + 18.0000i 1.46240 + 1.03407i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −14.1421 −0.807134 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(308\) 0 0
\(309\) 6.00000 8.48528i 0.341328 0.482711i
\(310\) 0 0
\(311\) −28.2843 −1.60385 −0.801927 0.597422i \(-0.796192\pi\)
−0.801927 + 0.597422i \(0.796192\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 5.65685 2.00000i 0.318728 0.112687i
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 4.00000i 0.223957i
\(320\) 0 0
\(321\) 14.0000 19.7990i 0.781404 1.10507i
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 2.82843i 0.156893i
\(326\) 0 0
\(327\) 16.9706 24.0000i 0.938474 1.32720i
\(328\) 0 0
\(329\) 5.65685i 0.311872i
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 8.00000 2.82843i 0.438397 0.154997i
\(334\) 0 0
\(335\) 2.82843 0.154533
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 14.1421 20.0000i 0.768095 1.08625i
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 12.0000 + 8.48528i 0.646058 + 0.456832i
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 5.65685i 0.302804i 0.988472 + 0.151402i \(0.0483789\pi\)
−0.988472 + 0.151402i \(0.951621\pi\)
\(350\) 0 0
\(351\) −14.1421 4.00000i −0.754851 0.213504i
\(352\) 0 0
\(353\) 31.1127i 1.65596i 0.560756 + 0.827981i \(0.310510\pi\)
−0.560756 + 0.827981i \(0.689490\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) 0 0
\(357\) −8.00000 5.65685i −0.423405 0.299392i
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 9.89949 + 7.00000i 0.519589 + 0.367405i
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 26.0000i 1.35719i 0.734513 + 0.678594i \(0.237411\pi\)
−0.734513 + 0.678594i \(0.762589\pi\)
\(368\) 0 0
\(369\) −32.0000 + 11.3137i −1.66585 + 0.588968i
\(370\) 0 0
\(371\) 20.0000i 1.03835i
\(372\) 0 0
\(373\) 14.1421i 0.732252i 0.930565 + 0.366126i \(0.119316\pi\)
−0.930565 + 0.366126i \(0.880684\pi\)
\(374\) 0 0
\(375\) −1.41421 1.00000i −0.0730297 0.0516398i
\(376\) 0 0
\(377\) 5.65685i 0.291343i
\(378\) 0 0
\(379\) 16.9706 0.871719 0.435860 0.900015i \(-0.356444\pi\)
0.435860 + 0.900015i \(0.356444\pi\)
\(380\) 0 0
\(381\) −2.00000 + 2.82843i −0.102463 + 0.144905i
\(382\) 0 0
\(383\) 14.1421 0.722629 0.361315 0.932444i \(-0.382328\pi\)
0.361315 + 0.932444i \(0.382328\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) −8.48528 24.0000i −0.431331 1.21999i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 24.0000i 1.21373i
\(392\) 0 0
\(393\) −6.00000 + 8.48528i −0.302660 + 0.428026i
\(394\) 0 0
\(395\) 2.00000i 0.100631i
\(396\) 0 0
\(397\) 8.48528i 0.425864i 0.977067 + 0.212932i \(0.0683013\pi\)
−0.977067 + 0.212932i \(0.931699\pi\)
\(398\) 0 0
\(399\) −11.3137 + 16.0000i −0.566394 + 0.801002i
\(400\) 0 0
\(401\) 11.3137i 0.564980i 0.959270 + 0.282490i \(0.0911603\pi\)
−0.959270 + 0.282490i \(0.908840\pi\)
\(402\) 0 0
\(403\) −16.9706 −0.845364
\(404\) 0 0
\(405\) 7.00000 5.65685i 0.347833 0.281091i
\(406\) 0 0
\(407\) −5.65685 −0.280400
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) −2.82843 + 4.00000i −0.139516 + 0.197305i
