Properties

Label 960.3.i.b.929.11
Level $960$
Weight $3$
Character 960.929
Analytic conductor $26.158$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(929,960)]
 
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([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.929");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.i (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.11
Character \(\chi\) \(=\) 960.929
Dual form 960.3.i.b.929.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.54191 - 1.59332i) q^{3} +(4.96076 - 0.625206i) q^{5} -5.93744i q^{7} +(3.92266 + 8.10017i) q^{9} +O(q^{10})\) \(q+(-2.54191 - 1.59332i) q^{3} +(4.96076 - 0.625206i) q^{5} -5.93744i q^{7} +(3.92266 + 8.10017i) q^{9} -5.99878 q^{11} +15.6160 q^{13} +(-13.6060 - 6.31486i) q^{15} +14.2275 q^{17} +29.6128i q^{19} +(-9.46024 + 15.0925i) q^{21} +21.5275 q^{23} +(24.2182 - 6.20299i) q^{25} +(2.93512 - 26.8400i) q^{27} -18.1383 q^{29} +8.55847 q^{31} +(15.2484 + 9.55799i) q^{33} +(-3.71212 - 29.4542i) q^{35} -13.4855 q^{37} +(-39.6947 - 24.8814i) q^{39} +18.8052i q^{41} -27.1036 q^{43} +(24.5236 + 37.7305i) q^{45} +37.4498 q^{47} +13.7469 q^{49} +(-36.1651 - 22.6690i) q^{51} +93.5909i q^{53} +(-29.7585 + 3.75047i) q^{55} +(47.1827 - 75.2731i) q^{57} +83.7790 q^{59} -39.3326i q^{61} +(48.0942 - 23.2905i) q^{63} +(77.4674 - 9.76324i) q^{65} -103.540 q^{67} +(-54.7210 - 34.3002i) q^{69} +14.5589i q^{71} -95.8968i q^{73} +(-71.4440 - 22.8200i) q^{75} +35.6174i q^{77} +133.154 q^{79} +(-50.2255 + 63.5484i) q^{81} -116.715i q^{83} +(70.5793 - 8.89513i) q^{85} +(46.1060 + 28.9001i) q^{87} -146.524i q^{89} -92.7193i q^{91} +(-21.7549 - 13.6364i) q^{93} +(18.5141 + 146.902i) q^{95} -19.9818i q^{97} +(-23.5312 - 48.5912i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) \) 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
\(3\) −2.54191 1.59332i −0.847305 0.531107i
\(4\) 0 0
\(5\) 4.96076 0.625206i 0.992152 0.125041i
\(6\) 0 0
\(7\) 5.93744i 0.848205i −0.905614 0.424103i \(-0.860590\pi\)
0.905614 0.424103i \(-0.139410\pi\)
\(8\) 0 0
\(9\) 3.92266 + 8.10017i 0.435851 + 0.900019i
\(10\) 0 0
\(11\) −5.99878 −0.545344 −0.272672 0.962107i \(-0.587907\pi\)
−0.272672 + 0.962107i \(0.587907\pi\)
\(12\) 0 0
\(13\) 15.6160 1.20123 0.600617 0.799537i \(-0.294922\pi\)
0.600617 + 0.799537i \(0.294922\pi\)
\(14\) 0 0
\(15\) −13.6060 6.31486i −0.907065 0.420991i
\(16\) 0 0
\(17\) 14.2275 0.836913 0.418457 0.908237i \(-0.362571\pi\)
0.418457 + 0.908237i \(0.362571\pi\)
\(18\) 0 0
\(19\) 29.6128i 1.55857i 0.626672 + 0.779283i \(0.284417\pi\)
−0.626672 + 0.779283i \(0.715583\pi\)
\(20\) 0 0
\(21\) −9.46024 + 15.0925i −0.450488 + 0.718688i
\(22\) 0 0
\(23\) 21.5275 0.935977 0.467989 0.883734i \(-0.344979\pi\)
0.467989 + 0.883734i \(0.344979\pi\)
\(24\) 0 0
\(25\) 24.2182 6.20299i 0.968729 0.248119i
\(26\) 0 0
\(27\) 2.93512 26.8400i 0.108708 0.994074i
\(28\) 0 0
\(29\) −18.1383 −0.625459 −0.312729 0.949842i \(-0.601243\pi\)
−0.312729 + 0.949842i \(0.601243\pi\)
\(30\) 0 0
\(31\) 8.55847 0.276080 0.138040 0.990427i \(-0.455920\pi\)
0.138040 + 0.990427i \(0.455920\pi\)
\(32\) 0 0
\(33\) 15.2484 + 9.55799i 0.462072 + 0.289636i
\(34\) 0 0
\(35\) −3.71212 29.4542i −0.106060 0.841548i
\(36\) 0 0
\(37\) −13.4855 −0.364473 −0.182236 0.983255i \(-0.558334\pi\)
−0.182236 + 0.983255i \(0.558334\pi\)
\(38\) 0 0
\(39\) −39.6947 24.8814i −1.01781 0.637984i
\(40\) 0 0
\(41\) 18.8052i 0.458664i 0.973348 + 0.229332i \(0.0736541\pi\)
−0.973348 + 0.229332i \(0.926346\pi\)
\(42\) 0 0
\(43\) −27.1036 −0.630317 −0.315158 0.949039i \(-0.602058\pi\)
−0.315158 + 0.949039i \(0.602058\pi\)
\(44\) 0 0
\(45\) 24.5236 + 37.7305i 0.544969 + 0.838456i
\(46\) 0 0
\(47\) 37.4498 0.796805 0.398403 0.917211i \(-0.369565\pi\)
0.398403 + 0.917211i \(0.369565\pi\)
\(48\) 0 0
\(49\) 13.7469 0.280548
\(50\) 0 0
\(51\) −36.1651 22.6690i −0.709120 0.444490i
\(52\) 0 0
\(53\) 93.5909i 1.76587i 0.469498 + 0.882933i \(0.344435\pi\)
−0.469498 + 0.882933i \(0.655565\pi\)
\(54\) 0 0
\(55\) −29.7585 + 3.75047i −0.541064 + 0.0681904i
\(56\) 0 0
\(57\) 47.1827 75.2731i 0.827766 1.32058i
\(58\) 0 0
\(59\) 83.7790 1.41998 0.709992 0.704210i \(-0.248699\pi\)
0.709992 + 0.704210i \(0.248699\pi\)
\(60\) 0 0
\(61\) 39.3326i 0.644797i −0.946604 0.322398i \(-0.895511\pi\)
0.946604 0.322398i \(-0.104489\pi\)
\(62\) 0 0
\(63\) 48.0942 23.2905i 0.763401 0.369691i
\(64\) 0 0
\(65\) 77.4674 9.76324i 1.19181 0.150204i
\(66\) 0 0
\(67\) −103.540 −1.54538 −0.772688 0.634785i \(-0.781088\pi\)
−0.772688 + 0.634785i \(0.781088\pi\)
\(68\) 0 0
\(69\) −54.7210 34.3002i −0.793058 0.497104i
\(70\) 0 0
\(71\) 14.5589i 0.205054i 0.994730 + 0.102527i \(0.0326928\pi\)
−0.994730 + 0.102527i \(0.967307\pi\)
\(72\) 0 0
\(73\) 95.8968i 1.31366i −0.754041 0.656828i \(-0.771898\pi\)
0.754041 0.656828i \(-0.228102\pi\)
\(74\) 0 0
\(75\) −71.4440 22.8200i −0.952587 0.304266i
\(76\) 0 0
\(77\) 35.6174i 0.462563i
\(78\) 0 0
