Properties

Label 384.3.h.g.65.2
Level $384$
Weight $3$
Character 384.65
Analytic conductor $10.463$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,3,Mod(65,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("384.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 274x^{12} + 1236x^{10} + 2703x^{8} + 2676x^{6} + 946x^{4} + 64x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.2
Root \(2.24296i\) of defining polynomial
Character \(\chi\) \(=\) 384.65
Dual form 384.3.h.g.65.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.75782 - 1.18087i) q^{3} +3.68481 q^{5} -5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} +O(q^{10})\) \(q+(-2.75782 - 1.18087i) q^{3} +3.68481 q^{5} -5.43855 q^{7} +(6.21110 + 6.51323i) q^{9} -1.16436 q^{11} -14.4222i q^{13} +(-10.1620 - 4.35127i) q^{15} +1.71276i q^{17} -28.8394i q^{19} +(14.9985 + 6.42221i) q^{21} +35.4225i q^{23} -11.4222 q^{25} +(-9.43781 - 25.2968i) q^{27} -33.6818 q^{29} -55.5336 q^{31} +(3.21110 + 1.37496i) q^{33} -20.0400 q^{35} +30.4222i q^{37} +(-17.0307 + 39.7738i) q^{39} -63.4196i q^{41} -43.0098i q^{43} +(22.8867 + 24.0000i) q^{45} -61.5301i q^{47} -19.4222 q^{49} +(2.02254 - 4.72347i) q^{51} -56.3093 q^{53} -4.29046 q^{55} +(-34.0555 + 79.5338i) q^{57} -49.6407 q^{59} -4.73338i q^{61} +(-33.7794 - 35.4225i) q^{63} -53.1430i q^{65} +7.08521i q^{67} +(41.8293 - 97.6888i) q^{69} +96.9527i q^{71} +68.5332 q^{73} +(31.5003 + 13.4881i) q^{75} +6.33245 q^{77} +9.72900 q^{79} +(-3.84441 + 80.9087i) q^{81} +38.9156 q^{83} +6.31118i q^{85} +(92.8883 + 39.7738i) q^{87} +0.675595i q^{89} +78.4358i q^{91} +(153.151 + 65.5778i) q^{93} -106.268i q^{95} +86.1110 q^{97} +(-7.23199 - 7.58378i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 48 q^{25} - 64 q^{33} - 80 q^{49} + 32 q^{57} - 288 q^{73} + 400 q^{81} + 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −2.75782 1.18087i −0.919272 0.393623i
\(4\) 0 0
\(5\) 3.68481 0.736961 0.368481 0.929635i \(-0.379878\pi\)
0.368481 + 0.929635i \(0.379878\pi\)
\(6\) 0 0
\(7\) −5.43855 −0.776935 −0.388468 0.921462i \(-0.626995\pi\)
−0.388468 + 0.921462i \(0.626995\pi\)
\(8\) 0 0
\(9\) 6.21110 + 6.51323i 0.690123 + 0.723693i
\(10\) 0 0
\(11\) −1.16436 −0.105851 −0.0529256 0.998598i \(-0.516855\pi\)
−0.0529256 + 0.998598i \(0.516855\pi\)
\(12\) 0 0
\(13\) 14.4222i 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) −10.1620 4.35127i −0.677468 0.290085i
\(16\) 0 0
\(17\) 1.71276i 0.100750i 0.998730 + 0.0503752i \(0.0160417\pi\)
−0.998730 + 0.0503752i \(0.983958\pi\)
\(18\) 0 0
\(19\) 28.8394i 1.51786i −0.651171 0.758931i \(-0.725722\pi\)
0.651171 0.758931i \(-0.274278\pi\)
\(20\) 0 0
\(21\) 14.9985 + 6.42221i 0.714215 + 0.305819i
\(22\) 0 0
\(23\) 35.4225i 1.54011i 0.637978 + 0.770055i \(0.279771\pi\)
−0.637978 + 0.770055i \(0.720229\pi\)
\(24\) 0 0
\(25\) −11.4222 −0.456888
\(26\) 0 0
\(27\) −9.43781 25.2968i −0.349549 0.936918i
\(28\) 0 0
\(29\) −33.6818 −1.16144 −0.580721 0.814102i \(-0.697229\pi\)
−0.580721 + 0.814102i \(0.697229\pi\)
\(30\) 0 0
\(31\) −55.5336 −1.79141 −0.895703 0.444654i \(-0.853327\pi\)
−0.895703 + 0.444654i \(0.853327\pi\)
\(32\) 0 0
\(33\) 3.21110 + 1.37496i 0.0973061 + 0.0416655i
\(34\) 0 0
\(35\) −20.0400 −0.572571
\(36\) 0 0
\(37\) 30.4222i 0.822222i 0.911585 + 0.411111i \(0.134859\pi\)
−0.911585 + 0.411111i \(0.865141\pi\)
\(38\) 0 0
\(39\) −17.0307 + 39.7738i −0.436685 + 1.01984i
\(40\) 0 0
\(41\) 63.4196i 1.54682i −0.633907 0.773409i \(-0.718550\pi\)
0.633907 0.773409i \(-0.281450\pi\)
\(42\) 0 0
\(43\) 43.0098i 1.00023i −0.865960 0.500114i \(-0.833291\pi\)
0.865960 0.500114i \(-0.166709\pi\)
\(44\) 0 0
\(45\) 22.8867 + 24.0000i 0.508594 + 0.533333i
\(46\) 0 0
\(47\) 61.5301i 1.30915i −0.755996 0.654576i \(-0.772847\pi\)
0.755996 0.654576i \(-0.227153\pi\)
\(48\) 0 0
\(49\) −19.4222 −0.396372
\(50\) 0 0
\(51\) 2.02254 4.72347i 0.0396577 0.0926171i
\(52\) 0 0
\(53\) −56.3093 −1.06244 −0.531219 0.847234i \(-0.678266\pi\)
−0.531219 + 0.847234i \(0.678266\pi\)
\(54\) 0 0
\(55\) −4.29046 −0.0780083
\(56\) 0 0
\(57\) −34.0555 + 79.5338i −0.597465 + 1.39533i
\(58\) 0 0
\(59\) −49.6407 −0.841368 −0.420684 0.907207i \(-0.638210\pi\)
−0.420684 + 0.907207i \(0.638210\pi\)
\(60\) 0 0
\(61\) 4.73338i 0.0775965i −0.999247 0.0387982i \(-0.987647\pi\)
0.999247 0.0387982i \(-0.0123530\pi\)
\(62\) 0 0
\(63\) −33.7794 35.4225i −0.536181 0.562262i
\(64\) 0 0
\(65\) 53.1430i 0.817585i
\(66\) 0 0
\(67\) 7.08521i 0.105749i 0.998601 + 0.0528747i \(0.0168384\pi\)
−0.998601 + 0.0528747i \(0.983162\pi\)
\(68\) 0 0
\(69\) 41.8293 97.6888i 0.606222 1.41578i
\(70\) 0 0
\(71\) 96.9527i 1.36553i 0.730638 + 0.682765i \(0.239223\pi\)
−0.730638 + 0.682765i \(0.760777\pi\)
\(72\) 0 0
\(73\) 68.5332 0.938811 0.469406 0.882983i \(-0.344468\pi\)
