Properties

Label 640.2.l.b.481.6
Level $640$
Weight $2$
Character 640.481
Analytic conductor $5.110$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,2,Mod(161,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 481.6
Root \(1.26868 - 0.624862i\) of defining polynomial
Character \(\chi\) \(=\) 640.481
Dual form 640.2.l.b.161.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.720673 - 0.720673i) q^{3} +(-0.707107 - 0.707107i) q^{5} +4.02840i q^{7} +1.96126i q^{9} +(0.646837 + 0.646837i) q^{11} +(-4.91492 + 4.91492i) q^{13} -1.01919 q^{15} -2.70862 q^{17} +(0.438397 - 0.438397i) q^{19} +(2.90316 + 2.90316i) q^{21} -3.60080i q^{23} +1.00000i q^{25} +(3.57545 + 3.57545i) q^{27} +(-2.00921 + 2.00921i) q^{29} +4.30994 q^{31} +0.932316 q^{33} +(2.84851 - 2.84851i) q^{35} +(0.743961 + 0.743961i) q^{37} +7.08410i q^{39} +0.603979i q^{41} +(5.03010 + 5.03010i) q^{43} +(1.38682 - 1.38682i) q^{45} +10.8177 q^{47} -9.22800 q^{49} +(-1.95203 + 1.95203i) q^{51} +(-4.07420 - 4.07420i) q^{53} -0.914766i q^{55} -0.631882i q^{57} +(-1.22845 - 1.22845i) q^{59} +(6.98912 - 6.98912i) q^{61} -7.90074 q^{63} +6.95074 q^{65} +(-5.24219 + 5.24219i) q^{67} +(-2.59500 - 2.59500i) q^{69} -13.7940i q^{71} +1.30876i q^{73} +(0.720673 + 0.720673i) q^{75} +(-2.60572 + 2.60572i) q^{77} +0.611127 q^{79} -0.730326 q^{81} +(-1.29471 + 1.29471i) q^{83} +(1.91529 + 1.91529i) q^{85} +2.89597i q^{87} +10.9236i q^{89} +(-19.7993 - 19.7993i) q^{91} +(3.10606 - 3.10606i) q^{93} -0.619987 q^{95} -12.7571 q^{97} +(-1.26862 + 1.26862i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} - 8 q^{15} + 8 q^{19} - 24 q^{27} + 16 q^{29} + 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} + 32 q^{51} - 16 q^{53} + 8 q^{59} - 16 q^{61} + 40 q^{63} - 40 q^{67} - 16 q^{69} - 16 q^{77}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 0.720673 0.720673i 0.416081 0.416081i −0.467770 0.883850i \(-0.654942\pi\)
0.883850 + 0.467770i \(0.154942\pi\)
\(4\) 0 0
\(5\) −0.707107 0.707107i −0.316228 0.316228i
\(6\) 0 0
\(7\) 4.02840i 1.52259i 0.648405 + 0.761296i \(0.275437\pi\)
−0.648405 + 0.761296i \(0.724563\pi\)
\(8\) 0 0
\(9\) 1.96126i 0.653754i
\(10\) 0 0
\(11\) 0.646837 + 0.646837i 0.195029 + 0.195029i 0.797865 0.602836i \(-0.205963\pi\)
−0.602836 + 0.797865i \(0.705963\pi\)
\(12\) 0 0
\(13\) −4.91492 + 4.91492i −1.36315 + 1.36315i −0.493286 + 0.869867i \(0.664204\pi\)
−0.869867 + 0.493286i \(0.835796\pi\)
\(14\) 0 0
\(15\) −1.01919 −0.263153
\(16\) 0 0
\(17\) −2.70862 −0.656938 −0.328469 0.944515i \(-0.606533\pi\)
−0.328469 + 0.944515i \(0.606533\pi\)
\(18\) 0 0
\(19\) 0.438397 0.438397i 0.100575 0.100575i −0.655029 0.755604i \(-0.727344\pi\)
0.755604 + 0.655029i \(0.227344\pi\)
\(20\) 0 0
\(21\) 2.90316 + 2.90316i 0.633521 + 0.633521i
\(22\) 0 0
\(23\) 3.60080i 0.750819i −0.926859 0.375410i \(-0.877502\pi\)
0.926859 0.375410i \(-0.122498\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 3.57545 + 3.57545i 0.688095 + 0.688095i
\(28\) 0 0
\(29\) −2.00921 + 2.00921i −0.373102 + 0.373102i −0.868606 0.495504i \(-0.834983\pi\)
0.495504 + 0.868606i \(0.334983\pi\)
\(30\) 0 0
\(31\) 4.30994 0.774087 0.387044 0.922061i \(-0.373496\pi\)
0.387044 + 0.922061i \(0.373496\pi\)
\(32\) 0 0
\(33\) 0.932316 0.162295
\(34\) 0 0
\(35\) 2.84851 2.84851i 0.481486 0.481486i
\(36\) 0 0
\(37\) 0.743961 + 0.743961i 0.122306 + 0.122306i 0.765611 0.643304i \(-0.222437\pi\)
−0.643304 + 0.765611i \(0.722437\pi\)
\(38\) 0 0
\(39\) 7.08410i 1.13436i
\(40\) 0 0
\(41\) 0.603979i 0.0943256i 0.998887 + 0.0471628i \(0.0150180\pi\)
−0.998887 + 0.0471628i \(0.984982\pi\)
\(42\) 0 0
\(43\) 5.03010 + 5.03010i 0.767083 + 0.767083i 0.977592 0.210509i \(-0.0675121\pi\)
−0.210509 + 0.977592i \(0.567512\pi\)
\(44\) 0 0
\(45\) 1.38682 1.38682i 0.206735 0.206735i
\(46\) 0 0
\(47\) 10.8177 1.57793 0.788963 0.614440i \(-0.210618\pi\)
0.788963 + 0.614440i \(0.210618\pi\)
\(48\) 0 0
\(49\) −9.22800 −1.31829
\(50\) 0 0
\(51\) −1.95203 + 1.95203i −0.273339 + 0.273339i
\(52\) 0 0
\(53\) −4.07420 4.07420i −0.559634 0.559634i 0.369569 0.929203i \(-0.379505\pi\)
−0.929203 + 0.369569i \(0.879505\pi\)
\(54\) 0 0
\(55\) 0.914766i 0.123347i
\(56\) 0 0
\(57\) 0.631882i 0.0836948i
\(58\) 0 0
\(59\) −1.22845 1.22845i −0.159931 0.159931i 0.622605 0.782536i \(-0.286074\pi\)
−0.782536 + 0.622605i \(0.786074\pi\)
\(60\) 0 0
\(61\) 6.98912 6.98912i 0.894865 0.894865i −0.100112 0.994976i \(-0.531920\pi\)
0.994976 + 0.100112i \(0.0319200\pi\)