\(79\) 133.154 1.68549 0.842747 0.538310i \(-0.180937\pi\)
0.842747 + 0.538310i \(0.180937\pi\)
\(80\) 0 0
\(81\) −50.2255 + 63.5484i −0.620068 + 0.784548i
\(82\) 0 0
\(83\) 116.715i 1.40621i −0.711087 0.703104i \(-0.751797\pi\)
0.711087 0.703104i \(-0.248203\pi\)
\(84\) 0 0
\(85\) 70.5793 8.89513i 0.830345 0.104649i
\(86\) 0 0
\(87\) 46.1060 + 28.9001i 0.529954 + 0.332185i
\(88\) 0 0
\(89\) 146.524i 1.64634i −0.567794 0.823170i \(-0.692203\pi\)
0.567794 0.823170i \(-0.307797\pi\)
\(90\) 0 0
\(91\) 92.7193i 1.01889i
\(92\) 0 0
\(93\) −21.7549 13.6364i −0.233924 0.146628i
\(94\) 0 0
\(95\) 18.5141 + 146.902i 0.194885 + 1.54633i
\(96\) 0 0
\(97\) 19.9818i 0.205998i −0.994681 0.102999i \(-0.967156\pi\)
0.994681 0.102999i \(-0.0328439\pi\)
\(98\) 0 0
\(99\) −23.5312 48.5912i −0.237688 0.490820i
\(100\) 0 0
\(101\) 115.654 1.14509 0.572543 0.819874i \(-0.305957\pi\)
0.572543 + 0.819874i \(0.305957\pi\)
\(102\) 0 0
\(103\) 108.400i 1.05243i −0.850352 0.526214i \(-0.823611\pi\)
0.850352 0.526214i \(-0.176389\pi\)
\(104\) 0 0
\(105\) −37.4941 + 80.7846i −0.357086 + 0.769377i
\(106\) 0 0
\(107\) 74.1440i 0.692935i −0.938062 0.346467i \(-0.887381\pi\)
0.938062 0.346467i \(-0.112619\pi\)
\(108\) 0 0
\(109\) 15.0213i 0.137810i −0.997623 0.0689051i \(-0.978049\pi\)
0.997623 0.0689051i \(-0.0219506\pi\)
\(110\) 0 0
\(111\) 34.2790 + 21.4867i 0.308820 + 0.193574i
\(112\) 0 0
\(113\) 164.159 1.45273 0.726366 0.687308i \(-0.241208\pi\)
0.726366 + 0.687308i \(0.241208\pi\)
\(114\) 0 0
\(115\) 106.793 13.4591i 0.928631 0.117036i
\(116\) 0 0
\(117\) 61.2564 + 126.493i 0.523559 + 1.08113i
\(118\) 0 0
\(119\) 84.4750i 0.709874i
\(120\) 0 0
\(121\) −85.0146 −0.702600
\(122\) 0 0
\(123\) 29.9627 47.8012i 0.243600 0.388628i
\(124\) 0 0
\(125\) 116.263 45.9129i 0.930101 0.367303i
\(126\) 0 0
\(127\) 68.6273i 0.540372i 0.962808 + 0.270186i \(0.0870852\pi\)
−0.962808 + 0.270186i \(0.912915\pi\)
\(128\) 0 0
\(129\) 68.8951 + 43.1848i 0.534070 + 0.334766i
\(130\) 0 0
\(131\) 176.512 1.34742 0.673712 0.738994i \(-0.264699\pi\)
0.673712 + 0.738994i \(0.264699\pi\)
\(132\) 0 0
\(133\) 175.824 1.32198
\(134\) 0 0
\(135\) −2.22010 134.982i −0.0164452 0.999865i
\(136\) 0 0
\(137\) 96.3424 0.703229 0.351615 0.936145i \(-0.385633\pi\)
0.351615 + 0.936145i \(0.385633\pi\)
\(138\) 0 0
\(139\) 20.1883i 0.145240i −0.997360 0.0726198i \(-0.976864\pi\)
0.997360 0.0726198i \(-0.0231360\pi\)
\(140\) 0 0
\(141\) −95.1943 59.6696i −0.675137 0.423189i
\(142\) 0 0
\(143\) −93.6773 −0.655086
\(144\) 0 0
\(145\) −89.9797 + 11.3402i −0.620550 + 0.0782080i
\(146\) 0 0
\(147\) −34.9433 21.9032i −0.237710 0.149001i
\(148\) 0 0
\(149\) −138.759 −0.931270 −0.465635 0.884977i \(-0.654174\pi\)
−0.465635 + 0.884977i \(0.654174\pi\)
\(150\) 0 0
\(151\) −123.501 −0.817888 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(152\) 0 0
\(153\) 55.8097 + 115.245i 0.364769 + 0.753238i
\(154\) 0 0
\(155\) 42.4565 5.35080i 0.273913 0.0345213i
\(156\) 0 0
\(157\) −233.338 −1.48623 −0.743116 0.669162i \(-0.766653\pi\)
−0.743116 + 0.669162i \(0.766653\pi\)
\(158\) 0 0
\(159\) 149.120 237.900i 0.937864 1.49623i
\(160\) 0 0
\(161\) 127.818i 0.793901i
\(162\) 0 0
\(163\) −98.7745 −0.605979 −0.302989 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(164\) 0 0
\(165\) 81.6193 + 37.8815i 0.494662 + 0.229585i
\(166\) 0 0
\(167\) 133.070 0.796825 0.398412 0.917206i \(-0.369561\pi\)
0.398412 + 0.917206i \(0.369561\pi\)
\(168\) 0 0
\(169\) 74.8609 0.442964
\(170\) 0 0
\(171\) −239.869 + 116.161i −1.40274 + 0.679302i
\(172\) 0 0
\(173\) 323.270i 1.86861i −0.356474 0.934305i \(-0.616021\pi\)
0.356474 0.934305i \(-0.383979\pi\)
\(174\) 0 0
\(175\) −36.8298 143.794i −0.210456 0.821681i
\(176\) 0 0
\(177\) −212.959 133.487i −1.20316 0.754163i
\(178\) 0 0
\(179\) 183.645 1.02595 0.512975 0.858403i \(-0.328543\pi\)
0.512975 + 0.858403i \(0.328543\pi\)
\(180\) 0 0
\(181\) 301.614i 1.66637i 0.552992 + 0.833187i \(0.313486\pi\)
−0.552992 + 0.833187i \(0.686514\pi\)
\(182\) 0 0
\(183\) −62.6695 + 99.9801i −0.342456 + 0.546339i
\(184\) 0 0
\(185\) −66.8983 + 8.43121i −0.361612 + 0.0455741i
\(186\) 0 0
\(187\) −85.3478 −0.456405
\(188\) 0 0
\(189\) −159.361 17.4271i −0.843178 0.0922068i
\(190\) 0 0
\(191\) 242.628i 1.27030i 0.772387 + 0.635152i \(0.219063\pi\)
−0.772387 + 0.635152i \(0.780937\pi\)
\(192\) 0 0
\(193\) 13.6897i 0.0709311i −0.999371 0.0354656i \(-0.988709\pi\)
0.999371 0.0354656i \(-0.0112914\pi\)
\(194\) 0 0
\(195\) −212.472 98.6132i −1.08960 0.505709i
\(196\) 0 0
\(197\) 361.518i 1.83511i 0.397604 + 0.917557i \(0.369842\pi\)
−0.397604 + 0.917557i \(0.630158\pi\)
\(198\) 0 0
\(199\) −196.862 −0.989258 −0.494629 0.869104i \(-0.664696\pi\)
−0.494629 + 0.869104i \(0.664696\pi\)
\(200\) 0 0
\(201\) 263.190 + 164.973i 1.30941 + 0.820761i
\(202\) 0 0
\(203\) 107.695i 0.530517i
\(204\) 0 0
\(205\) 11.7571 + 93.2881i 0.0573518 + 0.455064i
\(206\) 0 0
\(207\) 84.4449 + 174.376i 0.407946 + 0.842397i