0.469406 + 0.882983i \(0.344468\pi\)
\(74\) 0 0
\(75\) 31.5003 + 13.4881i 0.420005 + 0.179842i
\(76\) 0 0
\(77\) 6.33245 0.0822396
\(78\) 0 0
\(79\) 9.72900 0.123152 0.0615760 0.998102i \(-0.480387\pi\)
0.0615760 + 0.998102i \(0.480387\pi\)
\(80\) 0 0
\(81\) −3.84441 + 80.9087i −0.0474619 + 0.998873i
\(82\) 0 0
\(83\) 38.9156 0.468863 0.234431 0.972133i \(-0.424677\pi\)
0.234431 + 0.972133i \(0.424677\pi\)
\(84\) 0 0
\(85\) 6.31118i 0.0742492i
\(86\) 0 0
\(87\) 92.8883 + 39.7738i 1.06768 + 0.457170i
\(88\) 0 0
\(89\) 0.675595i 0.00759096i 0.999993 + 0.00379548i \(0.00120814\pi\)
−0.999993 + 0.00379548i \(0.998792\pi\)
\(90\) 0 0
\(91\) 78.4358i 0.861932i
\(92\) 0 0
\(93\) 153.151 + 65.5778i 1.64679 + 0.705138i
\(94\) 0 0
\(95\) 106.268i 1.11861i
\(96\) 0 0
\(97\) 86.1110 0.887743 0.443871 0.896091i \(-0.353605\pi\)
0.443871 + 0.896091i \(0.353605\pi\)
\(98\) 0 0
\(99\) −7.23199 7.58378i −0.0730504 0.0766038i
\(100\) 0 0
\(101\) −109.452 −1.08369 −0.541843 0.840480i \(-0.682273\pi\)
−0.541843 + 0.840480i \(0.682273\pi\)
\(102\) 0 0
\(103\) 160.014 1.55353 0.776767 0.629788i \(-0.216858\pi\)
0.776767 + 0.629788i \(0.216858\pi\)
\(104\) 0 0
\(105\) 55.2666 + 23.6646i 0.526349 + 0.225377i
\(106\) 0 0
\(107\) 184.345 1.72285 0.861423 0.507887i \(-0.169573\pi\)
0.861423 + 0.507887i \(0.169573\pi\)
\(108\) 0 0
\(109\) 68.7334i 0.630582i −0.948995 0.315291i \(-0.897898\pi\)
0.948995 0.315291i \(-0.102102\pi\)
\(110\) 0 0
\(111\) 35.9246 83.8989i 0.323645 0.755846i
\(112\) 0 0
\(113\) 71.3078i 0.631042i −0.948919 0.315521i \(-0.897821\pi\)
0.948919 0.315521i \(-0.102179\pi\)
\(114\) 0 0
\(115\) 130.525i 1.13500i
\(116\) 0 0
\(117\) 93.9352 89.5778i 0.802865 0.765622i
\(118\) 0 0
\(119\) 9.31491i 0.0782766i
\(120\) 0 0
\(121\) −119.644 −0.988796
\(122\) 0 0
\(123\) −74.8901 + 174.900i −0.608863 + 1.42195i
\(124\) 0 0
\(125\) −134.209 −1.07367
\(126\) 0 0
\(127\) 88.1648 0.694211 0.347106 0.937826i \(-0.387165\pi\)
0.347106 + 0.937826i \(0.387165\pi\)
\(128\) 0 0
\(129\) −50.7889 + 118.613i −0.393712 + 0.919482i
\(130\) 0 0
\(131\) −9.56071 −0.0729825 −0.0364913 0.999334i \(-0.511618\pi\)
−0.0364913 + 0.999334i \(0.511618\pi\)
\(132\) 0 0
\(133\) 156.844i 1.17928i
\(134\) 0 0
\(135\) −34.7765 93.2138i −0.257604 0.690472i
\(136\) 0 0
\(137\) 150.504i 1.09857i −0.835636 0.549284i \(-0.814901\pi\)
0.835636 0.549284i \(-0.185099\pi\)
\(138\) 0 0
\(139\) 51.5907i 0.371156i 0.982630 + 0.185578i \(0.0594158\pi\)
−0.982630 + 0.185578i \(0.940584\pi\)
\(140\) 0 0
\(141\) −72.6590 + 169.689i −0.515312 + 1.20347i
\(142\) 0 0
\(143\) 16.7927i 0.117431i
\(144\) 0 0
\(145\) −124.111 −0.855938
\(146\) 0 0
\(147\) 53.5629 + 22.9351i 0.364373 + 0.156021i
\(148\) 0 0
\(149\) −25.7936 −0.173112 −0.0865558 0.996247i \(-0.527586\pi\)
−0.0865558 + 0.996247i \(0.527586\pi\)
\(150\) 0 0
\(151\) 16.3156 0.108051 0.0540253 0.998540i \(-0.482795\pi\)
0.0540253 + 0.998540i \(0.482795\pi\)
\(152\) 0 0
\(153\) −11.1556 + 10.6381i −0.0729124 + 0.0695302i
\(154\) 0 0
\(155\) −204.630 −1.32020
\(156\) 0 0
\(157\) 251.489i 1.60184i −0.598772 0.800919i \(-0.704345\pi\)
0.598772 0.800919i \(-0.295655\pi\)
\(158\) 0 0
\(159\) 155.291 + 66.4938i 0.976670 + 0.418200i
\(160\) 0 0
\(161\) 192.647i 1.19657i
\(162\) 0 0
\(163\) 99.6915i 0.611604i 0.952095 + 0.305802i \(0.0989246\pi\)
−0.952095 + 0.305802i \(0.901075\pi\)
\(164\) 0 0
\(165\) 11.8323 + 5.06646i 0.0717108 + 0.0307058i
\(166\) 0 0
\(167\) 158.483i 0.948999i −0.880256 0.474499i \(-0.842629\pi\)
0.880256 0.474499i \(-0.157371\pi\)
\(168\) 0 0
\(169\) −39.0000 −0.230769
\(170\) 0 0
\(171\) 187.838 179.124i 1.09847 1.04751i
\(172\) 0 0
\(173\) −117.340 −0.678269 −0.339134 0.940738i \(-0.610134\pi\)
−0.339134 + 0.940738i \(0.610134\pi\)
\(174\) 0 0
\(175\) 62.1202 0.354973
\(176\) 0 0
\(177\) 136.900 + 58.6191i 0.773446 + 0.331181i
\(178\) 0 0
\(179\) 144.265 0.805948 0.402974 0.915212i \(-0.367977\pi\)
0.402974 + 0.915212i \(0.367977\pi\)
\(180\) 0 0
\(181\) 289.800i 1.60110i 0.599263 + 0.800552i \(0.295460\pi\)
−0.599263 + 0.800552i \(0.704540\pi\)
\(182\) 0 0
\(183\) −5.58950 + 13.0538i −0.0305437 + 0.0713323i
\(184\) 0 0
\(185\) 112.100i 0.605946i
\(186\) 0 0
\(187\) 1.99427i 0.0106646i
\(188\) 0 0
\(189\) 51.3280 + 137.578i 0.271577 + 0.727925i
\(190\) 0 0
\(191\) 151.005i 0.790602i 0.918552 + 0.395301i \(0.129360\pi\)
−0.918552 + 0.395301i \(0.870640\pi\)
\(192\) 0 0
\(193\) −2.22205 −0.0115132 −0.00575661 0.999983i \(-0.501832\pi\)
−0.00575661 + 0.999983i \(0.501832\pi\)
\(194\) 0 0
\(195\) −62.7549 + 146.559i −0.321820 + 0.751583i
\(196\) 0 0
\(197\) 174.742 0.887013 0.443507 0.896271i \(-0.353734\pi\)
0.443507 + 0.896271i \(0.353734\pi\)
\(198\) 0 0
\(199\) −140.556 −0.706312 −0.353156 0.935564i \(-0.614891\pi\)
−0.353156 + 0.935564i \(0.614891\pi\)