\(62\) 0 0
\(63\) −7.90074 −0.995400
\(64\) 0 0
\(65\) 6.95074 0.862134
\(66\) 0 0
\(67\) −5.24219 + 5.24219i −0.640435 + 0.640435i −0.950662 0.310227i \(-0.899595\pi\)
0.310227 + 0.950662i \(0.399595\pi\)
\(68\) 0 0
\(69\) −2.59500 2.59500i −0.312401 0.312401i
\(70\) 0 0
\(71\) 13.7940i 1.63704i −0.574475 0.818522i \(-0.694794\pi\)
0.574475 0.818522i \(-0.305206\pi\)
\(72\) 0 0
\(73\) 1.30876i 0.153179i 0.997063 + 0.0765895i \(0.0244031\pi\)
−0.997063 + 0.0765895i \(0.975597\pi\)
\(74\) 0 0
\(75\) 0.720673 + 0.720673i 0.0832162 + 0.0832162i
\(76\) 0 0
\(77\) −2.60572 + 2.60572i −0.296949 + 0.296949i
\(78\) 0 0
\(79\) 0.611127 0.0687571 0.0343786 0.999409i \(-0.489055\pi\)
0.0343786 + 0.999409i \(0.489055\pi\)
\(80\) 0 0
\(81\) −0.730326 −0.0811473
\(82\) 0 0
\(83\) −1.29471 + 1.29471i −0.142113 + 0.142113i −0.774584 0.632471i \(-0.782041\pi\)
0.632471 + 0.774584i \(0.282041\pi\)
\(84\) 0 0
\(85\) 1.91529 + 1.91529i 0.207742 + 0.207742i
\(86\) 0 0
\(87\) 2.89597i 0.310481i
\(88\) 0 0
\(89\) 10.9236i 1.15790i 0.815363 + 0.578950i \(0.196537\pi\)
−0.815363 + 0.578950i \(0.803463\pi\)
\(90\) 0 0
\(91\) −19.7993 19.7993i −2.07553 2.07553i
\(92\) 0 0
\(93\) 3.10606 3.10606i 0.322083 0.322083i
\(94\) 0 0
\(95\) −0.619987 −0.0636093
\(96\) 0 0
\(97\) −12.7571 −1.29528 −0.647642 0.761945i \(-0.724245\pi\)
−0.647642 + 0.761945i \(0.724245\pi\)
\(98\) 0 0
\(99\) −1.26862 + 1.26862i −0.127501 + 0.127501i
\(100\) 0 0
\(101\) 8.59804 + 8.59804i 0.855537 + 0.855537i 0.990809 0.135272i \(-0.0431908\pi\)
−0.135272 + 0.990809i \(0.543191\pi\)
\(102\) 0 0
\(103\) 12.0328i 1.18563i 0.805338 + 0.592815i \(0.201984\pi\)
−0.805338 + 0.592815i \(0.798016\pi\)
\(104\) 0 0
\(105\) 4.10569i 0.400674i
\(106\) 0 0
\(107\) 2.37309 + 2.37309i 0.229415 + 0.229415i 0.812448 0.583033i \(-0.198134\pi\)
−0.583033 + 0.812448i \(0.698134\pi\)
\(108\) 0 0
\(109\) 3.24479 3.24479i 0.310794 0.310794i −0.534423 0.845217i \(-0.679471\pi\)
0.845217 + 0.534423i \(0.179471\pi\)
\(110\) 0 0
\(111\) 1.07230 0.101779
\(112\) 0 0
\(113\) 17.3173 1.62907 0.814536 0.580114i \(-0.196992\pi\)
0.814536 + 0.580114i \(0.196992\pi\)
\(114\) 0 0
\(115\) −2.54615 + 2.54615i −0.237430 + 0.237430i
\(116\) 0 0
\(117\) −9.63944 9.63944i −0.891166 0.891166i
\(118\) 0 0
\(119\) 10.9114i 1.00025i
\(120\) 0 0
\(121\) 10.1632i 0.923928i
\(122\) 0 0
\(123\) 0.435271 + 0.435271i 0.0392471 + 0.0392471i
\(124\) 0 0
\(125\) 0.707107 0.707107i 0.0632456 0.0632456i
\(126\) 0 0
\(127\) −15.5438 −1.37929 −0.689645 0.724147i \(-0.742234\pi\)
−0.689645 + 0.724147i \(0.742234\pi\)
\(128\) 0 0
\(129\) 7.25011 0.638337
\(130\) 0 0
\(131\) 11.2770 11.2770i 0.985280 0.985280i −0.0146129 0.999893i \(-0.504652\pi\)
0.999893 + 0.0146129i \(0.00465160\pi\)
\(132\) 0 0
\(133\) 1.76604 + 1.76604i 0.153135 + 0.153135i
\(134\) 0 0
\(135\) 5.05645i 0.435190i
\(136\) 0 0
\(137\) 3.67273i 0.313782i −0.987616 0.156891i \(-0.949853\pi\)
0.987616 0.156891i \(-0.0501472\pi\)
\(138\) 0 0
\(139\) 5.23552 + 5.23552i 0.444071 + 0.444071i 0.893378 0.449307i \(-0.148329\pi\)
−0.449307 + 0.893378i \(0.648329\pi\)
\(140\) 0 0
\(141\) 7.79604 7.79604i 0.656545 0.656545i
\(142\) 0 0
\(143\) −6.35830 −0.531708
\(144\) 0 0
\(145\) 2.84146 0.235970
\(146\) 0 0
\(147\) −6.65037 + 6.65037i −0.548514 + 0.548514i
\(148\) 0 0
\(149\) 3.29391 + 3.29391i 0.269848 + 0.269848i 0.829039 0.559191i \(-0.188888\pi\)
−0.559191 + 0.829039i \(0.688888\pi\)
\(150\) 0 0
\(151\) 6.93206i 0.564123i −0.959396 0.282061i \(-0.908982\pi\)
0.959396 0.282061i \(-0.0910182\pi\)
\(152\) 0 0
\(153\) 5.31232i 0.429475i
\(154\) 0 0
\(155\) −3.04759 3.04759i −0.244788 0.244788i
\(156\) 0 0
\(157\) −5.65633 + 5.65633i −0.451425 + 0.451425i −0.895827 0.444403i \(-0.853416\pi\)
0.444403 + 0.895827i \(0.353416\pi\)
\(158\) 0 0
\(159\) −5.87233 −0.465706
\(160\) 0 0
\(161\) 14.5055 1.14319
\(162\) 0 0
\(163\) 10.9746 10.9746i 0.859593 0.859593i −0.131697 0.991290i \(-0.542043\pi\)
0.991290 + 0.131697i \(0.0420425\pi\)
\(164\) 0 0
\(165\) −0.659247 0.659247i −0.0513223 0.0513223i
\(166\) 0 0
\(167\) 11.7686i 0.910685i 0.890316 + 0.455343i \(0.150483\pi\)
−0.890316 + 0.455343i \(0.849517\pi\)
\(168\) 0 0
\(169\) 35.3128i 2.71637i
\(170\) 0 0
\(171\) 0.859811 + 0.859811i 0.0657514 + 0.0657514i
\(172\) 0 0
\(173\) 1.40225 1.40225i 0.106611 0.106611i −0.651789 0.758400i \(-0.725981\pi\)
0.758400 + 0.651789i \(0.225981\pi\)
\(174\) 0 0
\(175\) −4.02840 −0.304518