\(208\) 0 0
\(209\) 177.641i 0.849955i
\(210\) 0 0
\(211\) 202.277i 0.958658i 0.877635 + 0.479329i \(0.159120\pi\)
−0.877635 + 0.479329i \(0.840880\pi\)
\(212\) 0 0
\(213\) 23.1969 37.0073i 0.108906 0.173743i
\(214\) 0 0
\(215\) −134.454 + 16.9453i −0.625370 + 0.0788155i
\(216\) 0 0
\(217\) 50.8153i 0.234172i
\(218\) 0 0
\(219\) −152.794 + 243.762i −0.697692 + 1.11307i
\(220\) 0 0
\(221\) 222.178 1.00533
\(222\) 0 0
\(223\) 34.1739i 0.153246i −0.997060 0.0766232i \(-0.975586\pi\)
0.997060 0.0766232i \(-0.0244138\pi\)
\(224\) 0 0
\(225\) 145.245 + 171.840i 0.645534 + 0.763732i
\(226\) 0 0
\(227\) 316.667i 1.39501i 0.716580 + 0.697505i \(0.245706\pi\)
−0.716580 + 0.697505i \(0.754294\pi\)
\(228\) 0 0
\(229\) 236.008i 1.03060i −0.857009 0.515301i \(-0.827680\pi\)
0.857009 0.515301i \(-0.172320\pi\)
\(230\) 0 0
\(231\) 56.7499 90.5363i 0.245671 0.391932i
\(232\) 0 0
\(233\) −195.417 −0.838698 −0.419349 0.907825i \(-0.637742\pi\)
−0.419349 + 0.907825i \(0.637742\pi\)
\(234\) 0 0
\(235\) 185.780 23.4138i 0.790551 0.0996334i
\(236\) 0 0
\(237\) −338.466 212.157i −1.42813 0.895177i
\(238\) 0 0
\(239\) 9.71717i 0.0406576i 0.999793 + 0.0203288i \(0.00647130\pi\)
−0.999793 + 0.0203288i \(0.993529\pi\)
\(240\) 0 0
\(241\) 200.592 0.832331 0.416166 0.909289i \(-0.363374\pi\)
0.416166 + 0.909289i \(0.363374\pi\)
\(242\) 0 0
\(243\) 228.922 81.5091i 0.942066 0.335428i
\(244\) 0 0
\(245\) 68.1949 8.59462i 0.278346 0.0350801i
\(246\) 0 0
\(247\) 462.434i 1.87220i
\(248\) 0 0
\(249\) −185.965 + 296.680i −0.746847 + 1.19149i
\(250\) 0 0
\(251\) 20.8698 0.0831465 0.0415733 0.999135i \(-0.486763\pi\)
0.0415733 + 0.999135i \(0.486763\pi\)
\(252\) 0 0
\(253\) −129.139 −0.510429
\(254\) 0 0
\(255\) −193.579 89.8448i −0.759135 0.352333i
\(256\) 0 0
\(257\) −456.070 −1.77459 −0.887296 0.461199i \(-0.847419\pi\)
−0.887296 + 0.461199i \(0.847419\pi\)
\(258\) 0 0
\(259\) 80.0693i 0.309148i
\(260\) 0 0
\(261\) −71.1503 146.923i −0.272607 0.562925i
\(262\) 0 0
\(263\) −161.431 −0.613807 −0.306904 0.951741i \(-0.599293\pi\)
−0.306904 + 0.951741i \(0.599293\pi\)
\(264\) 0 0
\(265\) 58.5136 + 464.282i 0.220806 + 1.75201i
\(266\) 0 0
\(267\) −233.460 + 372.452i −0.874383 + 1.39495i
\(268\) 0 0
\(269\) 129.135 0.480054 0.240027 0.970766i \(-0.422844\pi\)
0.240027 + 0.970766i \(0.422844\pi\)
\(270\) 0 0
\(271\) −230.676 −0.851203 −0.425601 0.904911i \(-0.639937\pi\)
−0.425601 + 0.904911i \(0.639937\pi\)
\(272\) 0 0
\(273\) −147.732 + 235.684i −0.541141 + 0.863313i
\(274\) 0 0
\(275\) −145.280 + 37.2104i −0.528291 + 0.135310i
\(276\) 0 0
\(277\) −354.494 −1.27976 −0.639881 0.768474i \(-0.721016\pi\)
−0.639881 + 0.768474i \(0.721016\pi\)
\(278\) 0 0
\(279\) 33.5719 + 69.3251i 0.120329 + 0.248477i
\(280\) 0 0
\(281\) 350.785i 1.24834i −0.781287 0.624172i \(-0.785436\pi\)
0.781287 0.624172i \(-0.214564\pi\)
\(282\) 0 0
\(283\) 254.054 0.897718 0.448859 0.893602i \(-0.351830\pi\)
0.448859 + 0.893602i \(0.351830\pi\)
\(284\) 0 0
\(285\) 187.001 402.911i 0.656142 1.41372i
\(286\) 0 0
\(287\) 111.655 0.389041
\(288\) 0 0
\(289\) −86.5776 −0.299576
\(290\) 0 0
\(291\) −31.8375 + 50.7921i −0.109407 + 0.174543i
\(292\) 0 0
\(293\) 198.349i 0.676958i 0.940974 + 0.338479i \(0.109912\pi\)
−0.940974 + 0.338479i \(0.890088\pi\)
\(294\) 0 0
\(295\) 415.608 52.3791i 1.40884 0.177556i
\(296\) 0 0
\(297\) −17.6071 + 161.007i −0.0592833 + 0.542112i
\(298\) 0 0
\(299\) 336.174 1.12433
\(300\) 0 0
\(301\) 160.926i 0.534638i
\(302\) 0 0
\(303\) −293.982 184.274i −0.970237 0.608164i
\(304\) 0 0
\(305\) −24.5910 195.120i −0.0806261 0.639736i
\(306\) 0 0
\(307\) 229.921 0.748927 0.374464 0.927242i \(-0.377827\pi\)
0.374464 + 0.927242i \(0.377827\pi\)
\(308\) 0 0
\(309\) −172.716 + 275.544i −0.558952 + 0.891728i
\(310\) 0 0
\(311\) 43.0855i 0.138538i −0.997598 0.0692692i \(-0.977933\pi\)
0.997598 0.0692692i \(-0.0220668\pi\)
\(312\) 0 0
\(313\) 218.737i 0.698841i 0.936966 + 0.349420i \(0.113622\pi\)
−0.936966 + 0.349420i \(0.886378\pi\)
\(314\) 0 0
\(315\) 224.023 145.607i 0.711183 0.462246i
\(316\) 0 0
\(317\) 325.313i 1.02622i −0.858322 0.513111i \(-0.828493\pi\)
0.858322 0.513111i \(-0.171507\pi\)
\(318\) 0 0
\(319\) 108.808 0.341090
\(320\) 0 0
\(321\) −118.135 + 188.468i −0.368022 + 0.587127i
\(322\) 0 0
\(323\) 421.316i 1.30439i
\(324\) 0 0
\(325\) 378.193 96.8661i 1.16367 0.298050i
\(326\) 0 0
\(327\) −23.9338 + 38.1829i −0.0731919 + 0.116767i
\(328\) 0 0
\(329\) 222.356i 0.675854i
\(330\) 0 0
\(331\) 314.138i 0.949058i −0.880240 0.474529i \(-0.842618\pi\)
0.880240 0.474529i \(-0.157382\pi\)
\(332\) 0 0
\(333\) −52.8990 109.235i −0.158856 0.328033i
\(334\) 0 0
\(335\) −513.638 + 64.7339i −1.53325 + 0.193236i
\(336\) 0 0
\(337\) 325.048i 0.964534i −0.876024 0.482267i \(-0.839813\pi\)
0.876024 0.482267i \(-0.160187\pi\)
\(338\) 0 0
\(339\) −417.277 261.558i −1.23091 0.771556i
\(340\) 0 0
\(341\) −51.3404 −0.150558
\(342\) 0 0
\(343\) 372.555i 1.08617i
\(344\) 0 0