\(200\) 0 0
\(201\) 8.36669 19.5397i 0.0416253 0.0972124i
\(202\) 0 0
\(203\) 183.180 0.902366
\(204\) 0 0
\(205\) 233.689i 1.13995i
\(206\) 0 0
\(207\) −230.715 + 220.013i −1.11457 + 1.06286i
\(208\) 0 0
\(209\) 33.5796i 0.160668i
\(210\) 0 0
\(211\) 295.479i 1.40038i −0.713959 0.700188i \(-0.753100\pi\)
0.713959 0.700188i \(-0.246900\pi\)
\(212\) 0 0
\(213\) 114.488 267.378i 0.537504 1.25529i
\(214\) 0 0
\(215\) 158.483i 0.737129i
\(216\) 0 0
\(217\) 302.022 1.39181
\(218\) 0 0
\(219\) −189.002 80.9287i −0.863023 0.369537i
\(220\) 0 0
\(221\) 24.7017 0.111773
\(222\) 0 0
\(223\) −7.43282 −0.0333310 −0.0166655 0.999861i \(-0.505305\pi\)
−0.0166655 + 0.999861i \(0.505305\pi\)
\(224\) 0 0
\(225\) −70.9445 74.3955i −0.315309 0.330647i
\(226\) 0 0
\(227\) −305.995 −1.34799 −0.673997 0.738734i \(-0.735424\pi\)
−0.673997 + 0.738734i \(0.735424\pi\)
\(228\) 0 0
\(229\) 62.6443i 0.273556i −0.990602 0.136778i \(-0.956325\pi\)
0.990602 0.136778i \(-0.0436747\pi\)
\(230\) 0 0
\(231\) −17.4637 7.47779i −0.0756006 0.0323714i
\(232\) 0 0
\(233\) 270.130i 1.15936i 0.814845 + 0.579679i \(0.196822\pi\)
−0.814845 + 0.579679i \(0.803178\pi\)
\(234\) 0 0
\(235\) 226.727i 0.964794i
\(236\) 0 0
\(237\) −26.8308 11.4887i −0.113210 0.0484754i
\(238\) 0 0
\(239\) 344.910i 1.44314i 0.692342 + 0.721570i \(0.256579\pi\)
−0.692342 + 0.721570i \(0.743421\pi\)
\(240\) 0 0
\(241\) 23.9109 0.0992152 0.0496076 0.998769i \(-0.484203\pi\)
0.0496076 + 0.998769i \(0.484203\pi\)
\(242\) 0 0
\(243\) 106.145 218.592i 0.436809 0.899554i
\(244\) 0 0
\(245\) −71.5671 −0.292110
\(246\) 0 0
\(247\) −415.928 −1.68392
\(248\) 0 0
\(249\) −107.322 45.9542i −0.431013 0.184555i
\(250\) 0 0
\(251\) −395.470 −1.57558 −0.787788 0.615946i \(-0.788774\pi\)
−0.787788 + 0.615946i \(0.788774\pi\)
\(252\) 0 0
\(253\) 41.2447i 0.163023i
\(254\) 0 0
\(255\) 7.45267 17.4051i 0.0292262 0.0682552i
\(256\) 0 0
\(257\) 451.825i 1.75807i 0.476753 + 0.879037i \(0.341814\pi\)
−0.476753 + 0.879037i \(0.658186\pi\)
\(258\) 0 0
\(259\) 165.453i 0.638813i
\(260\) 0 0
\(261\) −209.201 219.378i −0.801538 0.840527i
\(262\) 0 0
\(263\) 167.798i 0.638014i −0.947752 0.319007i \(-0.896651\pi\)
0.947752 0.319007i \(-0.103349\pi\)
\(264\) 0 0
\(265\) −207.489 −0.782976
\(266\) 0 0
\(267\) 0.797788 1.86317i 0.00298797 0.00697815i
\(268\) 0 0
\(269\) 452.084 1.68061 0.840306 0.542113i \(-0.182375\pi\)
0.840306 + 0.542113i \(0.182375\pi\)
\(270\) 0 0
\(271\) −207.813 −0.766837 −0.383419 0.923575i \(-0.625253\pi\)
−0.383419 + 0.923575i \(0.625253\pi\)
\(272\) 0 0
\(273\) 92.6224 216.312i 0.339276 0.792350i
\(274\) 0 0
\(275\) 13.2996 0.0483622
\(276\) 0 0
\(277\) 216.111i 0.780184i 0.920776 + 0.390092i \(0.127557\pi\)
−0.920776 + 0.390092i \(0.872443\pi\)
\(278\) 0 0
\(279\) −344.925 361.703i −1.23629 1.29643i
\(280\) 0 0
\(281\) 499.107i 1.77618i −0.459669 0.888090i \(-0.652032\pi\)
0.459669 0.888090i \(-0.347968\pi\)
\(282\) 0 0
\(283\) 206.468i 0.729569i 0.931092 + 0.364785i \(0.118857\pi\)
−0.931092 + 0.364785i \(0.881143\pi\)
\(284\) 0 0
\(285\) −125.488 + 293.066i −0.440309 + 1.02830i
\(286\) 0 0
\(287\) 344.910i 1.20178i
\(288\) 0 0
\(289\) 286.066 0.989849
\(290\) 0 0
\(291\) −237.478 101.686i −0.816077 0.349436i
\(292\) 0 0
\(293\) 74.6785 0.254876 0.127438 0.991847i \(-0.459325\pi\)
0.127438 + 0.991847i \(0.459325\pi\)
\(294\) 0 0
\(295\) −182.916 −0.620055
\(296\) 0 0
\(297\) 10.9891 + 29.4547i 0.0370002 + 0.0991740i
\(298\) 0 0
\(299\) 510.871 1.70860
\(300\) 0 0
\(301\) 233.911i 0.777113i
\(302\) 0 0
\(303\) 301.849 + 129.249i 0.996202 + 0.426563i
\(304\) 0 0
\(305\) 17.4416i 0.0571856i
\(306\) 0 0
\(307\) 293.485i 0.955977i −0.878366 0.477988i \(-0.841366\pi\)
0.878366 0.477988i \(-0.158634\pi\)
\(308\) 0 0
\(309\) −441.289 188.955i −1.42812 0.611506i
\(310\) 0 0
\(311\) 44.7374i 0.143850i 0.997410 + 0.0719251i \(0.0229143\pi\)
−0.997410 + 0.0719251i \(0.977086\pi\)
\(312\) 0 0
\(313\) 69.2666 0.221299 0.110650 0.993859i \(-0.464707\pi\)
0.110650 + 0.993859i \(0.464707\pi\)
\(314\) 0 0
\(315\) −124.470 130.525i −0.395144 0.414365i
\(316\) 0 0
\(317\) −12.0916 −0.0381438 −0.0190719 0.999818i \(-0.506071\pi\)
−0.0190719 + 0.999818i \(0.506071\pi\)
\(318\) 0 0
\(319\) 39.2179 0.122940
\(320\) 0 0
\(321\) −508.389 217.687i −1.58377 0.678151i
\(322\) 0 0
\(323\) 49.3949 0.152925
\(324\) 0 0
\(325\) 164.733i 0.506872i
\(326\) 0 0
\(327\) −81.1650 + 189.554i −0.248211 + 0.579676i
\(328\) 0 0
\(329\) 334.635i 1.01713i
\(330\) 0 0
\(331\) 468.516i 1.41545i −0.706486 0.707727i \(-0.749721\pi\)
0.706486 0.707727i \(-0.250279\pi\)
\(332\) 0 0
\(333\) −198.147 + 188.955i −0.595036 + 0.567434i
\(334\) 0 0
\(335\) 26.1076i 0.0779332i
\(336\) 0 0
\(337\) −302.111 −0.896472 −0.448236 0.893915i \(-0.647948\pi\)