\(176\) 0 0
\(177\) −1.77063 −0.133088
\(178\) 0 0
\(179\) −9.66131 + 9.66131i −0.722120 + 0.722120i −0.969037 0.246917i \(-0.920583\pi\)
0.246917 + 0.969037i \(0.420583\pi\)
\(180\) 0 0
\(181\) −0.294844 0.294844i −0.0219156 0.0219156i 0.696064 0.717980i \(-0.254933\pi\)
−0.717980 + 0.696064i \(0.754933\pi\)
\(182\) 0 0
\(183\) 10.0737i 0.744672i
\(184\) 0 0
\(185\) 1.05212i 0.0773533i
\(186\) 0 0
\(187\) −1.75204 1.75204i −0.128122 0.128122i
\(188\) 0 0
\(189\) −14.4033 + 14.4033i −1.04769 + 1.04769i
\(190\) 0 0
\(191\) −16.9352 −1.22539 −0.612694 0.790320i \(-0.709914\pi\)
−0.612694 + 0.790320i \(0.709914\pi\)
\(192\) 0 0
\(193\) −16.5927 −1.19437 −0.597185 0.802103i \(-0.703714\pi\)
−0.597185 + 0.802103i \(0.703714\pi\)
\(194\) 0 0
\(195\) 5.00921 5.00921i 0.358717 0.358717i
\(196\) 0 0
\(197\) 2.38392 + 2.38392i 0.169847 + 0.169847i 0.786912 0.617065i \(-0.211678\pi\)
−0.617065 + 0.786912i \(0.711678\pi\)
\(198\) 0 0
\(199\) 10.1411i 0.718883i 0.933168 + 0.359442i \(0.117033\pi\)
−0.933168 + 0.359442i \(0.882967\pi\)
\(200\) 0 0
\(201\) 7.55581i 0.532946i
\(202\) 0 0
\(203\) −8.09392 8.09392i −0.568082 0.568082i
\(204\) 0 0
\(205\) 0.427078 0.427078i 0.0298284 0.0298284i
\(206\) 0 0
\(207\) 7.06211 0.490851
\(208\) 0 0
\(209\) 0.567143 0.0392301
\(210\) 0 0
\(211\) 2.81171 2.81171i 0.193566 0.193566i −0.603669 0.797235i \(-0.706295\pi\)
0.797235 + 0.603669i \(0.206295\pi\)
\(212\) 0 0
\(213\) −9.94095 9.94095i −0.681143 0.681143i
\(214\) 0 0
\(215\) 7.11363i 0.485146i
\(216\) 0 0
\(217\) 17.3621i 1.17862i
\(218\) 0 0
\(219\) 0.943190 + 0.943190i 0.0637349 + 0.0637349i
\(220\) 0 0
\(221\) 13.3127 13.3127i 0.895507 0.895507i
\(222\) 0 0
\(223\) 14.0502 0.940871 0.470436 0.882434i \(-0.344097\pi\)
0.470436 + 0.882434i \(0.344097\pi\)
\(224\) 0 0
\(225\) −1.96126 −0.130751
\(226\) 0 0
\(227\) 13.3495 13.3495i 0.886037 0.886037i −0.108103 0.994140i \(-0.534478\pi\)
0.994140 + 0.108103i \(0.0344776\pi\)
\(228\) 0 0
\(229\) −8.78589 8.78589i −0.580588 0.580588i 0.354477 0.935065i \(-0.384659\pi\)
−0.935065 + 0.354477i \(0.884659\pi\)
\(230\) 0 0
\(231\) 3.75574i 0.247110i
\(232\) 0 0
\(233\) 15.1472i 0.992329i 0.868229 + 0.496165i \(0.165259\pi\)
−0.868229 + 0.496165i \(0.834741\pi\)
\(234\) 0 0
\(235\) −7.64928 7.64928i −0.498984 0.498984i
\(236\) 0 0
\(237\) 0.440423 0.440423i 0.0286085 0.0286085i
\(238\) 0 0
\(239\) 17.9151 1.15883 0.579414 0.815033i \(-0.303281\pi\)
0.579414 + 0.815033i \(0.303281\pi\)
\(240\) 0 0
\(241\) 25.6594 1.65287 0.826433 0.563035i \(-0.190366\pi\)
0.826433 + 0.563035i \(0.190366\pi\)
\(242\) 0 0
\(243\) −11.2527 + 11.2527i −0.721859 + 0.721859i
\(244\) 0 0
\(245\) 6.52518 + 6.52518i 0.416879 + 0.416879i
\(246\) 0 0
\(247\) 4.30937i 0.274199i
\(248\) 0 0
\(249\) 1.86613i 0.118261i
\(250\) 0 0
\(251\) 5.95195 + 5.95195i 0.375684 + 0.375684i 0.869542 0.493858i \(-0.164414\pi\)
−0.493858 + 0.869542i \(0.664414\pi\)
\(252\) 0 0
\(253\) 2.32913 2.32913i 0.146431 0.146431i
\(254\) 0 0
\(255\) 2.76059 0.172875
\(256\) 0 0
\(257\) −4.17369 −0.260348 −0.130174 0.991491i \(-0.541554\pi\)
−0.130174 + 0.991491i \(0.541554\pi\)
\(258\) 0 0
\(259\) −2.99697 + 2.99697i −0.186223 + 0.186223i
\(260\) 0 0
\(261\) −3.94059 3.94059i −0.243917 0.243917i
\(262\) 0 0
\(263\) 9.14469i 0.563885i 0.959431 + 0.281943i \(0.0909788\pi\)
−0.959431 + 0.281943i \(0.909021\pi\)
\(264\) 0 0
\(265\) 5.76178i 0.353944i
\(266\) 0 0
\(267\) 7.87234 + 7.87234i 0.481780 + 0.481780i
\(268\) 0 0
\(269\) 8.40029 8.40029i 0.512175 0.512175i −0.403017 0.915192i \(-0.632039\pi\)
0.915192 + 0.403017i \(0.132039\pi\)
\(270\) 0 0
\(271\) 18.7794 1.14077 0.570383 0.821379i \(-0.306795\pi\)
0.570383 + 0.821379i \(0.306795\pi\)
\(272\) 0 0
\(273\) −28.5376 −1.72717
\(274\) 0 0
\(275\) −0.646837 + 0.646837i −0.0390057 + 0.0390057i
\(276\) 0 0
\(277\) 3.54167 + 3.54167i 0.212798 + 0.212798i 0.805455 0.592657i \(-0.201921\pi\)
−0.592657 + 0.805455i \(0.701921\pi\)
\(278\) 0 0
\(279\) 8.45291i 0.506062i
\(280\) 0 0
\(281\) 2.31811i 0.138287i −0.997607 0.0691433i \(-0.977973\pi\)
0.997607 0.0691433i \(-0.0220266\pi\)
\(282\) 0 0
\(283\) 1.63197 + 1.63197i 0.0970108 + 0.0970108i 0.753947 0.656936i \(-0.228148\pi\)
−0.656936 + 0.753947i \(0.728148\pi\)
\(284\) 0 0
\(285\) −0.446808 + 0.446808i −0.0264666 + 0.0264666i
\(286\) 0 0
\(287\) −2.43307 −0.143619
\(288\) 0 0
\(289\) −9.66335 −0.568433
\(290\) 0 0
\(291\) −9.19367 + 9.19367i −0.538943 + 0.538943i