\(345\) −292.902 135.943i −0.848992 0.394038i
\(346\) 0 0
\(347\) 415.143i 1.19638i 0.801355 + 0.598189i \(0.204113\pi\)
−0.801355 + 0.598189i \(0.795887\pi\)
\(348\) 0 0
\(349\) 179.846i 0.515318i −0.966236 0.257659i \(-0.917049\pi\)
0.966236 0.257659i \(-0.0829512\pi\)
\(350\) 0 0
\(351\) 45.8350 419.135i 0.130584 1.19412i
\(352\) 0 0
\(353\) −103.250 −0.292493 −0.146246 0.989248i \(-0.546719\pi\)
−0.146246 + 0.989248i \(0.546719\pi\)
\(354\) 0 0
\(355\) 9.10227 + 72.2229i 0.0256402 + 0.203445i
\(356\) 0 0
\(357\) −134.596 + 214.728i −0.377019 + 0.601480i
\(358\) 0 0
\(359\) 353.673i 0.985161i 0.870267 + 0.492580i \(0.163946\pi\)
−0.870267 + 0.492580i \(0.836054\pi\)
\(360\) 0 0
\(361\) −515.916 −1.42913
\(362\) 0 0
\(363\) 216.100 + 135.456i 0.595316 + 0.373156i
\(364\) 0 0
\(365\) −59.9552 475.721i −0.164261 1.30335i
\(366\) 0 0
\(367\) 640.613i 1.74554i 0.488133 + 0.872769i \(0.337678\pi\)
−0.488133 + 0.872769i \(0.662322\pi\)
\(368\) 0 0
\(369\) −152.325 + 73.7664i −0.412806 + 0.199909i
\(370\) 0 0
\(371\) 555.690 1.49782
\(372\) 0 0
\(373\) 503.724 1.35047 0.675233 0.737605i \(-0.264043\pi\)
0.675233 + 0.737605i \(0.264043\pi\)
\(374\) 0 0
\(375\) −368.684 68.5371i −0.983157 0.182766i
\(376\) 0 0
\(377\) −283.249 −0.751322
\(378\) 0 0
\(379\) 138.570i 0.365620i −0.983148 0.182810i \(-0.941481\pi\)
0.983148 0.182810i \(-0.0585194\pi\)
\(380\) 0 0
\(381\) 109.345 174.445i 0.286995 0.457860i
\(382\) 0 0
\(383\) 149.537 0.390437 0.195219 0.980760i \(-0.437458\pi\)
0.195219 + 0.980760i \(0.437458\pi\)
\(384\) 0 0
\(385\) 22.2682 + 176.689i 0.0578394 + 0.458933i
\(386\) 0 0
\(387\) −106.318 219.544i −0.274724 0.567297i
\(388\) 0 0
\(389\) 585.106 1.50413 0.752064 0.659090i \(-0.229058\pi\)
0.752064 + 0.659090i \(0.229058\pi\)
\(390\) 0 0
\(391\) 306.283 0.783332
\(392\) 0 0
\(393\) −448.679 281.241i −1.14168 0.715626i
\(394\) 0 0
\(395\) 660.545 83.2486i 1.67226 0.210756i
\(396\) 0 0
\(397\) −529.718 −1.33430 −0.667151 0.744922i \(-0.732487\pi\)
−0.667151 + 0.744922i \(0.732487\pi\)
\(398\) 0 0
\(399\) −446.929 280.144i −1.12012 0.702115i
\(400\) 0 0
\(401\) 134.714i 0.335944i −0.985792 0.167972i \(-0.946278\pi\)
0.985792 0.167972i \(-0.0537219\pi\)
\(402\) 0 0
\(403\) 133.649 0.331636
\(404\) 0 0
\(405\) −209.426 + 346.649i −0.517101 + 0.855924i
\(406\) 0 0
\(407\) 80.8966 0.198763
\(408\) 0 0
\(409\) 221.882 0.542499 0.271249 0.962509i \(-0.412563\pi\)
0.271249 + 0.962509i \(0.412563\pi\)
\(410\) 0 0
\(411\) −244.894 153.504i −0.595850 0.373490i
\(412\) 0 0
\(413\) 497.433i 1.20444i
\(414\) 0 0
\(415\) −72.9710 578.996i −0.175834 1.39517i
\(416\) 0 0
\(417\) −32.1664 + 51.3169i −0.0771378 + 0.123062i
\(418\) 0 0
\(419\) −358.418 −0.855413 −0.427707 0.903918i \(-0.640678\pi\)
−0.427707 + 0.903918i \(0.640678\pi\)
\(420\) 0 0
\(421\) 572.660i 1.36024i −0.733102 0.680119i \(-0.761928\pi\)
0.733102 0.680119i \(-0.238072\pi\)
\(422\) 0 0
\(423\) 146.903 + 303.350i 0.347288 + 0.717140i
\(424\) 0 0
\(425\) 344.566 88.2531i 0.810742 0.207654i
\(426\) 0 0
\(427\) −233.535 −0.546920
\(428\) 0 0
\(429\) 238.120 + 149.258i 0.555057 + 0.347921i
\(430\) 0 0
\(431\) 190.360i 0.441672i 0.975311 + 0.220836i \(0.0708785\pi\)
−0.975311 + 0.220836i \(0.929122\pi\)
\(432\) 0 0
\(433\) 725.584i 1.67571i 0.545891 + 0.837857i \(0.316191\pi\)
−0.545891 + 0.837857i \(0.683809\pi\)
\(434\) 0 0
\(435\) 246.789 + 114.541i 0.567332 + 0.263312i
\(436\) 0 0
\(437\) 637.488i 1.45878i
\(438\) 0 0
\(439\) 460.268 1.04845 0.524223 0.851581i \(-0.324356\pi\)
0.524223 + 0.851581i \(0.324356\pi\)
\(440\) 0 0
\(441\) 53.9242 + 111.352i 0.122277 + 0.252499i
\(442\) 0 0
\(443\) 121.292i 0.273796i −0.990585 0.136898i \(-0.956287\pi\)
0.990585 0.136898i \(-0.0437133\pi\)
\(444\) 0 0
\(445\) −91.6078 726.872i −0.205860 1.63342i
\(446\) 0 0
\(447\) 352.714 + 221.088i 0.789070 + 0.494604i
\(448\) 0 0
\(449\) 12.4215i 0.0276648i −0.999904 0.0138324i \(-0.995597\pi\)
0.999904 0.0138324i \(-0.00440313\pi\)
\(450\) 0 0
\(451\) 112.808i 0.250129i
\(452\) 0 0
\(453\) 313.929 + 196.777i 0.693000 + 0.434386i
\(454\) 0 0
\(455\) −57.9686 459.958i −0.127404 1.01090i
\(456\) 0 0
\(457\) 438.321i 0.959126i −0.877507 0.479563i \(-0.840795\pi\)
0.877507 0.479563i \(-0.159205\pi\)
\(458\) 0 0
\(459\) 41.7595 381.867i 0.0909792 0.831953i
\(460\) 0 0
\(461\) −290.164 −0.629424 −0.314712 0.949187i \(-0.601908\pi\)
−0.314712 + 0.949187i \(0.601908\pi\)
\(462\) 0 0
\(463\) 342.841i 0.740477i 0.928937 + 0.370239i \(0.120724\pi\)
−0.928937 + 0.370239i \(0.879276\pi\)
\(464\) 0 0
\(465\) −116.446 54.0455i −0.250422 0.116227i
\(466\) 0 0
\(467\) 217.840i 0.466466i 0.972421 + 0.233233i \(0.0749305\pi\)
−0.972421 + 0.233233i \(0.925069\pi\)
\(468\) 0 0
\(469\) 614.764i 1.31080i
\(470\) 0 0
\(471\) 593.126 + 371.783i 1.25929 + 0.789348i
\(472\) 0 0
\(473\) 162.589 0.343739
\(474\) 0 0
\(475\) 183.688 + 717.169i 0.386711 + 1.50983i
\(476\) 0 0
\(477\) −758.103 + 367.125i −1.58931 + 0.769654i