−0.448236 + 0.893915i \(0.647948\pi\)
\(338\) 0 0
\(339\) −84.2050 + 196.654i −0.248392 + 0.580099i
\(340\) 0 0
\(341\) 64.6613 0.189623
\(342\) 0 0
\(343\) 372.117 1.08489
\(344\) 0 0
\(345\) 154.133 359.964i 0.446762 1.04337i
\(346\) 0 0
\(347\) −419.740 −1.20963 −0.604813 0.796368i \(-0.706752\pi\)
−0.604813 + 0.796368i \(0.706752\pi\)
\(348\) 0 0
\(349\) 27.0446i 0.0774916i −0.999249 0.0387458i \(-0.987664\pi\)
0.999249 0.0387458i \(-0.0123363\pi\)
\(350\) 0 0
\(351\) −364.836 + 136.114i −1.03942 + 0.387789i
\(352\) 0 0
\(353\) 47.9572i 0.135856i 0.997690 + 0.0679281i \(0.0216388\pi\)
−0.997690 + 0.0679281i \(0.978361\pi\)
\(354\) 0 0
\(355\) 357.252i 1.00634i
\(356\) 0 0
\(357\) −10.9997 + 25.6888i −0.0308114 + 0.0719575i
\(358\) 0 0
\(359\) 26.1076i 0.0727232i −0.999339 0.0363616i \(-0.988423\pi\)
0.999339 0.0363616i \(-0.0115768\pi\)
\(360\) 0 0
\(361\) −470.711 −1.30391
\(362\) 0 0
\(363\) 329.957 + 141.284i 0.908972 + 0.389212i
\(364\) 0 0
\(365\) 252.532 0.691868
\(366\) 0 0
\(367\) 101.338 0.276126 0.138063 0.990423i \(-0.455912\pi\)
0.138063 + 0.990423i \(0.455912\pi\)
\(368\) 0 0
\(369\) 413.066 393.905i 1.11942 1.06749i
\(370\) 0 0
\(371\) 306.241 0.825446
\(372\) 0 0
\(373\) 286.644i 0.768483i −0.923233 0.384242i \(-0.874463\pi\)
0.923233 0.384242i \(-0.125537\pi\)
\(374\) 0 0
\(375\) 370.123 + 158.483i 0.986995 + 0.422621i
\(376\) 0 0
\(377\) 485.766i 1.28850i
\(378\) 0 0
\(379\) 257.560i 0.679579i 0.940502 + 0.339789i \(0.110356\pi\)
−0.940502 + 0.339789i \(0.889644\pi\)
\(380\) 0 0
\(381\) −243.142 104.111i −0.638169 0.273257i
\(382\) 0 0
\(383\) 311.325i 0.812859i −0.913682 0.406429i \(-0.866774\pi\)
0.913682 0.406429i \(-0.133226\pi\)
\(384\) 0 0
\(385\) 23.3338 0.0606074
\(386\) 0 0
\(387\) 280.133 267.138i 0.723858 0.690280i
\(388\) 0 0
\(389\) −13.1287 −0.0337500 −0.0168750 0.999858i \(-0.505372\pi\)
−0.0168750 + 0.999858i \(0.505372\pi\)
\(390\) 0 0
\(391\) −60.6702 −0.155167
\(392\) 0 0
\(393\) 26.3667 + 11.2899i 0.0670908 + 0.0287276i
\(394\) 0 0
\(395\) 35.8495 0.0907582
\(396\) 0 0
\(397\) 498.244i 1.25502i −0.778607 0.627511i \(-0.784074\pi\)
0.778607 0.627511i \(-0.215926\pi\)
\(398\) 0 0
\(399\) 185.213 432.548i 0.464192 1.08408i
\(400\) 0 0
\(401\) 523.495i 1.30547i 0.757585 + 0.652736i \(0.226379\pi\)
−0.757585 + 0.652736i \(0.773621\pi\)
\(402\) 0 0
\(403\) 800.916i 1.98739i
\(404\) 0 0
\(405\) −14.1659 + 298.133i −0.0349775 + 0.736131i
\(406\) 0 0
\(407\) 35.4225i 0.0870332i
\(408\) 0 0
\(409\) 581.822 1.42255 0.711274 0.702915i \(-0.248119\pi\)
0.711274 + 0.702915i \(0.248119\pi\)
\(410\) 0 0
\(411\) −177.725 + 415.062i −0.432421 + 1.00988i
\(412\) 0 0
\(413\) 269.973 0.653688
\(414\) 0 0
\(415\) 143.396 0.345534
\(416\) 0 0
\(417\) 60.9218 142.278i 0.146095 0.341194i
\(418\) 0 0
\(419\) −256.600 −0.612410 −0.306205 0.951966i \(-0.599059\pi\)
−0.306205 + 0.951966i \(0.599059\pi\)
\(420\) 0 0
\(421\) 72.1110i 0.171285i 0.996326 + 0.0856425i \(0.0272943\pi\)
−0.996326 + 0.0856425i \(0.972706\pi\)
\(422\) 0 0
\(423\) 400.760 382.170i 0.947423 0.903475i
\(424\) 0 0
\(425\) 19.5635i 0.0460317i
\(426\) 0 0
\(427\) 25.7427i 0.0602874i
\(428\) 0 0
\(429\) 19.8300 46.3112i 0.0462237 0.107951i
\(430\) 0 0
\(431\) 714.091i 1.65682i −0.560119 0.828412i \(-0.689245\pi\)
0.560119 0.828412i \(-0.310755\pi\)
\(432\) 0 0
\(433\) −192.200 −0.443880 −0.221940 0.975060i \(-0.571239\pi\)
−0.221940 + 0.975060i \(0.571239\pi\)
\(434\) 0 0
\(435\) 342.275 + 146.559i 0.786840 + 0.336917i
\(436\) 0 0
\(437\) 1021.56 2.33768
\(438\) 0 0
\(439\) 504.093 1.14827 0.574137 0.818759i \(-0.305338\pi\)
0.574137 + 0.818759i \(0.305338\pi\)
\(440\) 0 0
\(441\) −120.633 126.501i −0.273545 0.286851i
\(442\) 0 0
\(443\) 97.6254 0.220373 0.110187 0.993911i \(-0.464855\pi\)
0.110187 + 0.993911i \(0.464855\pi\)
\(444\) 0 0
\(445\) 2.48944i 0.00559424i
\(446\) 0 0
\(447\) 71.1341 + 30.4589i 0.159137 + 0.0681407i
\(448\) 0 0
\(449\) 306.869i 0.683450i 0.939800 + 0.341725i \(0.111011\pi\)
−0.939800 + 0.341725i \(0.888989\pi\)
\(450\) 0 0
\(451\) 73.8435i 0.163733i
\(452\) 0 0
\(453\) −44.9955 19.2666i −0.0993279 0.0425312i
\(454\) 0 0
\(455\) 289.021i 0.635211i
\(456\) 0 0
\(457\) 117.378 0.256844 0.128422 0.991720i \(-0.459009\pi\)
0.128422 + 0.991720i \(0.459009\pi\)
\(458\) 0 0
\(459\) 43.3273 16.1647i 0.0943949 0.0352172i
\(460\) 0 0
\(461\) −604.144 −1.31051 −0.655254 0.755409i \(-0.727438\pi\)
−0.655254 + 0.755409i \(0.727438\pi\)
\(462\) 0 0
\(463\) 523.249 1.13013 0.565063 0.825048i \(-0.308852\pi\)
0.565063 + 0.825048i \(0.308852\pi\)
\(464\) 0 0
\(465\) 564.333 + 241.641i 1.21362 + 0.519659i
\(466\) 0 0
\(467\) −219.340 −0.469679 −0.234840 0.972034i \(-0.575456\pi\)
−0.234840 + 0.972034i \(0.575456\pi\)
\(468\) 0 0
\(469\) 38.5332i 0.0821604i
\(470\) 0 0