\(412\) 0 0
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 2.00000i 0.0981761i
\(416\) 0 0
\(417\) 24.0000 + 16.9706i 1.17529 + 0.831052i
\(418\) 0 0
\(419\) 30.0000i 1.46560i 0.680446 + 0.732798i \(0.261786\pi\)
−0.680446 + 0.732798i \(0.738214\pi\)
\(420\) 0 0
\(421\) 22.6274i 1.10279i 0.834243 + 0.551396i \(0.185905\pi\)
−0.834243 + 0.551396i \(0.814095\pi\)
\(422\) 0 0
\(423\) −2.82843 8.00000i −0.137523 0.388973i
\(424\) 0 0
\(425\) 2.82843i 0.137199i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.00000 + 5.65685i 0.386244 + 0.273115i
\(430\) 0 0
\(431\) −5.65685 −0.272481 −0.136241 0.990676i \(-0.543502\pi\)
−0.136241 + 0.990676i \(0.543502\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) −2.82843 2.00000i −0.135613 0.0958927i
\(436\) 0 0
\(437\) −48.0000 −2.29615
\(438\) 0 0
\(439\) 6.00000i 0.286364i 0.989696 + 0.143182i \(0.0457335\pi\)
−0.989696 + 0.143182i \(0.954267\pi\)
\(440\) 0 0
\(441\) 3.00000 + 8.48528i 0.142857 + 0.404061i
\(442\) 0 0
\(443\) 14.0000i 0.665160i −0.943075 0.332580i \(-0.892081\pi\)
0.943075 0.332580i \(-0.107919\pi\)
\(444\) 0 0
\(445\) 5.65685i 0.268161i
\(446\) 0 0
\(447\) −19.7990 14.0000i −0.936460 0.662177i
\(448\) 0 0
\(449\) 5.65685i 0.266963i 0.991051 + 0.133482i \(0.0426157\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) 0 0
\(451\) 22.6274 1.06548
\(452\) 0 0
\(453\) 2.00000 2.82843i 0.0939682 0.132891i
\(454\) 0 0
\(455\) 5.65685 0.265197
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 0 0
\(459\) −14.1421 4.00000i −0.660098 0.186704i
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 6.00000 8.48528i 0.278243 0.393496i
\(466\) 0 0
\(467\) 2.00000i 0.0925490i 0.998929 + 0.0462745i \(0.0147349\pi\)
−0.998929 + 0.0462745i \(0.985265\pi\)
\(468\) 0 0
\(469\) 5.65685i 0.261209i
\(470\) 0 0
\(471\) −19.7990 + 28.0000i −0.912289 + 1.29017i
\(472\) 0 0
\(473\) 16.9706i 0.780307i
\(474\) 0 0
\(475\) 5.65685 0.259554
\(476\) 0 0
\(477\) 10.0000 + 28.2843i 0.457869 + 1.29505i
\(478\) 0 0
\(479\) −22.6274 −1.03387 −0.516937 0.856024i \(-0.672928\pi\)
−0.516937 + 0.856024i \(0.672928\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 16.9706 24.0000i 0.772187 1.09204i
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) −12.0000 8.48528i −0.542659 0.383718i
\(490\) 0 0
\(491\) 30.0000i 1.35388i 0.736038 + 0.676941i \(0.236695\pi\)
−0.736038 + 0.676941i \(0.763305\pi\)
\(492\) 0 0
\(493\) 5.65685i 0.254772i
\(494\) 0 0
\(495\) −5.65685 + 2.00000i −0.254257 + 0.0898933i
\(496\) 0 0
\(497\) 11.3137i 0.507489i
\(498\) 0 0
\(499\) 16.9706 0.759707 0.379853 0.925047i \(-0.375974\pi\)
0.379853 + 0.925047i \(0.375974\pi\)
\(500\) 0 0
\(501\) 28.0000 + 19.7990i 1.25095 + 0.884554i
\(502\) 0 0
\(503\) −19.7990 −0.882793 −0.441397 0.897312i \(-0.645517\pi\)
−0.441397 + 0.897312i \(0.645517\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 7.07107 + 5.00000i 0.314037 + 0.222058i
\(508\) 0 0
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 20.0000i 0.884748i
\(512\) 0 0