\(478\) 0 0
\(479\) 577.121i 1.20485i 0.798177 + 0.602423i \(0.205798\pi\)
−0.798177 + 0.602423i \(0.794202\pi\)
\(480\) 0 0
\(481\) −210.590 −0.437817
\(482\) 0 0
\(483\) −203.655 + 324.902i −0.421646 + 0.672676i
\(484\) 0 0
\(485\) −12.4928 99.1251i −0.0257583 0.204382i
\(486\) 0 0
\(487\) 689.685i 1.41619i 0.706117 + 0.708095i \(0.250445\pi\)
−0.706117 + 0.708095i \(0.749555\pi\)
\(488\) 0 0
\(489\) 251.076 + 157.380i 0.513449 + 0.321840i
\(490\) 0 0
\(491\) −834.980 −1.70057 −0.850285 0.526323i \(-0.823570\pi\)
−0.850285 + 0.526323i \(0.823570\pi\)
\(492\) 0 0
\(493\) −258.063 −0.523455
\(494\) 0 0
\(495\) −147.112 226.337i −0.297196 0.457247i
\(496\) 0 0
\(497\) 86.4422 0.173928
\(498\) 0 0
\(499\) 145.508i 0.291600i 0.989314 + 0.145800i \(0.0465755\pi\)
−0.989314 + 0.145800i \(0.953424\pi\)
\(500\) 0 0
\(501\) −338.252 212.023i −0.675153 0.423199i
\(502\) 0 0
\(503\) −244.686 −0.486453 −0.243227 0.969969i \(-0.578206\pi\)
−0.243227 + 0.969969i \(0.578206\pi\)
\(504\) 0 0
\(505\) 573.730 72.3074i 1.13610 0.143183i
\(506\) 0 0
\(507\) −190.290 119.278i −0.375326 0.235261i
\(508\) 0 0
\(509\) 413.015 0.811425 0.405713 0.914001i \(-0.367023\pi\)
0.405713 + 0.914001i \(0.367023\pi\)
\(510\) 0 0
\(511\) −569.381 −1.11425
\(512\) 0 0
\(513\) 794.807 + 86.9170i 1.54933 + 0.169429i
\(514\) 0 0
\(515\) −67.7724 537.747i −0.131597 1.04417i
\(516\) 0 0
\(517\) −224.653 −0.434533
\(518\) 0 0
\(519\) −515.072 + 821.724i −0.992432 + 1.58328i
\(520\) 0 0
\(521\) 189.285i 0.363312i 0.983362 + 0.181656i \(0.0581457\pi\)
−0.983362 + 0.181656i \(0.941854\pi\)
\(522\) 0 0
\(523\) −235.324 −0.449950 −0.224975 0.974365i \(-0.572230\pi\)
−0.224975 + 0.974365i \(0.572230\pi\)
\(524\) 0 0
\(525\) −135.492 + 424.194i −0.258080 + 0.807989i
\(526\) 0 0
\(527\) 121.766 0.231055
\(528\) 0 0
\(529\) −65.5678 −0.123947
\(530\) 0 0
\(531\) 328.636 + 678.625i 0.618901 + 1.27801i
\(532\) 0 0
\(533\) 293.663i 0.550963i
\(534\) 0 0
\(535\) −46.3552 367.810i −0.0866453 0.687496i
\(536\) 0 0
\(537\) −466.810 292.606i −0.869293 0.544890i
\(538\) 0 0
\(539\) −82.4644 −0.152995
\(540\) 0 0
\(541\) 789.168i 1.45872i 0.684130 + 0.729360i \(0.260182\pi\)
−0.684130 + 0.729360i \(0.739818\pi\)
\(542\) 0 0
\(543\) 480.567 766.676i 0.885022 1.41193i
\(544\) 0 0
\(545\) −9.39140 74.5170i −0.0172319 0.136729i
\(546\) 0 0
\(547\) −905.745 −1.65584 −0.827921 0.560845i \(-0.810476\pi\)
−0.827921 + 0.560845i \(0.810476\pi\)
\(548\) 0 0
\(549\) 318.601 154.288i 0.580329 0.281035i
\(550\) 0 0
\(551\) 537.125i 0.974819i
\(552\) 0 0
\(553\) 790.593i 1.42964i
\(554\) 0 0
\(555\) 183.483 + 85.1590i 0.330601 + 0.153440i
\(556\) 0 0
\(557\) 352.250i 0.632406i 0.948692 + 0.316203i \(0.102408\pi\)
−0.948692 + 0.316203i \(0.897592\pi\)
\(558\) 0 0
\(559\) −423.251 −0.757158
\(560\) 0 0
\(561\) 216.947 + 135.986i 0.386714 + 0.242400i
\(562\) 0 0
\(563\) 1028.02i 1.82597i −0.407989 0.912987i \(-0.633770\pi\)
0.407989 0.912987i \(-0.366230\pi\)
\(564\) 0 0
\(565\) 814.352 102.633i 1.44133 0.181651i
\(566\) 0 0
\(567\) 377.314 + 298.211i 0.665457 + 0.525945i
\(568\) 0 0
\(569\) 612.176i 1.07588i 0.842983 + 0.537941i \(0.180798\pi\)
−0.842983 + 0.537941i \(0.819202\pi\)
\(570\) 0 0
\(571\) 284.705i 0.498607i −0.968425 0.249304i \(-0.919798\pi\)
0.968425 0.249304i \(-0.0802017\pi\)
\(572\) 0 0
\(573\) 386.585 616.740i 0.674668 1.07633i
\(574\) 0 0
\(575\) 521.357 133.535i 0.906709 0.232234i
\(576\) 0 0
\(577\) 510.925i 0.885485i −0.896649 0.442743i \(-0.854006\pi\)
0.896649 0.442743i \(-0.145994\pi\)
\(578\) 0 0
\(579\) −21.8121 + 34.7981i −0.0376720 + 0.0601003i
\(580\) 0 0
\(581\) −692.989 −1.19275
\(582\) 0 0
\(583\) 561.432i 0.963005i
\(584\) 0 0
\(585\) 382.962 + 589.202i 0.654636 + 1.00718i
\(586\) 0 0
\(587\) 425.221i 0.724396i −0.932101 0.362198i \(-0.882026\pi\)
0.932101 0.362198i \(-0.117974\pi\)
\(588\) 0 0
\(589\) 253.440i 0.430289i
\(590\) 0 0
\(591\) 576.014 918.947i 0.974642 1.55490i
\(592\) 0 0
\(593\) 639.053 1.07766 0.538831 0.842414i \(-0.318866\pi\)
0.538831 + 0.842414i \(0.318866\pi\)
\(594\) 0 0
\(595\) −52.8142 419.060i −0.0887634 0.704302i
\(596\) 0 0
\(597\) 500.407 + 313.665i 0.838203 + 0.525402i
\(598\) 0 0
\(599\) 609.450i 1.01745i −0.860930 0.508723i \(-0.830118\pi\)
0.860930 0.508723i \(-0.169882\pi\)
\(600\) 0 0
\(601\) 443.553 0.738025 0.369012 0.929425i \(-0.379696\pi\)
0.369012 + 0.929425i \(0.379696\pi\)
\(602\) 0 0
\(603\) −406.153 838.694i −0.673554 1.39087i
\(604\) 0 0
\(605\) −421.737 + 53.1516i −0.697086 + 0.0878539i
\(606\) 0 0
\(607\) 483.158i 0.795977i −0.917390 0.397988i \(-0.869708\pi\)
0.917390 0.397988i \(-0.130292\pi\)
\(608\) 0 0
\(609\) 171.593 273.751i 0.281761 0.449510i
\(610\) 0 0
\(611\) 584.818 0.957150
\(612\) 0 0
\(613\) 115.677 0.188706 0.0943532 0.995539i \(-0.469922\pi\)
0.0943532 + 0.995539i \(0.469922\pi\)
\(614\) 0 0
\(615\) 118.752 255.863i 0.193093 0.416038i
\(616\) 0 0
\(617\) 91.9068 0.148958 0.0744788 0.997223i \(-0.476271\pi\)