\(471\) −296.975 + 693.560i −0.630520 + 1.47253i
\(472\) 0 0
\(473\) 50.0791i 0.105875i
\(474\) 0 0
\(475\) 329.409i 0.693494i
\(476\) 0 0
\(477\) −349.743 366.755i −0.733213 0.768879i
\(478\) 0 0
\(479\) 302.010i 0.630501i −0.949008 0.315251i \(-0.897911\pi\)
0.949008 0.315251i \(-0.102089\pi\)
\(480\) 0 0
\(481\) 438.755 0.912173
\(482\) 0 0
\(483\) −227.491 + 531.285i −0.470995 + 1.09997i
\(484\) 0 0
\(485\) 317.302 0.654232
\(486\) 0 0
\(487\) 208.115 0.427340 0.213670 0.976906i \(-0.431458\pi\)
0.213670 + 0.976906i \(0.431458\pi\)
\(488\) 0 0
\(489\) 117.722 274.931i 0.240741 0.562231i
\(490\) 0 0
\(491\) 584.291 1.19000 0.595001 0.803725i \(-0.297152\pi\)
0.595001 + 0.803725i \(0.297152\pi\)
\(492\) 0 0
\(493\) 57.6888i 0.117016i
\(494\) 0 0
\(495\) −26.6485 27.9447i −0.0538353 0.0564540i
\(496\) 0 0
\(497\) 527.282i 1.06093i
\(498\) 0 0
\(499\) 352.161i 0.705733i −0.935674 0.352867i \(-0.885207\pi\)
0.935674 0.352867i \(-0.114793\pi\)
\(500\) 0 0
\(501\) −187.147 + 437.066i −0.373547 + 0.872388i
\(502\) 0 0
\(503\) 130.538i 0.259519i 0.991545 + 0.129760i \(0.0414205\pi\)
−0.991545 + 0.129760i \(0.958579\pi\)
\(504\) 0 0
\(505\) −403.310 −0.798634
\(506\) 0 0
\(507\) 107.555 + 46.0538i 0.212140 + 0.0908360i
\(508\) 0 0
\(509\) 738.353 1.45059 0.725297 0.688436i \(-0.241702\pi\)
0.725297 + 0.688436i \(0.241702\pi\)
\(510\) 0 0
\(511\) −372.721 −0.729396
\(512\) 0 0
\(513\) −729.544 + 272.181i −1.42211 + 0.530567i
\(514\) 0 0
\(515\) 589.621 1.14489
\(516\) 0 0
\(517\) 71.6435i 0.138575i
\(518\) 0 0
\(519\) 323.603 + 138.564i 0.623513 + 0.266982i
\(520\) 0 0
\(521\) 318.449i 0.611227i 0.952156 + 0.305613i \(0.0988615\pi\)
−0.952156 + 0.305613i \(0.901139\pi\)
\(522\) 0 0
\(523\) 817.186i 1.56250i 0.624220 + 0.781249i \(0.285417\pi\)
−0.624220 + 0.781249i \(0.714583\pi\)
\(524\) 0 0
\(525\) −171.316 73.3557i −0.326316 0.139725i
\(526\) 0 0
\(527\) 95.1155i 0.180485i
\(528\) 0 0
\(529\) −725.755 −1.37194
\(530\) 0 0
\(531\) −308.323 323.321i −0.580647 0.608892i
\(532\) 0 0
\(533\) −914.650 −1.71604
\(534\) 0 0
\(535\) 679.274 1.26967
\(536\) 0 0
\(537\) −397.855 170.357i −0.740885 0.317239i
\(538\) 0 0
\(539\) 22.6145 0.0419564
\(540\) 0 0
\(541\) 322.022i 0.595235i 0.954685 + 0.297617i \(0.0961919\pi\)
−0.954685 + 0.297617i \(0.903808\pi\)
\(542\) 0 0
\(543\) 342.215 799.215i 0.630231 1.47185i
\(544\) 0 0
\(545\) 253.269i 0.464714i
\(546\) 0 0
\(547\) 276.716i 0.505880i −0.967482 0.252940i \(-0.918602\pi\)
0.967482 0.252940i \(-0.0813975\pi\)
\(548\) 0 0
\(549\) 30.8296 29.3995i 0.0561560 0.0535511i
\(550\) 0 0
\(551\) 971.364i 1.76291i
\(552\) 0 0
\(553\) −52.9116 −0.0956811
\(554\) 0 0
\(555\) 132.375 309.151i 0.238514 0.557029i
\(556\) 0 0
\(557\) −579.033 −1.03956 −0.519778 0.854301i \(-0.673985\pi\)
−0.519778 + 0.854301i \(0.673985\pi\)
\(558\) 0 0
\(559\) −620.296 −1.10965
\(560\) 0 0
\(561\) −2.35497 + 5.49984i −0.00419781 + 0.00980364i
\(562\) 0 0
\(563\) 494.751 0.878776 0.439388 0.898297i \(-0.355195\pi\)
0.439388 + 0.898297i \(0.355195\pi\)
\(564\) 0 0
\(565\) 262.755i 0.465054i
\(566\) 0 0
\(567\) 20.9080 440.026i 0.0368748 0.776060i
\(568\) 0 0
\(569\) 229.386i 0.403138i −0.979474 0.201569i \(-0.935396\pi\)
0.979474 0.201569i \(-0.0646041\pi\)
\(570\) 0 0
\(571\) 451.354i 0.790462i −0.918582 0.395231i \(-0.870665\pi\)
0.918582 0.395231i \(-0.129335\pi\)
\(572\) 0 0
\(573\) 178.317 416.444i 0.311199 0.726779i
\(574\) 0 0
\(575\) 404.603i 0.703658i
\(576\) 0 0
\(577\) 792.644 1.37373 0.686867 0.726783i \(-0.258986\pi\)
0.686867 + 0.726783i \(0.258986\pi\)
\(578\) 0 0
\(579\) 6.12801 + 2.62395i 0.0105838 + 0.00453186i
\(580\) 0 0
\(581\) −211.644 −0.364276
\(582\) 0 0
\(583\) 65.5645 0.112461
\(584\) 0 0
\(585\) 346.133 330.077i 0.591680 0.564234i
\(586\) 0 0
\(587\) −930.546 −1.58526 −0.792629 0.609704i \(-0.791288\pi\)
−0.792629 + 0.609704i \(0.791288\pi\)
\(588\) 0 0
\(589\) 1601.55i 2.71911i
\(590\) 0 0
\(591\) −481.905 206.347i −0.815407 0.349148i
\(592\) 0 0
\(593\) 147.392i 0.248554i −0.992248 0.124277i \(-0.960339\pi\)
0.992248 0.124277i \(-0.0396611\pi\)
\(594\) 0 0
\(595\) 34.3236i 0.0576868i
\(596\) 0 0
\(597\) 387.628 + 165.978i 0.649293 + 0.278020i
\(598\) 0 0
\(599\) 423.233i 0.706566i 0.935516 + 0.353283i \(0.114935\pi\)
−0.935516 + 0.353283i \(0.885065\pi\)
\(600\) 0 0
\(601\) 145.156 0.241523 0.120762 0.992682i \(-0.461466\pi\)
0.120762 + 0.992682i \(0.461466\pi\)
\(602\) 0 0
\(603\) −46.1476 + 44.0069i −0.0765300 + 0.0729800i
\(604\) 0 0
\(605\) −440.866 −0.728704
\(606\) 0 0
\(607\) −503.791 −0.829968 −0.414984 0.909829i \(-0.636213\pi\)
−0.414984 + 0.909829i \(0.636213\pi\)
\(608\) 0 0
\(609\) −505.177 216.312i −0.829520 0.355192i
\(610\) 0 0
\(611\) −887.400 −1.45237
\(612\) 0 0