\(513\) −8.00000 + 28.2843i −0.353209 + 1.24878i
\(514\) 0 0
\(515\) 6.00000i 0.264392i
\(516\) 0 0
\(517\) 5.65685i 0.248788i
\(518\) 0 0
\(519\) 25.4558 + 18.0000i 1.11739 + 0.790112i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 2.82843 0.123678 0.0618392 0.998086i \(-0.480303\pi\)
0.0618392 + 0.998086i \(0.480303\pi\)
\(524\) 0 0
\(525\) −2.00000 + 2.82843i −0.0872872 + 0.123443i
\(526\) 0 0
\(527\) −16.9706 −0.739249
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 28.2843 10.0000i 1.22743 0.433963i
\(532\) 0 0
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) 14.0000i 0.605273i
\(536\) 0 0
\(537\) 18.0000 25.4558i 0.776757 1.09850i
\(538\) 0 0
\(539\) 6.00000i 0.258438i
\(540\) 0 0
\(541\) 33.9411i 1.45924i −0.683851 0.729621i \(-0.739696\pi\)
0.683851 0.729621i \(-0.260304\pi\)
\(542\) 0 0
\(543\) −22.6274 + 32.0000i −0.971035 + 1.37325i
\(544\) 0 0
\(545\) 16.9706i 0.726939i
\(546\) 0 0
\(547\) 19.7990 0.846544 0.423272 0.906003i \(-0.360882\pi\)
0.423272 + 0.906003i \(0.360882\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −2.82843 + 4.00000i −0.120060 + 0.169791i
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 24.0000i 1.01509i
\(560\) 0 0
\(561\) 8.00000 + 5.65685i 0.337760 + 0.238833i
\(562\) 0 0
\(563\) 42.0000i 1.77009i 0.465506 + 0.885044i \(0.345872\pi\)
−0.465506 + 0.885044i \(0.654128\pi\)
\(564\) 0 0
\(565\) 14.1421i 0.594964i
\(566\) 0 0
\(567\) −11.3137 14.0000i −0.475131 0.587945i
\(568\) 0 0
\(569\) 16.9706i 0.711443i 0.934592 + 0.355722i \(0.115765\pi\)
−0.934592 + 0.355722i \(0.884235\pi\)
\(570\) 0 0
\(571\) −22.6274 −0.946928 −0.473464 0.880813i \(-0.656997\pi\)
−0.473464 + 0.880813i \(0.656997\pi\)
\(572\) 0 0
\(573\) 8.00000 + 5.65685i 0.334205 + 0.236318i
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −19.7990 14.0000i −0.822818 0.581820i
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 8.00000 2.82843i 0.330759 0.116941i
\(586\) 0 0
\(587\) 30.0000i 1.23823i −0.785299 0.619116i \(-0.787491\pi\)
0.785299 0.619116i \(-0.212509\pi\)
\(588\) 0 0
\(589\) 33.9411i 1.39852i
\(590\) 0 0
\(591\) 14.1421 + 10.0000i 0.581730 + 0.411345i
\(592\) 0 0
\(593\) 25.4558i 1.04535i −0.852533 0.522673i \(-0.824935\pi\)
0.852533 0.522673i \(-0.175065\pi\)
\(594\) 0 0
\(595\) 5.65685 0.231908
\(596\) 0 0
\(597\) 26.0000 36.7696i 1.06411 1.50488i
\(598\) 0 0
\(599\) −22.6274 −0.924531 −0.462266 0.886742i \(-0.652963\pi\)
−0.462266 + 0.886742i \(0.652963\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −2.82843 8.00000i −0.115182 0.325785i
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 26.0000i 1.05531i 0.849460 + 0.527654i \(0.176928\pi\)
−0.849460 + 0.527654i \(0.823072\pi\)
\(608\) 0 0
\(609\) −4.00000 + 5.65685i −0.162088 + 0.229227i
\(610\) 0 0
\(611\) 8.00000i 0.323645i
\(612\) 0 0
\(613\) 42.4264i 1.71359i −0.515660 0.856793i \(-0.672453\pi\)
0.515660 0.856793i \(-0.327547\pi\)
\(614\) 0 0
\(615\) 11.3137 16.0000i 0.456213 0.645182i
\(616\) 0 0
\(617\) 2.82843i 0.113868i 0.998378 + 0.0569341i \(0.0181325\pi\)