\(292\) 0 0
\(293\) 11.5789 + 11.5789i 0.676444 + 0.676444i 0.959194 0.282750i \(-0.0912466\pi\)
−0.282750 + 0.959194i \(0.591247\pi\)
\(294\) 0 0
\(295\) 1.73729i 0.101149i
\(296\) 0 0
\(297\) 4.62546i 0.268397i
\(298\) 0 0
\(299\) 17.6976 + 17.6976i 1.02348 + 1.02348i
\(300\) 0 0
\(301\) −20.2632 + 20.2632i −1.16795 + 1.16795i
\(302\) 0 0
\(303\) 12.3927 0.711945
\(304\) 0 0
\(305\) −9.88410 −0.565962
\(306\) 0 0
\(307\) −11.7116 + 11.7116i −0.668415 + 0.668415i −0.957349 0.288934i \(-0.906699\pi\)
0.288934 + 0.957349i \(0.406699\pi\)
\(308\) 0 0
\(309\) 8.67174 + 8.67174i 0.493318 + 0.493318i
\(310\) 0 0
\(311\) 11.2068i 0.635477i −0.948178 0.317739i \(-0.897077\pi\)
0.948178 0.317739i \(-0.102923\pi\)
\(312\) 0 0
\(313\) 7.50635i 0.424284i −0.977239 0.212142i \(-0.931956\pi\)
0.977239 0.212142i \(-0.0680439\pi\)
\(314\) 0 0
\(315\) 5.58667 + 5.58667i 0.314773 + 0.314773i
\(316\) 0 0
\(317\) −16.2854 + 16.2854i −0.914680 + 0.914680i −0.996636 0.0819564i \(-0.973883\pi\)
0.0819564 + 0.996636i \(0.473883\pi\)
\(318\) 0 0
\(319\) −2.59927 −0.145531
\(320\) 0 0
\(321\) 3.42044 0.190910
\(322\) 0 0
\(323\) −1.18745 + 1.18745i −0.0660716 + 0.0660716i
\(324\) 0 0
\(325\) −4.91492 4.91492i −0.272631 0.272631i
\(326\) 0 0
\(327\) 4.67686i 0.258631i
\(328\) 0 0
\(329\) 43.5781i 2.40254i
\(330\) 0 0
\(331\) 19.3846 + 19.3846i 1.06547 + 1.06547i 0.997701 + 0.0677707i \(0.0215886\pi\)
0.0677707 + 0.997701i \(0.478411\pi\)
\(332\) 0 0
\(333\) −1.45910 + 1.45910i −0.0799582 + 0.0799582i
\(334\) 0 0
\(335\) 7.41357 0.405047
\(336\) 0 0
\(337\) −7.82991 −0.426522 −0.213261 0.976995i \(-0.568408\pi\)
−0.213261 + 0.976995i \(0.568408\pi\)
\(338\) 0 0
\(339\) 12.4801 12.4801i 0.677825 0.677825i
\(340\) 0 0
\(341\) 2.78783 + 2.78783i 0.150969 + 0.150969i
\(342\) 0 0
\(343\) 8.97529i 0.484620i
\(344\) 0 0
\(345\) 3.66988i 0.197580i
\(346\) 0 0
\(347\) −8.91753 8.91753i −0.478718 0.478718i 0.426004 0.904721i \(-0.359921\pi\)
−0.904721 + 0.426004i \(0.859921\pi\)
\(348\) 0 0
\(349\) 6.69072 6.69072i 0.358146 0.358146i −0.504983 0.863129i \(-0.668501\pi\)
0.863129 + 0.504983i \(0.168501\pi\)
\(350\) 0 0
\(351\) −35.1461 −1.87596
\(352\) 0 0
\(353\) −2.05215 −0.109225 −0.0546126 0.998508i \(-0.517392\pi\)
−0.0546126 + 0.998508i \(0.517392\pi\)
\(354\) 0 0
\(355\) −9.75382 + 9.75382i −0.517679 + 0.517679i
\(356\) 0 0
\(357\) −7.86357 7.86357i −0.416184 0.416184i
\(358\) 0 0
\(359\) 9.52634i 0.502781i −0.967886 0.251391i \(-0.919112\pi\)
0.967886 0.251391i \(-0.0808879\pi\)
\(360\) 0 0
\(361\) 18.6156i 0.979769i
\(362\) 0 0
\(363\) −7.32435 7.32435i −0.384429 0.384429i
\(364\) 0 0
\(365\) 0.925435 0.925435i 0.0484395 0.0484395i
\(366\) 0 0
\(367\) −3.39736 −0.177341 −0.0886703 0.996061i \(-0.528262\pi\)
−0.0886703 + 0.996061i \(0.528262\pi\)
\(368\) 0 0
\(369\) −1.18456 −0.0616657
\(370\) 0 0
\(371\) 16.4125 16.4125i 0.852094 0.852094i
\(372\) 0 0
\(373\) 22.4895 + 22.4895i 1.16446 + 1.16446i 0.983488 + 0.180971i \(0.0579241\pi\)
0.180971 + 0.983488i \(0.442076\pi\)
\(374\) 0 0
\(375\) 1.01919i 0.0526305i
\(376\) 0 0
\(377\) 19.7502i 1.01719i
\(378\) 0 0
\(379\) −14.9819 14.9819i −0.769567 0.769567i 0.208463 0.978030i \(-0.433154\pi\)
−0.978030 + 0.208463i \(0.933154\pi\)
\(380\) 0 0
\(381\) −11.2020 + 11.2020i −0.573896 + 0.573896i
\(382\) 0 0
\(383\) 26.1197 1.33466 0.667328 0.744764i \(-0.267438\pi\)
0.667328 + 0.744764i \(0.267438\pi\)
\(384\) 0 0
\(385\) 3.68504 0.187807
\(386\) 0 0
\(387\) −9.86534 + 9.86534i −0.501483 + 0.501483i
\(388\) 0 0
\(389\) 2.08395 + 2.08395i 0.105660 + 0.105660i 0.757961 0.652300i \(-0.226196\pi\)
−0.652300 + 0.757961i \(0.726196\pi\)
\(390\) 0 0
\(391\) 9.75322i 0.493241i
\(392\) 0 0
\(393\) 16.2541i 0.819912i
\(394\) 0 0
\(395\) −0.432132 0.432132i −0.0217429 0.0217429i
\(396\) 0 0
\(397\) −1.43282 + 1.43282i −0.0719114 + 0.0719114i −0.742148 0.670236i \(-0.766193\pi\)
0.670236 + 0.742148i \(0.266193\pi\)
\(398\) 0 0
\(399\) 2.54547 0.127433
\(400\) 0 0
\(401\) −29.9853 −1.49739 −0.748697 0.662912i \(-0.769320\pi\)
−0.748697 + 0.662912i \(0.769320\pi\)
\(402\) 0 0
\(403\) −21.1830 + 21.1830i −1.05520 + 1.05520i
\(404\) 0 0
\(405\) 0.516418 + 0.516418i 0.0256610 + 0.0256610i
\(406\) 0 0
\(407\) 0.962443i 0.0477065i
\(408\) 0 0
\(409\) 4.17833i 0.206605i 0.994650 + 0.103302i \(0.0329410\pi\)
−0.994650 + 0.103302i \(0.967059\pi\)
\(410\) 0 0
\(411\) −2.64684 2.64684i −0.130559 0.130559i
\(412\) 0 0