\(613\) 245.400i 0.400326i −0.979763 0.200163i \(-0.935853\pi\)
0.979763 0.200163i \(-0.0641471\pi\)
\(614\) 0 0
\(615\) −275.956 + 644.471i −0.448708 + 1.04792i
\(616\) 0 0
\(617\) 87.0366i 0.141064i −0.997510 0.0705321i \(-0.977530\pi\)
0.997510 0.0705321i \(-0.0224697\pi\)
\(618\) 0 0
\(619\) 203.477i 0.328718i −0.986401 0.164359i \(-0.947444\pi\)
0.986401 0.164359i \(-0.0525556\pi\)
\(620\) 0 0
\(621\) 896.076 334.311i 1.44296 0.538343i
\(622\) 0 0
\(623\) 3.67426i 0.00589768i
\(624\) 0 0
\(625\) −208.978 −0.334365
\(626\) 0 0
\(627\) 39.6530 92.6063i 0.0632425 0.147697i
\(628\) 0 0
\(629\) −52.1059 −0.0828392
\(630\) 0 0
\(631\) −975.796 −1.54643 −0.773214 0.634145i \(-0.781352\pi\)
−0.773214 + 0.634145i \(0.781352\pi\)
\(632\) 0 0
\(633\) −348.922 + 814.877i −0.551219 + 1.28733i
\(634\) 0 0
\(635\) 324.870 0.511607
\(636\) 0 0
\(637\) 280.111i 0.439735i
\(638\) 0 0
\(639\) −631.475 + 602.183i −0.988224 + 0.942383i
\(640\) 0 0
\(641\) 797.821i 1.24465i −0.782759 0.622325i \(-0.786188\pi\)
0.782759 0.622325i \(-0.213812\pi\)
\(642\) 0 0
\(643\) 94.7058i 0.147287i −0.997285 0.0736437i \(-0.976537\pi\)
0.997285 0.0736437i \(-0.0234628\pi\)
\(644\) 0 0
\(645\) −187.147 + 437.066i −0.290151 + 0.677622i
\(646\) 0 0
\(647\) 367.344i 0.567765i −0.958859 0.283882i \(-0.908378\pi\)
0.958859 0.283882i \(-0.0916225\pi\)
\(648\) 0 0
\(649\) 57.7998 0.0890599
\(650\) 0 0
\(651\) −832.921 356.648i −1.27945 0.547846i
\(652\) 0 0
\(653\) 680.024 1.04138 0.520692 0.853745i \(-0.325674\pi\)
0.520692 + 0.853745i \(0.325674\pi\)
\(654\) 0 0
\(655\) −35.2294 −0.0537853
\(656\) 0 0
\(657\) 425.667 + 446.373i 0.647895 + 0.679411i
\(658\) 0 0
\(659\) −687.246 −1.04286 −0.521431 0.853293i \(-0.674602\pi\)
−0.521431 + 0.853293i \(0.674602\pi\)
\(660\) 0 0
\(661\) 1121.58i 1.69679i −0.529364 0.848395i \(-0.677569\pi\)
0.529364 0.848395i \(-0.322431\pi\)
\(662\) 0 0
\(663\) −68.1229 29.1695i −0.102749 0.0439962i
\(664\) 0 0
\(665\) 577.941i 0.869085i
\(666\) 0 0
\(667\) 1193.10i 1.78875i
\(668\) 0 0
\(669\) 20.4984 + 8.77718i 0.0306403 + 0.0131198i
\(670\) 0 0
\(671\) 5.51138i 0.00821369i
\(672\) 0 0
\(673\) 563.177 0.836816 0.418408 0.908259i \(-0.362588\pi\)
0.418408 + 0.908259i \(0.362588\pi\)
\(674\) 0 0
\(675\) 107.801 + 288.945i 0.159705 + 0.428067i
\(676\) 0 0
\(677\) −998.773 −1.47529 −0.737646 0.675188i \(-0.764063\pi\)
−0.737646 + 0.675188i \(0.764063\pi\)
\(678\) 0 0
\(679\) −468.319 −0.689719
\(680\) 0 0
\(681\) 843.877 + 361.339i 1.23917 + 0.530601i
\(682\) 0 0
\(683\) 147.020 0.215257 0.107628 0.994191i \(-0.465674\pi\)
0.107628 + 0.994191i \(0.465674\pi\)
\(684\) 0 0
\(685\) 554.577i 0.809601i
\(686\) 0 0
\(687\) −73.9746 + 172.761i −0.107678 + 0.251472i
\(688\) 0 0
\(689\) 812.104i 1.17867i
\(690\) 0 0
\(691\) 862.689i 1.24846i −0.781239 0.624232i \(-0.785412\pi\)
0.781239 0.624232i \(-0.214588\pi\)
\(692\) 0 0
\(693\) 39.3315 + 41.2447i 0.0567554 + 0.0595162i
\(694\) 0 0
\(695\) 190.102i 0.273528i
\(696\) 0 0
\(697\) 108.622 0.155843
\(698\) 0 0
\(699\) 318.988 744.970i 0.456349 1.06576i
\(700\) 0 0
\(701\) 468.489 0.668315 0.334158 0.942517i \(-0.391548\pi\)
0.334158 + 0.942517i \(0.391548\pi\)
\(702\) 0 0
\(703\) 877.358 1.24802
\(704\) 0 0
\(705\) −267.734 + 625.270i −0.379765 + 0.886908i
\(706\) 0 0
\(707\) 595.261 0.841954
\(708\) 0 0
\(709\) 1020.60i 1.43949i −0.694238 0.719745i \(-0.744259\pi\)
0.694238 0.719745i \(-0.255741\pi\)
\(710\) 0 0
\(711\) 60.4278 + 63.3673i 0.0849899 + 0.0891241i
\(712\) 0 0
\(713\) 1967.14i 2.75896i
\(714\) 0 0
\(715\) 61.8778i 0.0865424i
\(716\) 0 0
\(717\) 407.294 951.199i 0.568052 1.32664i
\(718\) 0 0
\(719\) 477.286i 0.663819i 0.943311 + 0.331909i \(0.107693\pi\)
−0.943311 + 0.331909i \(0.892307\pi\)
\(720\) 0 0
\(721\) −870.244 −1.20700
\(722\) 0 0
\(723\) −65.9418 28.2356i −0.0912058 0.0390534i
\(724\) 0 0
\(725\) 384.721 0.530649
\(726\) 0 0
\(727\) −1042.27 −1.43365 −0.716827 0.697251i \(-0.754406\pi\)
−0.716827 + 0.697251i \(0.754406\pi\)
\(728\) 0 0
\(729\) −550.855 + 477.493i −0.755631 + 0.654997i
\(730\) 0 0
\(731\) 73.6654 0.100773
\(732\) 0 0
\(733\) 952.333i 1.29923i 0.760265 + 0.649613i \(0.225069\pi\)
−0.760265 + 0.649613i \(0.774931\pi\)
\(734\) 0 0
\(735\) 197.369 + 84.5112i 0.268529 + 0.114981i
\(736\) 0 0
\(737\) 8.24976i 0.0111937i
\(738\) 0 0
\(739\) 25.2442i 0.0341599i −0.999854 0.0170800i \(-0.994563\pi\)
0.999854 0.0170800i \(-0.00543698\pi\)
\(740\) 0 0
\(741\) 1147.05 + 491.156i 1.54798 + 0.662828i
\(742\) 0 0
\(743\) 706.613i 0.951027i −0.879708 0.475514i \(-0.842262\pi\)
0.879708 0.475514i \(-0.157738\pi\)
\(744\) 0 0
\(745\) −95.0446 −0.127577
\(746\) 0 0
\(747\) 241.709 + 253.466i 0.323573 + 0.339313i
\(748\) 0 0
\(749\) −1002.57 −1.33854
\(750\) 0 0
\(751\) −1414.08 −1.88293 −0.941466 0.337108i \(-0.890551\pi\)