−0.998378 + 0.0569341i \(0.981868\pi\)
\(618\) 0 0
\(619\) −39.5980 −1.59158 −0.795789 0.605575i \(-0.792943\pi\)
−0.795789 + 0.605575i \(0.792943\pi\)
\(620\) 0 0
\(621\) 12.0000 42.4264i 0.481543 1.70251i
\(622\) 0 0
\(623\) −11.3137 −0.453274
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.3137 16.0000i 0.451826 0.638978i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 2.00000i 0.0796187i −0.999207 0.0398094i \(-0.987325\pi\)
0.999207 0.0398094i \(-0.0126751\pi\)
\(632\) 0 0
\(633\) −16.0000 11.3137i −0.635943 0.449680i
\(634\) 0 0
\(635\) 2.00000i 0.0793676i
\(636\) 0 0
\(637\) 8.48528i 0.336199i
\(638\) 0 0
\(639\) 5.65685 + 16.0000i 0.223782 + 0.632950i
\(640\) 0 0
\(641\) 22.6274i 0.893729i 0.894602 + 0.446865i \(0.147459\pi\)
−0.894602 + 0.446865i \(0.852541\pi\)
\(642\) 0 0
\(643\) 14.1421 0.557711 0.278856 0.960333i \(-0.410045\pi\)
0.278856 + 0.960333i \(0.410045\pi\)
\(644\) 0 0
\(645\) 12.0000 + 8.48528i 0.472500 + 0.334108i
\(646\) 0 0
\(647\) 19.7990 0.778379 0.389189 0.921158i \(-0.372755\pi\)
0.389189 + 0.921158i \(0.372755\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) −16.9706 12.0000i −0.665129 0.470317i
\(652\) 0 0
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) 6.00000i 0.234439i
\(656\) 0 0
\(657\) 10.0000 + 28.2843i 0.390137 + 1.10347i
\(658\) 0 0
\(659\) 30.0000i 1.16863i 0.811525 + 0.584317i \(0.198638\pi\)
−0.811525 + 0.584317i \(0.801362\pi\)
\(660\) 0 0
\(661\) 11.3137i 0.440052i 0.975494 + 0.220026i \(0.0706143\pi\)
−0.975494 + 0.220026i \(0.929386\pi\)
\(662\) 0 0
\(663\) −11.3137 8.00000i −0.439388 0.310694i
\(664\) 0 0
\(665\) 11.3137i 0.438727i
\(666\) 0 0
\(667\) −16.9706 −0.657103
\(668\) 0 0
\(669\) 14.0000 19.7990i 0.541271 0.765473i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 0 0
\(675\) −1.41421 + 5.00000i −0.0544331 + 0.192450i
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 28.0000i 1.07454i
\(680\) 0 0
\(681\) −10.0000 + 14.1421i −0.383201 + 0.541928i
\(682\) 0 0
\(683\) 22.0000i 0.841807i −0.907106 0.420903i \(-0.861713\pi\)
0.907106 0.420903i \(-0.138287\pi\)
\(684\) 0 0
\(685\) 2.82843i 0.108069i
\(686\) 0 0
\(687\) 28.2843 40.0000i 1.07911 1.52610i
\(688\) 0 0
\(689\) 28.2843i 1.07754i
\(690\) 0 0
\(691\) −11.3137 −0.430394 −0.215197 0.976571i \(-0.569039\pi\)
−0.215197 + 0.976571i \(0.569039\pi\)
\(692\) 0 0
\(693\) 4.00000 + 11.3137i 0.151947 + 0.429772i
\(694\) 0 0
\(695\) −16.9706 −0.643730
\(696\) 0 0
\(697\) −32.0000 −1.21209
\(698\) 0 0
\(699\) −8.48528 + 12.0000i −0.320943 + 0.453882i
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 16.0000i 0.603451i
\(704\) 0 0
\(705\) 4.00000 + 2.82843i 0.150649 + 0.106525i
\(706\) 0 0
\(707\) 36.0000i 1.35392i
\(708\) 0 0
\(709\) 5.65685i 0.212448i −0.994342 0.106224i \(-0.966124\pi\)
0.994342 0.106224i \(-0.0338760\pi\)
\(710\) 0 0
\(711\) 5.65685 2.00000i 0.212149 0.0750059i
\(712\) 0 0
\(713\) 50.9117i 1.90666i
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) 0 0
\(717\) 32.0000 + 22.6274i 1.19506 + 0.845036i