0.0744788 + 0.997223i \(0.476271\pi\)
\(618\) 0 0
\(619\) 748.627i 1.20941i 0.796448 + 0.604706i \(0.206710\pi\)
−0.796448 + 0.604706i \(0.793290\pi\)
\(620\) 0 0
\(621\) 63.1857 577.797i 0.101748 0.930430i
\(622\) 0 0
\(623\) −869.979 −1.39643
\(624\) 0 0
\(625\) 548.046 300.451i 0.876873 0.480721i
\(626\) 0 0
\(627\) −283.038 + 451.547i −0.451417 + 0.720171i
\(628\) 0 0
\(629\) −191.865 −0.305032
\(630\) 0 0
\(631\) −856.682 −1.35766 −0.678829 0.734296i \(-0.737512\pi\)
−0.678829 + 0.734296i \(0.737512\pi\)
\(632\) 0 0
\(633\) 322.292 514.170i 0.509150 0.812276i
\(634\) 0 0
\(635\) 42.9062 + 340.443i 0.0675687 + 0.536131i
\(636\) 0 0
\(637\) 214.672 0.337004
\(638\) 0 0
\(639\) −117.929 + 57.1094i −0.184553 + 0.0893730i
\(640\) 0 0
\(641\) 29.0978i 0.0453944i −0.999742 0.0226972i \(-0.992775\pi\)
0.999742 0.0226972i \(-0.00722536\pi\)
\(642\) 0 0
\(643\) 873.920 1.35913 0.679564 0.733616i \(-0.262169\pi\)
0.679564 + 0.733616i \(0.262169\pi\)
\(644\) 0 0
\(645\) 368.771 + 171.156i 0.571738 + 0.265357i
\(646\) 0 0
\(647\) 1152.62 1.78149 0.890743 0.454507i \(-0.150184\pi\)
0.890743 + 0.454507i \(0.150184\pi\)
\(648\) 0 0
\(649\) −502.572 −0.774379
\(650\) 0 0
\(651\) −80.9652 + 129.168i −0.124370 + 0.198415i
\(652\) 0 0
\(653\) 143.894i 0.220358i 0.993912 + 0.110179i \(0.0351424\pi\)
−0.993912 + 0.110179i \(0.964858\pi\)
\(654\) 0 0
\(655\) 875.635 110.357i 1.33685 0.168483i
\(656\) 0 0
\(657\) 776.781 376.170i 1.18231 0.572557i
\(658\) 0 0
\(659\) −410.806 −0.623379 −0.311689 0.950184i \(-0.600895\pi\)
−0.311689 + 0.950184i \(0.600895\pi\)
\(660\) 0 0
\(661\) 1129.32i 1.70851i 0.519855 + 0.854254i \(0.325986\pi\)
−0.519855 + 0.854254i \(0.674014\pi\)
\(662\) 0 0
\(663\) −564.757 354.000i −0.851820 0.533937i
\(664\) 0 0
\(665\) 872.220 109.926i 1.31161 0.165302i
\(666\) 0 0
\(667\) −390.472 −0.585415
\(668\) 0 0
\(669\) −54.4500 + 86.8672i −0.0813902 + 0.129846i
\(670\) 0 0
\(671\) 235.948i 0.351636i
\(672\) 0 0
\(673\) 630.561i 0.936940i 0.883479 + 0.468470i \(0.155195\pi\)
−0.883479 + 0.468470i \(0.844805\pi\)
\(674\) 0 0
\(675\) −95.4047 668.224i −0.141340 0.989961i
\(676\) 0 0
\(677\) 412.121i 0.608746i 0.952553 + 0.304373i \(0.0984470\pi\)
−0.952553 + 0.304373i \(0.901553\pi\)
\(678\) 0 0
\(679\) −118.641 −0.174729
\(680\) 0 0
\(681\) 504.552 804.941i 0.740899 1.18200i
\(682\) 0 0
\(683\) 538.782i 0.788847i −0.918929 0.394423i \(-0.870944\pi\)
0.918929 0.394423i \(-0.129056\pi\)
\(684\) 0 0
\(685\) 477.931 60.2338i 0.697710 0.0879326i
\(686\) 0 0
\(687\) −376.036 + 599.912i −0.547360 + 0.873234i
\(688\) 0 0
\(689\) 1461.52i 2.12122i
\(690\) 0 0
\(691\) 617.899i 0.894210i −0.894482 0.447105i \(-0.852455\pi\)
0.894482 0.447105i \(-0.147545\pi\)
\(692\) 0 0
\(693\) −288.507 + 139.715i −0.416316 + 0.201609i
\(694\) 0 0
\(695\) −12.6218 100.149i −0.0181609 0.144100i
\(696\) 0 0
\(697\) 267.552i 0.383862i
\(698\) 0 0
\(699\) 496.732 + 311.361i 0.710633 + 0.445438i
\(700\) 0 0
\(701\) 583.091 0.831799 0.415900 0.909410i \(-0.363467\pi\)
0.415900 + 0.909410i \(0.363467\pi\)
\(702\) 0 0
\(703\) 399.343i 0.568055i
\(704\) 0 0
\(705\) −509.542 236.491i −0.722754 0.335448i
\(706\) 0 0
\(707\) 686.687i 0.971268i
\(708\) 0 0
\(709\) 291.529i 0.411184i −0.978638 0.205592i \(-0.934088\pi\)
0.978638 0.205592i \(-0.0659119\pi\)
\(710\) 0 0
\(711\) 522.317 + 1078.57i 0.734623 + 1.51698i
\(712\) 0 0
\(713\) 184.242 0.258404
\(714\) 0 0
\(715\) −464.710 + 58.5675i −0.649944 + 0.0819127i
\(716\) 0 0
\(717\) 15.4826 24.7002i 0.0215935 0.0344494i
\(718\) 0 0
\(719\) 1189.83i 1.65483i 0.561588 + 0.827417i \(0.310191\pi\)
−0.561588 + 0.827417i \(0.689809\pi\)
\(720\) 0 0
\(721\) −643.619 −0.892675
\(722\) 0 0
\(723\) −509.887 319.607i −0.705238 0.442057i
\(724\) 0 0
\(725\) −439.278 + 112.512i −0.605900 + 0.155188i
\(726\) 0 0
\(727\) 332.813i 0.457790i −0.973451 0.228895i \(-0.926489\pi\)
0.973451 0.228895i \(-0.0735112\pi\)
\(728\) 0 0
\(729\) −711.770 157.557i −0.976365 0.216128i
\(730\) 0 0
\(731\) −385.617 −0.527520
\(732\) 0 0
\(733\) 430.132 0.586810 0.293405 0.955988i \(-0.405212\pi\)
0.293405 + 0.955988i \(0.405212\pi\)
\(734\) 0 0
\(735\) −187.039 86.8095i −0.254475 0.118108i
\(736\) 0 0
\(737\) 621.115 0.842762
\(738\) 0 0
\(739\) 824.829i 1.11614i −0.829793 0.558071i \(-0.811542\pi\)
0.829793 0.558071i \(-0.188458\pi\)
\(740\) 0 0
\(741\) 736.807 1175.47i 0.994341 1.58633i
\(742\) 0 0
\(743\) 848.265 1.14168 0.570838 0.821063i \(-0.306619\pi\)
0.570838 + 0.821063i \(0.306619\pi\)
\(744\) 0 0
\(745\) −688.351 + 86.7531i −0.923961 + 0.116447i
\(746\) 0 0
\(747\) 945.414 457.834i 1.26561 0.612897i
\(748\) 0 0
\(749\) −440.225 −0.587751
\(750\) 0 0
\(751\) −63.2095 −0.0841671 −0.0420836 0.999114i \(-0.513400\pi\)
−0.0420836 + 0.999114i \(0.513400\pi\)
\(752\) 0 0
\(753\) −53.0492 33.2522i −0.0704504 0.0441597i
\(754\) 0 0
\(755\) −612.659 + 77.2136i −0.811469 + 0.102270i
\(756\) 0 0