−0.941466 + 0.337108i \(0.890551\pi\)
\(752\) 0 0
\(753\) 1090.63 + 466.997i 1.44838 + 0.620182i
\(754\) 0 0
\(755\) 60.1200 0.0796291
\(756\) 0 0
\(757\) 309.177i 0.408425i 0.978927 + 0.204212i \(0.0654633\pi\)
−0.978927 + 0.204212i \(0.934537\pi\)
\(758\) 0 0
\(759\) −48.7046 + 113.745i −0.0641694 + 0.149862i
\(760\) 0 0
\(761\) 400.757i 0.526618i 0.964712 + 0.263309i \(0.0848139\pi\)
−0.964712 + 0.263309i \(0.915186\pi\)
\(762\) 0 0
\(763\) 373.810i 0.489921i
\(764\) 0 0
\(765\) −41.1062 + 39.1994i −0.0537336 + 0.0512410i
\(766\) 0 0
\(767\) 715.928i 0.933414i
\(768\) 0 0
\(769\) 804.533 1.04621 0.523104 0.852269i \(-0.324774\pi\)
0.523104 + 0.852269i \(0.324774\pi\)
\(770\) 0 0
\(771\) 533.546 1246.05i 0.692018 1.61615i
\(772\) 0 0
\(773\) 122.227 0.158120 0.0790599 0.996870i \(-0.474808\pi\)
0.0790599 + 0.996870i \(0.474808\pi\)
\(774\) 0 0
\(775\) 634.316 0.818472
\(776\) 0 0
\(777\) −195.378 + 456.288i −0.251451 + 0.587243i
\(778\) 0 0
\(779\) −1828.98 −2.34786
\(780\) 0 0
\(781\) 112.888i 0.144543i
\(782\) 0 0
\(783\) 317.883 + 852.042i 0.405981 + 1.08818i
\(784\) 0 0
\(785\) 926.687i 1.18049i
\(786\) 0 0
\(787\) 574.295i 0.729727i 0.931061 + 0.364863i \(0.118884\pi\)
−0.931061 + 0.364863i \(0.881116\pi\)
\(788\) 0 0
\(789\) −198.147 + 462.755i −0.251137 + 0.586509i
\(790\) 0 0
\(791\) 387.811i 0.490279i
\(792\) 0 0
\(793\) −68.2658 −0.0860856
\(794\) 0 0
\(795\) 572.216 + 245.017i 0.719768 + 0.308197i
\(796\) 0 0
\(797\) −1000.03 −1.25474 −0.627371 0.778721i \(-0.715869\pi\)
−0.627371 + 0.778721i \(0.715869\pi\)
\(798\) 0 0
\(799\) 105.386 0.131898
\(800\) 0 0
\(801\) −4.40031 + 4.19619i −0.00549352 + 0.00523869i
\(802\) 0 0
\(803\) −79.7976 −0.0993744
\(804\) 0 0
\(805\) 709.867i 0.881822i
\(806\) 0 0
\(807\) −1246.77 533.852i −1.54494 0.661527i
\(808\) 0 0
\(809\) 583.032i 0.720682i −0.932821 0.360341i \(-0.882660\pi\)
0.932821 0.360341i \(-0.117340\pi\)
\(810\) 0 0
\(811\) 911.183i 1.12353i −0.827297 0.561765i \(-0.810122\pi\)
0.827297 0.561765i \(-0.189878\pi\)
\(812\) 0 0
\(813\) 573.110 + 245.400i 0.704932 + 0.301844i
\(814\) 0 0
\(815\) 367.344i 0.450728i
\(816\) 0 0
\(817\) −1240.38 −1.51821
\(818\) 0 0
\(819\) −510.871 + 487.173i −0.623774 + 0.594839i
\(820\) 0 0
\(821\) −914.486 −1.11387 −0.556934 0.830557i \(-0.688022\pi\)
−0.556934 + 0.830557i \(0.688022\pi\)
\(822\) 0 0
\(823\) 686.707 0.834395 0.417197 0.908816i \(-0.363012\pi\)
0.417197 + 0.908816i \(0.363012\pi\)
\(824\) 0 0
\(825\) −36.6779 15.7051i −0.0444580 0.0190365i
\(826\) 0 0
\(827\) 1107.30 1.33893 0.669466 0.742843i \(-0.266523\pi\)
0.669466 + 0.742843i \(0.266523\pi\)
\(828\) 0 0
\(829\) 366.866i 0.442541i −0.975212 0.221270i \(-0.928980\pi\)
0.975212 0.221270i \(-0.0710203\pi\)
\(830\) 0 0
\(831\) 255.199 595.995i 0.307098 0.717202i
\(832\) 0 0
\(833\) 33.2655i 0.0399346i
\(834\) 0 0
\(835\) 583.978i 0.699375i
\(836\) 0 0
\(837\) 524.115 + 1404.82i 0.626183 + 1.67840i
\(838\) 0 0
\(839\) 1085.11i 1.29334i −0.762771 0.646668i \(-0.776162\pi\)
0.762771 0.646668i \(-0.223838\pi\)
\(840\) 0 0
\(841\) 293.466 0.348949
\(842\) 0 0
\(843\) −589.379 + 1376.44i −0.699145 + 1.63279i
\(844\) 0 0
\(845\) −143.707 −0.170068
\(846\) 0 0
\(847\) 650.691 0.768230
\(848\) 0 0
\(849\) 243.812 569.401i 0.287175 0.670673i
\(850\) 0 0
\(851\) −1077.63 −1.26631
\(852\) 0 0
\(853\) 1039.75i 1.21894i 0.792810 + 0.609469i \(0.208617\pi\)
−0.792810 + 0.609469i \(0.791383\pi\)
\(854\) 0 0
\(855\) 692.145 660.039i 0.809527 0.771975i
\(856\) 0 0
\(857\) 1216.69i 1.41971i −0.704346 0.709857i \(-0.748760\pi\)
0.704346 0.709857i \(-0.251240\pi\)
\(858\) 0 0
\(859\) 318.230i 0.370466i 0.982695 + 0.185233i \(0.0593040\pi\)
−0.982695 + 0.185233i \(0.940696\pi\)
\(860\) 0 0
\(861\) 407.294 951.199i 0.473047 1.10476i
\(862\) 0 0
\(863\) 464.296i 0.538003i 0.963140 + 0.269001i \(0.0866936\pi\)
−0.963140 + 0.269001i \(0.913306\pi\)
\(864\) 0 0
\(865\) −432.377 −0.499858
\(866\) 0 0
\(867\) −788.919 337.807i −0.909941 0.389627i
\(868\) 0 0
\(869\) −11.3281 −0.0130358
\(870\) 0 0
\(871\) 102.184 0.117318
\(872\) 0 0
\(873\) 534.844 + 560.861i 0.612651 + 0.642453i
\(874\) 0 0
\(875\) 729.901 0.834172
\(876\) 0 0
\(877\) 1239.89i 1.41378i 0.707321 + 0.706892i \(0.249903\pi\)
−0.707321 + 0.706892i \(0.750097\pi\)
\(878\) 0 0
\(879\) −205.950 88.1855i −0.234300 0.100325i
\(880\) 0 0
\(881\) 49.3084i 0.0559687i 0.999608 + 0.0279843i \(0.00890885\pi\)
−0.999608 + 0.0279843i \(0.991091\pi\)
\(882\) 0 0
\(883\) 1359.44i 1.53957i 0.638303 + 0.769785i \(0.279637\pi\)
−0.638303 + 0.769785i \(0.720363\pi\)
\(884\) 0 0
\(885\) 504.450 + 216.000i 0.570000 + 0.244068i
\(886\) 0 0
\(887\) 1353.53i 1.52597i −0.646417 0.762984i \(-0.723733\pi\)
0.646417 0.762984i \(-0.276267\pi\)
\(888\) 0 0
\(889\) −479.489 −0.539357