\(718\) 0 0
\(719\) 33.9411 1.26579 0.632895 0.774237i \(-0.281866\pi\)
0.632895 + 0.774237i \(0.281866\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) −19.7990 14.0000i −0.736332 0.520666i
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 0 0
\(729\) −23.0000 14.1421i −0.851852 0.523783i
\(730\) 0 0
\(731\) 24.0000i 0.887672i
\(732\) 0 0
\(733\) 42.4264i 1.56706i −0.621357 0.783528i \(-0.713418\pi\)
0.621357 0.783528i \(-0.286582\pi\)
\(734\) 0 0
\(735\) −4.24264 3.00000i −0.156492 0.110657i
\(736\) 0 0
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) −39.5980 −1.45664 −0.728318 0.685240i \(-0.759697\pi\)
−0.728318 + 0.685240i \(0.759697\pi\)
\(740\) 0 0
\(741\) −16.0000 + 22.6274i −0.587775 + 0.831239i
\(742\) 0 0
\(743\) 2.82843 0.103765 0.0518825 0.998653i \(-0.483478\pi\)
0.0518825 + 0.998653i \(0.483478\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) −5.65685 + 2.00000i −0.206973 + 0.0731762i
\(748\) 0 0
\(749\) 28.0000 1.02310
\(750\) 0 0
\(751\) 10.0000i 0.364905i −0.983215 0.182453i \(-0.941596\pi\)
0.983215 0.182453i \(-0.0584036\pi\)
\(752\) 0 0
\(753\) 18.0000 25.4558i 0.655956 0.927663i
\(754\) 0 0
\(755\) 2.00000i 0.0727875i
\(756\) 0 0
\(757\) 14.1421i 0.514005i −0.966411 0.257002i \(-0.917265\pi\)
0.966411 0.257002i \(-0.0827348\pi\)
\(758\) 0 0
\(759\) −16.9706 + 24.0000i −0.615992 + 0.871145i
\(760\) 0 0
\(761\) 33.9411i 1.23036i −0.788385 0.615182i \(-0.789082\pi\)
0.788385 0.615182i \(-0.210918\pi\)
\(762\) 0 0
\(763\) 33.9411 1.22875
\(764\) 0 0
\(765\) 8.00000 2.82843i 0.289241 0.102262i
\(766\) 0 0
\(767\) 28.2843 1.02129
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 19.7990 28.0000i 0.713043 1.00840i
\(772\) 0 0
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 0 0
\(775\) 6.00000i 0.215526i
\(776\) 0 0
\(777\) 8.00000 + 5.65685i 0.286998 + 0.202939i
\(778\) 0 0
\(779\) 64.0000i 2.29304i
\(780\) 0 0
\(781\) 11.3137i 0.404836i
\(782\) 0 0
\(783\) −2.82843 + 10.0000i −0.101080 + 0.357371i
\(784\) 0 0
\(785\) 19.7990i 0.706656i
\(786\) 0 0
\(787\) 19.7990 0.705758 0.352879 0.935669i \(-0.385203\pi\)
0.352879 + 0.935669i \(0.385203\pi\)
\(788\) 0 0
\(789\) 20.0000 + 14.1421i 0.712019 + 0.503473i
\(790\) 0 0
\(791\) 28.2843 1.00567
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −14.1421 10.0000i −0.501570 0.354663i
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 8.00000i 0.283020i
\(800\) 0 0
\(801\) −16.0000 + 5.65685i −0.565332 + 0.199875i
\(802\) 0 0
\(803\) 20.0000i 0.705785i
\(804\) 0 0
\(805\) 16.9706i 0.598134i
\(806\) 0 0
\(807\) −8.48528 6.00000i −0.298696 0.211210i
\(808\) 0 0
\(809\) 28.2843i 0.994422i −0.867630 0.497211i \(-0.834357\pi\)
0.867630 0.497211i \(-0.165643\pi\)
\(810\) 0 0
\(811\) 22.6274 0.794556 0.397278 0.917698i \(-0.369955\pi\)
0.397278 + 0.917698i \(0.369955\pi\)
\(812\) 0 0
\(813\) 10.0000 14.1421i 0.350715 0.495986i
\(814\) 0 0
\(815\) 8.48528 0.297226
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 0 0
\(819\) −5.65685 16.0000i −0.197666 0.559085i
\(820\) 0 0