\(757\) −1255.62 −1.65868 −0.829340 0.558744i \(-0.811283\pi\)
−0.829340 + 0.558744i \(0.811283\pi\)
\(758\) 0 0
\(759\) 328.259 + 205.759i 0.432489 + 0.271093i
\(760\) 0 0
\(761\) 819.598i 1.07700i 0.842625 + 0.538501i \(0.181009\pi\)
−0.842625 + 0.538501i \(0.818991\pi\)
\(762\) 0 0
\(763\) −89.1880 −0.116891
\(764\) 0 0
\(765\) 348.910 + 536.812i 0.456092 + 0.701715i
\(766\) 0 0
\(767\) 1308.30 1.70573
\(768\) 0 0
\(769\) 275.299 0.357997 0.178998 0.983849i \(-0.442714\pi\)
0.178998 + 0.983849i \(0.442714\pi\)
\(770\) 0 0
\(771\) 1159.29 + 726.667i 1.50362 + 0.942499i
\(772\) 0 0
\(773\) 310.696i 0.401935i −0.979598 0.200968i \(-0.935591\pi\)
0.979598 0.200968i \(-0.0644086\pi\)
\(774\) 0 0
\(775\) 207.271 53.0881i 0.267446 0.0685007i
\(776\) 0 0
\(777\) 127.576 203.529i 0.164191 0.261942i
\(778\) 0 0
\(779\) −556.875 −0.714858
\(780\) 0 0
\(781\) 87.3354i 0.111825i
\(782\) 0 0
\(783\) −53.2381 + 486.832i −0.0679924 + 0.621752i
\(784\) 0 0
\(785\) −1157.54 + 145.885i −1.47457 + 0.185840i
\(786\) 0 0
\(787\) −882.041 −1.12076 −0.560382 0.828234i \(-0.689346\pi\)
−0.560382 + 0.828234i \(0.689346\pi\)
\(788\) 0 0
\(789\) 410.344 + 257.212i 0.520082 + 0.325997i
\(790\) 0 0
\(791\) 974.682i 1.23221i
\(792\) 0 0
\(793\) 614.220i 0.774552i
\(794\) 0 0
\(795\) 591.014 1273.40i 0.743414 1.60176i
\(796\) 0 0
\(797\) 572.739i 0.718619i 0.933218 + 0.359309i \(0.116988\pi\)
−0.933218 + 0.359309i \(0.883012\pi\)
\(798\) 0 0
\(799\) 532.818 0.666857
\(800\) 0 0
\(801\) 1186.87 574.765i 1.48174 0.717559i
\(802\) 0 0
\(803\) 575.264i 0.716394i
\(804\) 0 0
\(805\) −79.9125 634.074i −0.0992702 0.787670i
\(806\) 0 0
\(807\) −328.249 205.753i −0.406752 0.254960i
\(808\) 0 0
\(809\) 1122.50i 1.38751i −0.720210 0.693756i \(-0.755955\pi\)
0.720210 0.693756i \(-0.244045\pi\)
\(810\) 0 0
\(811\) 76.3000i 0.0940814i −0.998893 0.0470407i \(-0.985021\pi\)
0.998893 0.0470407i \(-0.0149791\pi\)
\(812\) 0 0
\(813\) 586.358 + 367.541i 0.721228 + 0.452080i
\(814\) 0 0
\(815\) −489.997 + 61.7544i −0.601223 + 0.0757723i
\(816\) 0 0
\(817\) 802.613i 0.982391i
\(818\) 0 0
\(819\) 751.042 363.706i 0.917023 0.444085i
\(820\) 0 0
\(821\) 499.762 0.608724 0.304362 0.952557i \(-0.401557\pi\)
0.304362 + 0.952557i \(0.401557\pi\)
\(822\) 0 0
\(823\) 1081.64i 1.31426i 0.753777 + 0.657131i \(0.228230\pi\)
−0.753777 + 0.657131i \(0.771770\pi\)
\(824\) 0 0
\(825\) 428.577 + 136.892i 0.519487 + 0.165930i
\(826\) 0 0
\(827\) 221.353i 0.267658i 0.991004 + 0.133829i \(0.0427272\pi\)
−0.991004 + 0.133829i \(0.957273\pi\)
\(828\) 0 0
\(829\) 1010.54i 1.21899i −0.792789 0.609496i \(-0.791372\pi\)
0.792789 0.609496i \(-0.208628\pi\)
\(830\) 0 0
\(831\) 901.094 + 564.823i 1.08435 + 0.679691i
\(832\) 0 0
\(833\) 195.584 0.234794
\(834\) 0 0
\(835\) 660.127 83.1959i 0.790571 0.0996358i
\(836\) 0 0
\(837\) 25.1201 229.709i 0.0300121 0.274443i
\(838\) 0 0
\(839\) 1554.32i 1.85259i −0.376803 0.926294i \(-0.622976\pi\)
0.376803 0.926294i \(-0.377024\pi\)
\(840\) 0 0
\(841\) −512.002 −0.608802
\(842\) 0 0
\(843\) −558.913 + 891.665i −0.663004 + 1.05773i
\(844\) 0 0
\(845\) 371.367 46.8035i 0.439488 0.0553887i
\(846\) 0 0
\(847\) 504.769i 0.595949i
\(848\) 0 0
\(849\) −645.784 404.790i −0.760641 0.476785i
\(850\) 0 0
\(851\) −290.309 −0.341138
\(852\) 0 0
\(853\) −79.6953 −0.0934295 −0.0467147 0.998908i \(-0.514875\pi\)
−0.0467147 + 0.998908i \(0.514875\pi\)
\(854\) 0 0
\(855\) −1117.31 + 726.212i −1.30679 + 0.849371i
\(856\) 0 0
\(857\) 169.805 0.198139 0.0990693 0.995081i \(-0.468413\pi\)
0.0990693 + 0.995081i \(0.468413\pi\)
\(858\) 0 0
\(859\) 811.495i 0.944698i −0.881412 0.472349i \(-0.843406\pi\)
0.881412 0.472349i \(-0.156594\pi\)
\(860\) 0 0
\(861\) −283.817 177.902i −0.329636 0.206622i
\(862\) 0 0
\(863\) 925.135 1.07200 0.535999 0.844218i \(-0.319935\pi\)
0.535999 + 0.844218i \(0.319935\pi\)
\(864\) 0 0
\(865\) −202.110 1603.66i −0.233653 1.85394i
\(866\) 0 0
\(867\) 220.073 + 137.946i 0.253833 + 0.159107i
\(868\) 0 0
\(869\) −798.762 −0.919173
\(870\) 0 0
\(871\) −1616.89 −1.85636
\(872\) 0 0
\(873\) 161.856 78.3819i 0.185402 0.0897845i
\(874\) 0 0
\(875\) −272.605 690.302i −0.311548 0.788917i
\(876\) 0 0
\(877\) 425.026 0.484636 0.242318 0.970197i \(-0.422092\pi\)
0.242318 + 0.970197i \(0.422092\pi\)
\(878\) 0 0
\(879\) 316.033 504.185i 0.359537 0.573590i
\(880\) 0 0
\(881\) 41.8218i 0.0474709i 0.999718 + 0.0237354i \(0.00755593\pi\)
−0.999718 + 0.0237354i \(0.992444\pi\)
\(882\) 0 0
\(883\) −549.203 −0.621973 −0.310987 0.950414i \(-0.600659\pi\)
−0.310987 + 0.950414i \(0.600659\pi\)
\(884\) 0 0
\(885\) −1139.90 529.053i −1.28802 0.597800i
\(886\) 0 0
\(887\) −1315.99 −1.48364 −0.741819 0.670600i \(-0.766037\pi\)
−0.741819 + 0.670600i \(0.766037\pi\)
\(888\) 0 0
\(889\) 407.470 0.458346
\(890\) 0 0
\(891\) 301.292 381.213i 0.338150 0.427848i
\(892\) 0 0
\(893\) 1108.99i 1.24187i
\(894\) 0 0
\(895\) 911.019 114.816i 1.01790 0.128286i
\(896\) 0 0
\(897\) −854.526 535.633i −0.952648 0.597139i