\(890\) 0 0
\(891\) 4.47629 94.2072i 0.00502390 0.105732i
\(892\) 0 0
\(893\) −1774.49 −1.98711
\(894\) 0 0
\(895\) 531.587 0.593952
\(896\) 0 0
\(897\) −1408.89 603.271i −1.57067 0.672543i
\(898\) 0 0
\(899\) 1870.47 2.08061
\(900\) 0 0
\(901\) 96.4441i 0.107041i
\(902\) 0 0
\(903\) 276.218 645.083i 0.305889 0.714378i
\(904\) 0 0
\(905\) 1067.86i 1.17995i
\(906\) 0 0
\(907\) 1442.68i 1.59061i 0.606213 + 0.795303i \(0.292688\pi\)
−0.606213 + 0.795303i \(0.707312\pi\)
\(908\) 0 0
\(909\) −679.819 712.888i −0.747876 0.784255i
\(910\) 0 0
\(911\) 128.701i 0.141274i −0.997502 0.0706372i \(-0.977497\pi\)
0.997502 0.0706372i \(-0.0225033\pi\)
\(912\) 0 0
\(913\) −45.3120 −0.0496297
\(914\) 0 0
\(915\) −20.5962 + 48.1007i −0.0225095 + 0.0525691i
\(916\) 0 0
\(917\) 51.9964 0.0567027
\(918\) 0 0
\(919\) −837.294 −0.911092 −0.455546 0.890212i \(-0.650556\pi\)
−0.455546 + 0.890212i \(0.650556\pi\)
\(920\) 0 0
\(921\) −346.567 + 809.377i −0.376294 + 0.878803i
\(922\) 0 0
\(923\) 1398.27 1.51492
\(924\) 0 0
\(925\) 347.489i 0.375663i
\(926\) 0 0
\(927\) 993.864 + 1042.21i 1.07213 + 1.12428i
\(928\) 0 0
\(929\) 671.610i 0.722939i −0.932384 0.361469i \(-0.882275\pi\)
0.932384 0.361469i \(-0.117725\pi\)
\(930\) 0 0
\(931\) 560.125i 0.601638i
\(932\) 0 0
\(933\) 52.8290 123.378i 0.0566227 0.132238i
\(934\) 0 0
\(935\) 7.34851i 0.00785937i
\(936\) 0 0
\(937\) −598.511 −0.638753 −0.319376 0.947628i \(-0.603473\pi\)
−0.319376 + 0.947628i \(0.603473\pi\)
\(938\) 0 0
\(939\) −191.025 81.7947i −0.203434 0.0871083i
\(940\) 0 0
\(941\) 308.432 0.327770 0.163885 0.986479i \(-0.447597\pi\)
0.163885 + 0.986479i \(0.447597\pi\)
\(942\) 0 0
\(943\) 2246.48 2.38227
\(944\) 0 0
\(945\) 189.134 + 506.947i 0.200141 + 0.536452i
\(946\) 0 0
\(947\) 119.140 0.125808 0.0629040 0.998020i \(-0.479964\pi\)
0.0629040 + 0.998020i \(0.479964\pi\)
\(948\) 0 0
\(949\) 988.400i 1.04152i
\(950\) 0 0
\(951\) 33.3464 + 14.2786i 0.0350645 + 0.0150143i
\(952\) 0 0
\(953\) 1061.63i 1.11399i −0.830516 0.556995i \(-0.811954\pi\)
0.830516 0.556995i \(-0.188046\pi\)
\(954\) 0 0
\(955\) 556.424i 0.582643i
\(956\) 0 0
\(957\) −108.156 46.3112i −0.113015 0.0483920i
\(958\) 0 0
\(959\) 818.522i 0.853516i
\(960\) 0 0
\(961\) 2122.98 2.20913
\(962\) 0 0
\(963\) 1144.98 + 1200.68i 1.18898 + 1.24681i
\(964\) 0 0
\(965\) −8.18783 −0.00848479
\(966\) 0 0
\(967\) −167.506 −0.173223 −0.0866114 0.996242i \(-0.527604\pi\)
−0.0866114 + 0.996242i \(0.527604\pi\)
\(968\) 0 0
\(969\) −136.222 58.3288i −0.140580 0.0601949i
\(970\) 0 0
\(971\) −1724.86 −1.77638 −0.888189 0.459479i \(-0.848036\pi\)
−0.888189 + 0.459479i \(0.848036\pi\)
\(972\) 0 0
\(973\) 280.579i 0.288364i
\(974\) 0 0
\(975\) 194.528 454.304i 0.199516 0.465953i
\(976\) 0 0
\(977\) 166.328i 0.170243i 0.996371 + 0.0851216i \(0.0271279\pi\)
−0.996371 + 0.0851216i \(0.972872\pi\)
\(978\) 0 0
\(979\) 0.786639i 0.000803512i
\(980\) 0 0
\(981\) 447.677 426.910i 0.456347 0.435178i
\(982\) 0 0
\(983\) 1532.48i 1.55899i −0.626411 0.779493i \(-0.715477\pi\)
0.626411 0.779493i \(-0.284523\pi\)
\(984\) 0 0
\(985\) 643.889 0.653694
\(986\) 0 0
\(987\) 395.159 922.861i 0.400364 0.935016i
\(988\) 0 0
\(989\) 1523.52 1.54046
\(990\) 0 0
\(991\) −152.339 −0.153722 −0.0768612 0.997042i \(-0.524490\pi\)
−0.0768612 + 0.997042i \(0.524490\pi\)
\(992\) 0 0
\(993\) −553.255 + 1292.08i −0.557155 + 1.30119i
\(994\) 0 0
\(995\) −517.922 −0.520524
\(996\) 0 0
\(997\) 626.688i 0.628574i 0.949328 + 0.314287i \(0.101765\pi\)
−0.949328 + 0.314287i \(0.898235\pi\)
\(998\) 0 0
\(999\) 769.584 287.119i 0.770355 0.287406i
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 384.3.h.g.65.2 yes 16
3.2 odd 2 inner 384.3.h.g.65.13 yes 16
4.3 odd 2 inner 384.3.h.g.65.16 yes 16
8.3 odd 2 inner 384.3.h.g.65.1 16
8.5 even 2 inner 384.3.h.g.65.15 yes 16
12.11 even 2 inner 384.3.h.g.65.3 yes 16
16.3 odd 4 768.3.e.o.257.4 8
16.5 even 4 768.3.e.p.257.4 8
16.11 odd 4 768.3.e.p.257.5 8
16.13 even 4 768.3.e.o.257.5 8
24.5 odd 2 inner 384.3.h.g.65.4 yes 16
24.11 even 2 inner 384.3.h.g.65.14 yes 16
48.5 odd 4 768.3.e.p.257.3 8
48.11 even 4 768.3.e.p.257.6 8
48.29 odd 4 768.3.e.o.257.6 8
48.35 even 4 768.3.e.o.257.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.h.g.65.1 16 8.3 odd 2 inner
384.3.h.g.65.2 yes 16 1.1 even 1 trivial
384.3.h.g.65.3 yes 16 12.11 even 2 inner
384.3.h.g.65.4 yes 16 24.5 odd 2 inner
384.3.h.g.65.13 yes 16 3.2 odd 2 inner
384.3.h.g.65.14 yes 16 24.11 even 2 inner
384.3.h.g.65.15 yes 16 8.5 even 2 inner
384.3.h.g.65.16 yes 16 4.3 odd 2 inner
768.3.e.o.257.3 8 48.35 even 4
768.3.e.o.257.4 8 16.3 odd 4
768.3.e.o.257.5 8 16.13 even 4
768.3.e.o.257.6 8 48.29 odd 4
768.3.e.p.257.3 8 48.5 odd 4
768.3.e.p.257.4 8 16.5 even 4
768.3.e.p.257.5 8 16.11 odd 4
768.3.e.p.257.6 8 48.11 even 4