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 0 0
\(825\) 2.00000 2.82843i 0.0696311 0.0984732i
\(826\) 0 0
\(827\) 6.00000i 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 0 0
\(829\) 39.5980i 1.37529i 0.726045 + 0.687647i \(0.241356\pi\)
−0.726045 + 0.687647i \(0.758644\pi\)
\(830\) 0 0
\(831\) 25.4558 36.0000i 0.883053 1.24883i
\(832\) 0 0
\(833\) 8.48528i 0.293998i
\(834\) 0 0
\(835\) −19.7990 −0.685172
\(836\) 0 0
\(837\) −30.0000 8.48528i −1.03695 0.293294i
\(838\) 0 0
\(839\) 45.2548 1.56237 0.781185 0.624299i \(-0.214615\pi\)
0.781185 + 0.624299i \(0.214615\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 11.3137 16.0000i 0.389665 0.551069i
\(844\) 0 0
\(845\) −5.00000 −0.172005
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) −12.0000 8.48528i −0.411839 0.291214i
\(850\) 0 0
\(851\) 24.0000i 0.822709i
\(852\) 0 0
\(853\) 19.7990i 0.677905i −0.940804 0.338952i \(-0.889927\pi\)
0.940804 0.338952i \(-0.110073\pi\)
\(854\) 0 0
\(855\) −5.65685 16.0000i −0.193460 0.547188i
\(856\) 0 0
\(857\) 19.7990i 0.676321i −0.941089 0.338160i \(-0.890195\pi\)
0.941089 0.338160i \(-0.109805\pi\)
\(858\) 0 0
\(859\) −16.9706 −0.579028 −0.289514 0.957174i \(-0.593494\pi\)
−0.289514 + 0.957174i \(0.593494\pi\)
\(860\) 0 0
\(861\) −32.0000 22.6274i −1.09056 0.771140i
\(862\) 0 0
\(863\) −19.7990 −0.673965 −0.336983 0.941511i \(-0.609406\pi\)
−0.336983 + 0.941511i \(0.609406\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 12.7279 + 9.00000i 0.432263 + 0.305656i
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 8.00000i 0.271070i
\(872\) 0 0
\(873\) −14.0000 39.5980i −0.473828 1.34019i
\(874\) 0 0
\(875\) 2.00000i 0.0676123i
\(876\) 0 0
\(877\) 8.48528i 0.286528i 0.989685 + 0.143264i \(0.0457597\pi\)
−0.989685 + 0.143264i \(0.954240\pi\)
\(878\) 0 0
\(879\) −8.48528 6.00000i −0.286201 0.202375i
\(880\) 0 0
\(881\) 45.2548i 1.52467i 0.647180 + 0.762337i \(0.275948\pi\)
−0.647180 + 0.762337i \(0.724052\pi\)
\(882\) 0 0
\(883\) 25.4558 0.856657 0.428329 0.903623i \(-0.359103\pi\)
0.428329 + 0.903623i \(0.359103\pi\)
\(884\) 0 0
\(885\) −10.0000 + 14.1421i −0.336146 + 0.475383i
\(886\) 0 0
\(887\) −14.1421 −0.474846 −0.237423 0.971406i \(-0.576303\pi\)
−0.237423 + 0.971406i \(0.576303\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 11.3137 + 14.0000i 0.379023 + 0.469018i
\(892\) 0 0
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 18.0000i 0.601674i
\(896\) 0 0
\(897\) 24.0000 33.9411i 0.801337 1.13326i
\(898\) 0 0
\(899\) 12.0000i 0.400222i
\(900\) 0 0
\(901\) 28.2843i 0.942286i
\(902\) 0 0
\(903\) 16.9706 24.0000i 0.564745 0.798670i
\(904\) 0 0
\(905\) 22.6274i 0.752161i
\(906\) 0 0
\(907\) −14.1421 −0.469582 −0.234791 0.972046i \(-0.575441\pi\)
−0.234791 + 0.972046i \(0.575441\pi\)
\(908\) 0 0
\(909\) 18.0000 + 50.9117i 0.597022 + 1.68863i
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 26.0000i 0.857661i −0.903385 0.428830i \(-0.858926\pi\)
0.903385 0.428830i \(-0.141074\pi\)
\(920\) 0 0
\(921\) −20.0000 14.1421i −0.659022 0.465999i