\(413\) 4.94870 4.94870i 0.243510 0.243510i
\(414\) 0 0
\(415\) 1.83100 0.0898801
\(416\) 0 0
\(417\) 7.54620 0.369539
\(418\) 0 0
\(419\) 24.4667 24.4667i 1.19528 1.19528i 0.219712 0.975565i \(-0.429488\pi\)
0.975565 0.219712i \(-0.0705119\pi\)
\(420\) 0 0
\(421\) −25.6017 25.6017i −1.24775 1.24775i −0.956711 0.291039i \(-0.905999\pi\)
−0.291039 0.956711i \(-0.594001\pi\)
\(422\) 0 0
\(423\) 21.2164i 1.03158i
\(424\) 0 0
\(425\) 2.70862i 0.131388i
\(426\) 0 0
\(427\) 28.1550 + 28.1550i 1.36251 + 1.36251i
\(428\) 0 0
\(429\) −4.58226 + 4.58226i −0.221233 + 0.221233i
\(430\) 0 0
\(431\) 17.6126 0.848367 0.424184 0.905576i \(-0.360561\pi\)
0.424184 + 0.905576i \(0.360561\pi\)
\(432\) 0 0
\(433\) 27.0568 1.30027 0.650133 0.759820i \(-0.274713\pi\)
0.650133 + 0.759820i \(0.274713\pi\)
\(434\) 0 0
\(435\) 2.04776 2.04776i 0.0981827 0.0981827i
\(436\) 0 0
\(437\) −1.57858 1.57858i −0.0755138 0.0755138i
\(438\) 0 0
\(439\) 22.9965i 1.09756i 0.835967 + 0.548780i \(0.184908\pi\)
−0.835967 + 0.548780i \(0.815092\pi\)
\(440\) 0 0
\(441\) 18.0985i 0.861834i
\(442\) 0 0
\(443\) 13.7715 + 13.7715i 0.654303 + 0.654303i 0.954026 0.299723i \(-0.0968943\pi\)
−0.299723 + 0.954026i \(0.596894\pi\)
\(444\) 0 0
\(445\) 7.72415 7.72415i 0.366160 0.366160i
\(446\) 0 0
\(447\) 4.74766 0.224557
\(448\) 0 0
\(449\) 6.88838 0.325083 0.162541 0.986702i \(-0.448031\pi\)
0.162541 + 0.986702i \(0.448031\pi\)
\(450\) 0 0
\(451\) −0.390676 + 0.390676i −0.0183962 + 0.0183962i
\(452\) 0 0
\(453\) −4.99575 4.99575i −0.234721 0.234721i
\(454\) 0 0
\(455\) 28.0004i 1.31268i
\(456\) 0 0
\(457\) 2.52622i 0.118171i −0.998253 0.0590857i \(-0.981181\pi\)
0.998253 0.0590857i \(-0.0188185\pi\)
\(458\) 0 0
\(459\) −9.68454 9.68454i −0.452036 0.452036i
\(460\) 0 0
\(461\) −9.23502 + 9.23502i −0.430118 + 0.430118i −0.888668 0.458550i \(-0.848369\pi\)
0.458550 + 0.888668i \(0.348369\pi\)
\(462\) 0 0
\(463\) −11.2676 −0.523652 −0.261826 0.965115i \(-0.584325\pi\)
−0.261826 + 0.965115i \(0.584325\pi\)
\(464\) 0 0
\(465\) −4.39263 −0.203703
\(466\) 0 0
\(467\) 25.8291 25.8291i 1.19523 1.19523i 0.219650 0.975579i \(-0.429508\pi\)
0.975579 0.219650i \(-0.0704916\pi\)
\(468\) 0 0
\(469\) −21.1176 21.1176i −0.975122 0.975122i
\(470\) 0 0
\(471\) 8.15273i 0.375658i
\(472\) 0 0
\(473\) 6.50731i 0.299206i
\(474\) 0 0
\(475\) 0.438397 + 0.438397i 0.0201150 + 0.0201150i
\(476\) 0 0
\(477\) 7.99056 7.99056i 0.365863 0.365863i
\(478\) 0 0
\(479\) −15.7261 −0.718545 −0.359273 0.933233i \(-0.616975\pi\)
−0.359273 + 0.933233i \(0.616975\pi\)
\(480\) 0 0
\(481\) −7.31301 −0.333445
\(482\) 0 0
\(483\) 10.4537 10.4537i 0.475660 0.475660i
\(484\) 0 0
\(485\) 9.02061 + 9.02061i 0.409605 + 0.409605i
\(486\) 0 0
\(487\) 35.3717i 1.60284i −0.598100 0.801422i \(-0.704077\pi\)
0.598100 0.801422i \(-0.295923\pi\)
\(488\) 0 0
\(489\) 15.8181i 0.715320i
\(490\) 0 0
\(491\) −7.95703 7.95703i −0.359096 0.359096i 0.504384 0.863480i \(-0.331720\pi\)
−0.863480 + 0.504384i \(0.831720\pi\)
\(492\) 0 0
\(493\) 5.44221 5.44221i 0.245105 0.245105i
\(494\) 0 0
\(495\) 1.79409 0.0806385
\(496\) 0 0
\(497\) 55.5677 2.49255
\(498\) 0 0
\(499\) −11.5864 + 11.5864i −0.518677 + 0.518677i −0.917171 0.398494i \(-0.869533\pi\)
0.398494 + 0.917171i \(0.369533\pi\)
\(500\) 0 0
\(501\) 8.48135 + 8.48135i 0.378919 + 0.378919i
\(502\) 0 0
\(503\) 23.5051i 1.04804i −0.851706 0.524020i \(-0.824432\pi\)
0.851706 0.524020i \(-0.175568\pi\)
\(504\) 0 0
\(505\) 12.1595i 0.541089i
\(506\) 0 0
\(507\) −25.4490 25.4490i −1.13023 1.13023i
\(508\) 0 0
\(509\) 3.08381 3.08381i 0.136687 0.136687i −0.635452 0.772140i \(-0.719186\pi\)
0.772140 + 0.635452i \(0.219186\pi\)
\(510\) 0 0
\(511\) −5.27222 −0.233229
\(512\) 0 0
\(513\) 3.13493 0.138411
\(514\) 0 0
\(515\) 8.50850 8.50850i 0.374929 0.374929i
\(516\) 0 0
\(517\) 6.99730 + 6.99730i 0.307741 + 0.307741i
\(518\) 0 0
\(519\) 2.02113i 0.0887178i
\(520\) 0 0
\(521\) 11.5762i 0.507161i −0.967314 0.253580i \(-0.918392\pi\)
0.967314 0.253580i \(-0.0816083\pi\)
\(522\) 0 0
\(523\) −3.97900 3.97900i −0.173990 0.173990i 0.614740 0.788730i \(-0.289261\pi\)
−0.788730 + 0.614740i \(0.789261\pi\)
\(524\) 0 0
\(525\) −2.90316 + 2.90316i −0.126704 + 0.126704i
\(526\) 0 0
\(527\) −11.6740 −0.508527
\(528\) 0 0
\(529\) 10.0342 0.436271
\(530\) 0 0
\(531\) 2.40932 2.40932i 0.104555 0.104555i
\(532\) 0 0
\(533\) −2.96851 2.96851i −0.128580 0.128580i
\(534\) 0 0
\(535\) 3.35605i 0.145095i
\(536\) 0 0