\(898\) 0 0
\(899\) −155.236 −0.172676
\(900\) 0 0
\(901\) 1331.57i 1.47788i
\(902\) 0 0
\(903\) 256.407 409.060i 0.283950 0.453001i
\(904\) 0 0
\(905\) 188.570 + 1496.23i 0.208365 + 1.65329i
\(906\) 0 0
\(907\) 103.221 0.113804 0.0569022 0.998380i \(-0.481878\pi\)
0.0569022 + 0.998380i \(0.481878\pi\)
\(908\) 0 0
\(909\) 453.670 + 936.815i 0.499087 + 1.03060i
\(910\) 0 0
\(911\) 937.234i 1.02880i 0.857551 + 0.514399i \(0.171985\pi\)
−0.857551 + 0.514399i \(0.828015\pi\)
\(912\) 0 0
\(913\) 700.149i 0.766867i
\(914\) 0 0
\(915\) −248.380 + 535.158i −0.271454 + 0.584873i
\(916\) 0 0
\(917\) 1048.03i 1.14289i
\(918\) 0 0
\(919\) 565.469 0.615309 0.307655 0.951498i \(-0.400456\pi\)
0.307655 + 0.951498i \(0.400456\pi\)
\(920\) 0 0
\(921\) −584.439 366.337i −0.634570 0.397761i
\(922\) 0 0
\(923\) 227.352i 0.246318i
\(924\) 0 0
\(925\) −326.595 + 83.6504i −0.353076 + 0.0904328i
\(926\) 0 0
\(927\) 878.060 425.217i 0.947206 0.458702i
\(928\) 0 0
\(929\) 561.412i 0.604318i −0.953258 0.302159i \(-0.902293\pi\)
0.953258 0.302159i \(-0.0977074\pi\)
\(930\) 0 0
\(931\) 407.083i 0.437253i
\(932\) 0 0
\(933\) −68.6490 + 109.520i −0.0735787 + 0.117384i
\(934\) 0 0
\(935\) −423.390 + 53.3599i −0.452823 + 0.0570694i
\(936\) 0 0
\(937\) 1293.38i 1.38034i 0.723647 + 0.690170i \(0.242464\pi\)
−0.723647 + 0.690170i \(0.757536\pi\)
\(938\) 0 0
\(939\) 348.519 556.011i 0.371159 0.592131i
\(940\) 0 0
\(941\) −1509.76 −1.60442 −0.802209 0.597043i \(-0.796342\pi\)
−0.802209 + 0.597043i \(0.796342\pi\)
\(942\) 0 0
\(943\) 404.829i 0.429299i
\(944\) 0 0
\(945\) −801.445 + 13.1817i −0.848090 + 0.0139489i
\(946\) 0 0
\(947\) 368.756i 0.389394i 0.980863 + 0.194697i \(0.0623723\pi\)
−0.980863 + 0.194697i \(0.937628\pi\)
\(948\) 0 0
\(949\) 1497.53i 1.57801i
\(950\) 0 0
\(951\) −518.327 + 826.917i −0.545034 + 0.869523i
\(952\) 0 0
\(953\) −1580.92 −1.65889 −0.829443 0.558591i \(-0.811342\pi\)
−0.829443 + 0.558591i \(0.811342\pi\)
\(954\) 0 0
\(955\) 151.692 + 1203.62i 0.158840 + 1.26033i
\(956\) 0 0
\(957\) −276.580 173.366i −0.289007 0.181155i
\(958\) 0 0
\(959\) 572.027i 0.596483i
\(960\) 0 0
\(961\) −887.753 −0.923780
\(962\) 0 0
\(963\) 600.579 290.841i 0.623654 0.302016i
\(964\) 0 0
\(965\) −8.55888 67.9113i −0.00886931 0.0703744i
\(966\) 0 0
\(967\) 653.389i 0.675686i −0.941202 0.337843i \(-0.890303\pi\)
0.941202 0.337843i \(-0.109697\pi\)
\(968\) 0 0
\(969\) 671.292 1070.95i 0.692768 1.10521i
\(970\) 0 0
\(971\) −414.882 −0.427273 −0.213636 0.976913i \(-0.568531\pi\)
−0.213636 + 0.976913i \(0.568531\pi\)
\(972\) 0 0
\(973\) −119.867 −0.123193
\(974\) 0 0
\(975\) −1115.67 356.358i −1.14428 0.365495i
\(976\) 0 0
\(977\) −829.934 −0.849472 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(978\) 0 0
\(979\) 878.968i 0.897822i
\(980\) 0 0
\(981\) 121.675 58.9234i 0.124032 0.0600646i
\(982\) 0 0
\(983\) 43.3930 0.0441435 0.0220717 0.999756i \(-0.492974\pi\)
0.0220717 + 0.999756i \(0.492974\pi\)
\(984\) 0 0
\(985\) 226.023 + 1793.40i 0.229465 + 1.82071i
\(986\) 0 0
\(987\) −354.284 + 565.210i −0.358951 + 0.572654i
\(988\) 0 0
\(989\) −583.472 −0.589962
\(990\) 0 0
\(991\) 325.761 0.328719 0.164360 0.986400i \(-0.447444\pi\)
0.164360 + 0.986400i \(0.447444\pi\)
\(992\) 0 0
\(993\) −500.523 + 798.513i −0.504052 + 0.804142i
\(994\) 0 0
\(995\) −976.586 + 123.079i −0.981493 + 0.123698i
\(996\) 0 0
\(997\) 487.259 0.488725 0.244363 0.969684i \(-0.421421\pi\)
0.244363 + 0.969684i \(0.421421\pi\)
\(998\) 0 0
\(999\) −39.5815 + 361.951i −0.0396212 + 0.362313i
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 960.3.i.b.929.11 yes 64
3.2 odd 2 inner 960.3.i.b.929.14 yes 64
4.3 odd 2 inner 960.3.i.b.929.55 yes 64
5.4 even 2 inner 960.3.i.b.929.56 yes 64
8.3 odd 2 inner 960.3.i.b.929.10 yes 64
8.5 even 2 inner 960.3.i.b.929.54 yes 64
12.11 even 2 inner 960.3.i.b.929.50 yes 64
15.14 odd 2 inner 960.3.i.b.929.49 yes 64
20.19 odd 2 inner 960.3.i.b.929.12 yes 64
24.5 odd 2 inner 960.3.i.b.929.51 yes 64
24.11 even 2 inner 960.3.i.b.929.15 yes 64
40.19 odd 2 inner 960.3.i.b.929.53 yes 64
40.29 even 2 inner 960.3.i.b.929.9 64
60.59 even 2 inner 960.3.i.b.929.13 yes 64
120.29 odd 2 inner 960.3.i.b.929.16 yes 64
120.59 even 2 inner 960.3.i.b.929.52 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.3.i.b.929.9 64 40.29 even 2 inner
960.3.i.b.929.10 yes 64 8.3 odd 2 inner
960.3.i.b.929.11 yes 64 1.1 even 1 trivial
960.3.i.b.929.12 yes 64 20.19 odd 2 inner
960.3.i.b.929.13 yes 64 60.59 even 2 inner
960.3.i.b.929.14 yes 64 3.2 odd 2 inner
960.3.i.b.929.15 yes 64 24.11 even 2 inner
960.3.i.b.929.16 yes 64 120.29 odd 2 inner
960.3.i.b.929.49 yes 64 15.14 odd 2 inner
960.3.i.b.929.50 yes 64 12.11 even 2 inner
960.3.i.b.929.51 yes 64 24.5 odd 2 inner
960.3.i.b.929.52 yes 64 120.59 even 2 inner
960.3.i.b.929.53 yes 64 40.19 odd 2 inner
960.3.i.b.929.54 yes 64 8.5 even 2 inner
960.3.i.b.929.55 yes 64 4.3 odd 2 inner
960.3.i.b.929.56 yes 64 5.4 even 2 inner