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 2.82843i 0.0929981i
\(926\) 0 0
\(927\) 16.9706 6.00000i 0.557386 0.197066i
\(928\) 0 0
\(929\) 39.5980i 1.29917i −0.760290 0.649584i \(-0.774943\pi\)
0.760290 0.649584i \(-0.225057\pi\)
\(930\) 0 0
\(931\) 16.9706 0.556188
\(932\) 0 0
\(933\) −40.0000 28.2843i −1.30954 0.925985i
\(934\) 0 0
\(935\) −5.65685 −0.184999
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 14.1421 + 10.0000i 0.461511 + 0.326338i
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) 96.0000i 3.12619i
\(944\) 0 0
\(945\) 10.0000 + 2.82843i 0.325300 + 0.0920087i
\(946\) 0 0
\(947\) 2.00000i 0.0649913i 0.999472 + 0.0324956i \(0.0103455\pi\)
−0.999472 + 0.0324956i \(0.989654\pi\)
\(948\) 0 0
\(949\) 28.2843i 0.918146i
\(950\) 0 0
\(951\) 25.4558 + 18.0000i 0.825462 + 0.583690i
\(952\) 0 0
\(953\) 14.1421i 0.458109i −0.973414 0.229054i \(-0.926437\pi\)
0.973414 0.229054i \(-0.0735634\pi\)
\(954\) 0 0
\(955\) −5.65685 −0.183052
\(956\) 0 0
\(957\) 4.00000 5.65685i 0.129302 0.182860i
\(958\) 0 0
\(959\) −5.65685 −0.182669
\(960\) 0 0
\(961\) −5.00000 −0.161290
\(962\) 0 0
\(963\) 39.5980 14.0000i 1.27603 0.451144i
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 54.0000i 1.73652i −0.496107 0.868261i \(-0.665238\pi\)
0.496107 0.868261i \(-0.334762\pi\)
\(968\) 0 0
\(969\) −16.0000 + 22.6274i −0.513994 + 0.726897i
\(970\) 0 0
\(971\) 34.0000i 1.09111i −0.838074 0.545556i \(-0.816319\pi\)
0.838074 0.545556i \(-0.183681\pi\)
\(972\) 0 0
\(973\) 33.9411i 1.08810i
\(974\) 0 0
\(975\) −2.82843 + 4.00000i −0.0905822 + 0.128103i
\(976\) 0 0
\(977\) 36.7696i 1.17636i 0.808729 + 0.588181i \(0.200156\pi\)
−0.808729 + 0.588181i \(0.799844\pi\)
\(978\) 0 0
\(979\) 11.3137 0.361588
\(980\) 0 0
\(981\) 48.0000 16.9706i 1.53252 0.541828i
\(982\) 0 0
\(983\) −8.48528 −0.270638 −0.135319 0.990802i \(-0.543206\pi\)
−0.135319 + 0.990802i \(0.543206\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) 5.65685 8.00000i 0.180060 0.254643i
\(988\) 0 0
\(989\) 72.0000 2.28947
\(990\) 0 0
\(991\) 54.0000i 1.71537i 0.514178 + 0.857683i \(0.328097\pi\)
−0.514178 + 0.857683i \(0.671903\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 26.0000i 0.824255i
\(996\) 0 0
\(997\) 25.4558i 0.806195i −0.915157 0.403097i \(-0.867934\pi\)
0.915157 0.403097i \(-0.132066\pi\)
\(998\) 0 0
\(999\) 14.1421 + 4.00000i 0.447437 + 0.126554i
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.b.a.671.4 yes 4
3.2 odd 2 960.2.b.b.671.3 yes 4
4.3 odd 2 inner 960.2.b.a.671.1 4
8.3 odd 2 960.2.b.b.671.4 yes 4
8.5 even 2 960.2.b.b.671.1 yes 4
12.11 even 2 960.2.b.b.671.2 yes 4
24.5 odd 2 inner 960.2.b.a.671.2 yes 4
24.11 even 2 inner 960.2.b.a.671.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.2.b.a.671.1 4 4.3 odd 2 inner
960.2.b.a.671.2 yes 4 24.5 odd 2 inner
960.2.b.a.671.3 yes 4 24.11 even 2 inner
960.2.b.a.671.4 yes 4 1.1 even 1 trivial
960.2.b.b.671.1 yes 4 8.5 even 2
960.2.b.b.671.2 yes 4 12.11 even 2
960.2.b.b.671.3 yes 4 3.2 odd 2
960.2.b.b.671.4 yes 4 8.3 odd 2