\(537\) 13.9253i 0.600921i
\(538\) 0 0
\(539\) −5.96902 5.96902i −0.257104 0.257104i
\(540\) 0 0
\(541\) −17.2148 + 17.2148i −0.740123 + 0.740123i −0.972602 0.232478i \(-0.925316\pi\)
0.232478 + 0.972602i \(0.425316\pi\)
\(542\) 0 0
\(543\) −0.424973 −0.0182373
\(544\) 0 0
\(545\) −4.58882 −0.196564
\(546\) 0 0
\(547\) 20.3610 20.3610i 0.870573 0.870573i −0.121962 0.992535i \(-0.538919\pi\)
0.992535 + 0.121962i \(0.0389187\pi\)
\(548\) 0 0
\(549\) 13.7075 + 13.7075i 0.585021 + 0.585021i
\(550\) 0 0
\(551\) 1.76167i 0.0750495i
\(552\) 0 0
\(553\) 2.46186i 0.104689i
\(554\) 0 0
\(555\) −0.758234 0.758234i −0.0321852 0.0321852i
\(556\) 0 0
\(557\) 22.7029 22.7029i 0.961954 0.961954i −0.0373478 0.999302i \(-0.511891\pi\)
0.999302 + 0.0373478i \(0.0118910\pi\)
\(558\) 0 0
\(559\) −49.4451 −2.09130
\(560\) 0 0
\(561\) −2.52529 −0.106618
\(562\) 0 0
\(563\) −15.4153 + 15.4153i −0.649676 + 0.649676i −0.952915 0.303238i \(-0.901932\pi\)
0.303238 + 0.952915i \(0.401932\pi\)
\(564\) 0 0
\(565\) −12.2452 12.2452i −0.515158 0.515158i
\(566\) 0 0
\(567\) 2.94204i 0.123554i
\(568\) 0 0
\(569\) 22.6529i 0.949660i −0.880078 0.474830i \(-0.842510\pi\)
0.880078 0.474830i \(-0.157490\pi\)
\(570\) 0 0
\(571\) 13.4941 + 13.4941i 0.564710 + 0.564710i 0.930642 0.365931i \(-0.119250\pi\)
−0.365931 + 0.930642i \(0.619250\pi\)
\(572\) 0 0
\(573\) −12.2047 + 12.2047i −0.509860 + 0.509860i
\(574\) 0 0
\(575\) 3.60080 0.150164
\(576\) 0 0
\(577\) 6.08684 0.253398 0.126699 0.991941i \(-0.459562\pi\)
0.126699 + 0.991941i \(0.459562\pi\)
\(578\) 0 0
\(579\) −11.9579 + 11.9579i −0.496955 + 0.496955i
\(580\) 0 0
\(581\) −5.21561 5.21561i −0.216380 0.216380i
\(582\) 0 0
\(583\) 5.27068i 0.218289i
\(584\) 0 0
\(585\) 13.6322i 0.563623i
\(586\) 0 0
\(587\) −21.4418 21.4418i −0.884999 0.884999i 0.109039 0.994038i \(-0.465223\pi\)
−0.994038 + 0.109039i \(0.965223\pi\)
\(588\) 0 0
\(589\) 1.88946 1.88946i 0.0778540 0.0778540i
\(590\) 0 0
\(591\) 3.43605 0.141340
\(592\) 0 0
\(593\) −28.2005 −1.15806 −0.579028 0.815308i \(-0.696568\pi\)
−0.579028 + 0.815308i \(0.696568\pi\)
\(594\) 0 0
\(595\) −7.71554 + 7.71554i −0.316306 + 0.316306i
\(596\) 0 0
\(597\) 7.30841 + 7.30841i 0.299113 + 0.299113i
\(598\) 0 0
\(599\) 38.0516i 1.55475i −0.629039 0.777374i \(-0.716551\pi\)
0.629039 0.777374i \(-0.283449\pi\)
\(600\) 0 0
\(601\) 19.0716i 0.777947i 0.921249 + 0.388974i \(0.127170\pi\)
−0.921249 + 0.388974i \(0.872830\pi\)
\(602\) 0 0
\(603\) −10.2813 10.2813i −0.418687 0.418687i
\(604\) 0 0
\(605\) −7.18647 + 7.18647i −0.292172 + 0.292172i
\(606\) 0 0
\(607\) 5.73433 0.232749 0.116375 0.993205i \(-0.462873\pi\)
0.116375 + 0.993205i \(0.462873\pi\)
\(608\) 0 0
\(609\) −11.6661 −0.472736
\(610\) 0 0
\(611\) −53.1682 + 53.1682i −2.15096 + 2.15096i
\(612\) 0 0
\(613\) −5.36917 5.36917i −0.216859 0.216859i 0.590315 0.807173i \(-0.299004\pi\)
−0.807173 + 0.590315i \(0.799004\pi\)
\(614\) 0 0
\(615\) 0.615566i 0.0248220i
\(616\) 0 0
\(617\) 28.2915i 1.13897i −0.822000 0.569487i \(-0.807142\pi\)
0.822000 0.569487i \(-0.192858\pi\)
\(618\) 0 0
\(619\) −18.9669 18.9669i −0.762345 0.762345i 0.214401 0.976746i \(-0.431220\pi\)
−0.976746 + 0.214401i \(0.931220\pi\)
\(620\) 0 0
\(621\) 12.8745 12.8745i 0.516635 0.516635i
\(622\) 0 0
\(623\) −44.0046 −1.76301
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0.408725 0.408725i 0.0163229 0.0163229i
\(628\) 0 0
\(629\) −2.01511 2.01511i −0.0803477 0.0803477i
\(630\) 0 0
\(631\) 41.7662i 1.66269i 0.555758 + 0.831344i \(0.312428\pi\)
−0.555758 + 0.831344i \(0.687572\pi\)
\(632\) 0 0
\(633\) 4.05265i 0.161078i
\(634\) 0 0
\(635\) 10.9911 + 10.9911i 0.436170 + 0.436170i
\(636\) 0 0
\(637\) 45.3549 45.3549i 1.79703 1.79703i
\(638\) 0 0
\(639\) 27.0536 1.07022
\(640\) 0 0
\(641\) −2.85195 −0.112645 −0.0563227 0.998413i \(-0.517938\pi\)
−0.0563227 + 0.998413i \(0.517938\pi\)
\(642\) 0 0
\(643\) 31.8921 31.8921i 1.25770 1.25770i 0.305516 0.952187i \(-0.401171\pi\)
0.952187 0.305516i \(-0.0988288\pi\)
\(644\) 0 0
\(645\) −5.12660 5.12660i −0.201860 0.201860i
\(646\) 0 0
\(647\) 7.83402i 0.307987i 0.988072 + 0.153994i \(0.0492135\pi\)
−0.988072 + 0.153994i \(0.950786\pi\)
\(648\) 0 0
\(649\) 1.58922i 0.0623823i
\(650\) 0 0
\(651\) 12.5124 + 12.5124i 0.490401 + 0.490401i
\(652\) 0 0
\(653\) 12.6822 12.6822i 0.496292 0.496292i −0.413989 0.910282i \(-0.635865\pi\)
0.910282 + 0.413989i \(0.135865\pi\)
\(654\) 0 0
\(655\) −15.9482 −0.623146
\(656\) 0 0
\(657\) −2.56682 −0.100141
\(658\) 0 0
\(659\) −12.9694 + 12.9694i −0.505217 + 0.505217i −0.913055 0.407837i \(-0.866283\pi\)
0.407837 + 0.913055i \(0.366283\pi\)
\(660\) 0 0
\(661\) 6.85796 + 6.85796i 0.266744 + 0.266744i 0.827787 0.561043i \(-0.189600\pi\)
−0.561043 + 0.827787i \(0.689600\pi\)
\(662\) 0 0
\(663\) 19.1882i 0.745206i
\(664\) 0 0
\(665\) 2.49756i 0.0968511i
\(666\) 0 0
\(667\) 7.23478 + 7.23478i 0.280132 + 0.280132i
\(668\) 0 0
\(669\) 10.1256 10.1256i 0.391478 0.391478i
\(670\) 0 0
\(671\) 9.04164 0.349049
\(672\) 0 0
\(673\) 23.1277 0.891508 0.445754 0.895155i \(-0.352936\pi\)
0.445754 + 0.895155i \(0.352936\pi\)
\(674\) 0 0
\(675\) −3.57545 + 3.57545i −0.137619 + 0.137619i
\(676\) 0 0
\(677\) 20.2521 + 20.2521i 0.778352 + 0.778352i 0.979550 0.201199i \(-0.0644837\pi\)
−0.201199 + 0.979550i \(0.564484\pi\)
\(678\) 0 0
\(679\) 51.3905i 1.97219i
\(680\) 0 0
\(681\) 19.2412i 0.737326i
\(682\) 0 0
\(683\) −26.2957 26.2957i −1.00618 1.00618i −0.999981 0.00619708i \(-0.998027\pi\)
−0.00619708 0.999981i \(-0.501973\pi\)
\(684\) 0 0
\(685\) −2.59701 + 2.59701i −0.0992267 + 0.0992267i
\(686\) 0 0
\(687\) −12.6635 −0.483143
\(688\) 0 0
\(689\) 40.0487 1.52573
\(690\) 0 0
\(691\) 4.91230 4.91230i 0.186873 0.186873i −0.607470 0.794343i \(-0.707815\pi\)
0.794343 + 0.607470i \(0.207815\pi\)
\(692\) 0 0
\(693\) −5.11049 5.11049i −0.194132 0.194132i
\(694\) 0 0
\(695\) 7.40414i 0.280855i
\(696\) 0 0
\(697\) 1.63595i 0.0619661i
\(698\) 0 0
\(699\) 10.9162 + 10.9162i 0.412889 + 0.412889i
\(700\) 0 0
\(701\) 12.3598 12.3598i 0.466824 0.466824i −0.434060 0.900884i \(-0.642919\pi\)
0.900884 + 0.434060i \(0.142919\pi\)
\(702\) 0 0
\(703\) 0.652300 0.0246020
\(704\) 0 0
\(705\) −11.0253 −0.415235
\(706\) 0 0
\(707\) −34.6363 + 34.6363i −1.30263 + 1.30263i
\(708\) 0 0
\(709\) 26.6076 + 26.6076i 0.999270 + 0.999270i 1.00000 0.000729493i \(-0.000232205\pi\)
−0.000729493 1.00000i \(0.500232\pi\)
\(710\) 0 0
\(711\) 1.19858i 0.0449502i
\(712\) 0 0
\(713\) 15.5192i 0.581200i
\(714\) 0 0
\(715\) 4.49600 + 4.49600i 0.168141 + 0.168141i
\(716\) 0 0
\(717\) 12.9109 12.9109i 0.482166 0.482166i
\(718\) 0 0
\(719\) 50.9765 1.90110 0.950551 0.310570i \(-0.100520\pi\)
0.950551 + 0.310570i \(0.100520\pi\)
\(720\) 0 0
\(721\) −48.4731 −1.80523
\(722\) 0 0
\(723\) 18.4920 18.4920i 0.687726 0.687726i
\(724\) 0 0
\(725\) −2.00921 2.00921i −0.0746203 0.0746203i
\(726\) 0 0
\(727\) 13.2824i 0.492616i −0.969192 0.246308i \(-0.920782\pi\)
0.969192 0.246308i \(-0.0792175\pi\)
\(728\) 0 0
\(729\) 14.0280i 0.519556i
\(730\) 0 0
\(731\) −13.6246 13.6246i −0.503926 0.503926i
\(732\) 0 0
\(733\) −21.3075 + 21.3075i −0.787012 + 0.787012i −0.981003 0.193991i \(-0.937857\pi\)
0.193991 + 0.981003i \(0.437857\pi\)
\(734\) 0 0
\(735\) 9.40505 0.346910
\(736\) 0 0
\(737\) −6.78168 −0.249807
\(738\) 0 0
\(739\) −5.83841 + 5.83841i −0.214769 + 0.214769i −0.806290 0.591521i \(-0.798528\pi\)
0.591521 + 0.806290i \(0.298528\pi\)
\(740\) 0 0
\(741\) 3.10565 + 3.10565i 0.114089 + 0.114089i
\(742\) 0 0
\(743\) 3.25778i 0.119516i 0.998213 + 0.0597582i \(0.0190330\pi\)
−0.998213 + 0.0597582i \(0.980967\pi\)
\(744\) 0 0
\(745\) 4.65829i 0.170667i
\(746\) 0 0
\(747\) −2.53926 2.53926i −0.0929068 0.0929068i
\(748\) 0 0
\(749\) −9.55975 + 9.55975i −0.349306 + 0.349306i
\(750\) 0 0
\(751\) 20.7322 0.756530 0.378265 0.925697i \(-0.376521\pi\)
0.378265 + 0.925697i \(0.376521\pi\)
\(752\) 0 0
\(753\) 8.57882 0.312630
\(754\) 0 0
\(755\) −4.90171 + 4.90171i −0.178391 + 0.178391i
\(756\) 0 0
\(757\) −23.3278 23.3278i −0.847862 0.847862i 0.142004 0.989866i \(-0.454645\pi\)
−0.989866 + 0.142004i \(0.954645\pi\)
\(758\) 0 0
\(759\) 3.35709i 0.121854i
\(760\) 0 0
\(761\) 24.0242i 0.870878i 0.900218 + 0.435439i \(0.143407\pi\)
−0.900218 + 0.435439i \(0.856593\pi\)
\(762\) 0 0
\(763\) 13.0713 + 13.0713i 0.473213 + 0.473213i
\(764\) 0 0
\(765\) −3.75638 + 3.75638i −0.135812 + 0.135812i
\(766\) 0 0
\(767\) 12.0755 0.436021
\(768\) 0 0
\(769\) −2.70862 −0.0976754 −0.0488377 0.998807i \(-0.515552\pi\)
−0.0488377 + 0.998807i \(0.515552\pi\)
\(770\) 0 0
\(771\) −3.00787 + 3.00787i −0.108326 + 0.108326i
\(772\) 0 0
\(773\) −22.9473 22.9473i −0.825358 0.825358i 0.161513 0.986871i \(-0.448363\pi\)
−0.986871 + 0.161513i \(0.948363\pi\)
\(774\) 0 0
\(775\) 4.30994i 0.154817i
\(776\) 0 0
\(777\) 4.31967i 0.154967i
\(778\) 0 0
\(779\) 0.264783 + 0.264783i 0.00948682 + 0.00948682i
\(780\) 0 0
\(781\) 8.92246 8.92246i 0.319271 0.319271i
\(782\) 0 0
\(783\) −14.3677 −0.513459
\(784\) 0 0
\(785\) 7.99926 0.285506
\(786\) 0 0
\(787\) 14.0592 14.0592i 0.501157 0.501157i −0.410641 0.911797i \(-0.634695\pi\)
0.911797 + 0.410641i \(0.134695\pi\)
\(788\) 0 0
\(789\) 6.59033 + 6.59033i 0.234622 + 0.234622i
\(790\) 0 0
\(791\) 69.7609i 2.48041i
\(792\) 0 0
\(793\) 68.7019i 2.43967i
\(794\) 0 0
\(795\) 4.15236 + 4.15236i 0.147269 + 0.147269i
\(796\) 0 0
\(797\) −35.4258 + 35.4258i −1.25485 + 1.25485i −0.301326 + 0.953521i \(0.597429\pi\)
−0.953521 + 0.301326i \(0.902571\pi\)
\(798\) 0 0
\(799\) −29.3011 −1.03660
\(800\) 0 0
\(801\) −21.4240 −0.756981
\(802\) 0 0
\(803\) −0.846556 + 0.846556i −0.0298743 + 0.0298743i
\(804\) 0 0
\(805\) −10.2569 10.2569i −0.361509 0.361509i
\(806\) 0 0
\(807\) 12.1077i 0.426212i
\(808\) 0 0
\(809\) 16.9217i 0.594935i 0.954732 + 0.297467i \(0.0961420\pi\)
−0.954732 + 0.297467i \(0.903858\pi\)
\(810\) 0 0
\(811\) 20.4270 + 20.4270i 0.717288 + 0.717288i 0.968049 0.250761i \(-0.0806807\pi\)
−0.250761 + 0.968049i \(0.580681\pi\)
\(812\) 0 0
\(813\) 13.5338 13.5338i 0.474651 0.474651i
\(814\) 0 0
\(815\) −15.5204 −0.543655
\(816\) 0 0
\(817\) 4.41036 0.154299
\(818\) 0 0
\(819\) 38.8315 38.8315i 1.35688 1.35688i
\(820\) 0 0
\(821\) −32.4563 32.4563i −1.13273 1.13273i −0.989721 0.143013i \(-0.954321\pi\)
−0.143013 0.989721i \(-0.545679\pi\)
\(822\) 0 0
\(823\) 6.50705i 0.226821i 0.993548 + 0.113411i \(0.0361776\pi\)
−0.993548 + 0.113411i \(0.963822\pi\)
\(824\) 0 0
\(825\) 0.932316i 0.0324591i
\(826\) 0 0
\(827\) −27.0528 27.0528i −0.940718 0.940718i 0.0576204 0.998339i \(-0.481649\pi\)
−0.998339 + 0.0576204i \(0.981649\pi\)
\(828\) 0 0
\(829\) 8.54216 8.54216i 0.296682 0.296682i −0.543031 0.839713i \(-0.682723\pi\)
0.839713 + 0.543031i \(0.182723\pi\)
\(830\) 0 0
\(831\) 5.10477 0.177083
\(832\) 0 0
\(833\) 24.9952 0.866032
\(834\) 0 0
\(835\) 8.32169 8.32169i 0.287984 0.287984i
\(836\) 0 0
\(837\) 15.4099 + 15.4099i 0.532646 + 0.532646i
\(838\) 0 0
\(839\) 24.4138i 0.842860i 0.906861 + 0.421430i \(0.138472\pi\)
−0.906861 + 0.421430i \(0.861528\pi\)
\(840\) 0 0
\(841\) 20.9261i 0.721590i
\(842\) 0 0
\(843\) −1.67060 1.67060i −0.0575384 0.0575384i
\(844\) 0 0
\(845\) −24.9700 + 24.9700i −0.858993 + 0.858993i
\(846\) 0 0
\(847\) 40.9414 1.40676
\(848\) 0 0
\(849\) 2.35224 0.0807287
\(850\) 0 0
\(851\) 2.67885 2.67885i 0.0918299 0.0918299i
\(852\) 0 0
\(853\) −8.23270 8.23270i −0.281882 0.281882i 0.551977 0.833859i \(-0.313874\pi\)
−0.833859 + 0.551977i \(0.813874\pi\)
\(854\) 0 0
\(855\) 1.21596i 0.0415848i
\(856\) 0 0
\(857\) 4.73909i 0.161884i −0.996719 0.0809421i \(-0.974207\pi\)
0.996719 0.0809421i \(-0.0257929\pi\)
\(858\) 0 0
\(859\) 9.65120 + 9.65120i 0.329295 + 0.329295i 0.852318 0.523024i \(-0.175196\pi\)
−0.523024 + 0.852318i \(0.675196\pi\)
\(860\) 0 0
\(861\) −1.75345 + 1.75345i −0.0597573 + 0.0597573i
\(862\) 0 0
\(863\) −3.80368 −0.129479 −0.0647393 0.997902i \(-0.520622\pi\)
−0.0647393 + 0.997902i \(0.520622\pi\)
\(864\) 0 0
\(865\) −1.98309 −0.0674269
\(866\) 0 0
\(867\) −6.96412 + 6.96412i −0.236514 + 0.236514i
\(868\) 0 0
\(869\) 0.395300 + 0.395300i 0.0134096 + 0.0134096i
\(870\) 0 0
\(871\) 51.5299i 1.74602i
\(872\) 0 0
\(873\) 25.0199i 0.846796i
\(874\) 0 0
\(875\) 2.84851 + 2.84851i 0.0962972 + 0.0962972i
\(876\) 0 0
\(877\) −2.38917 + 2.38917i −0.0806765 + 0.0806765i −0.746294 0.665617i \(-0.768168\pi\)
0.665617 + 0.746294i \(0.268168\pi\)
\(878\) 0 0
\(879\) 16.6891 0.562911
\(880\) 0 0
\(881\) −23.6195 −0.795762 −0.397881 0.917437i \(-0.630254\pi\)
−0.397881 + 0.917437i \(0.630254\pi\)
\(882\) 0 0
\(883\) 9.64752 9.64752i 0.324665 0.324665i −0.525889 0.850553i \(-0.676267\pi\)
0.850553 + 0.525889i \(0.176267\pi\)
\(884\) 0 0
\(885\) 1.25202 + 1.25202i 0.0420862 + 0.0420862i
\(886\) 0 0
\(887\) 8.91140i 0.299215i 0.988745 + 0.149608i \(0.0478011\pi\)
−0.988745 + 0.149608i \(0.952199\pi\)
\(888\) 0 0
\(889\) 62.6167i 2.10010i
\(890\) 0 0
\(891\) −0.472402 0.472402i −0.0158261 0.0158261i
\(892\) 0 0
\(893\) 4.74246 4.74246i 0.158700 0.158700i
\(894\) 0 0
\(895\) 13.6632 0.456709
\(896\) 0 0
\(897\) 25.5084 0.851702
\(898\) 0 0
\(899\) −8.65959 + 8.65959i −0.288813 + 0.288813i
\(900\) 0 0
\(901\) 11.0355 + 11.0355i 0.367645 + 0.367645i
\(902\) 0 0
\(903\) 29.2064i 0.971927i
\(904\) 0 0
\(905\) 0.416973i 0.0138606i
\(906\) 0 0
\(907\) −23.4874 23.4874i −0.779886 0.779886i 0.199925 0.979811i \(-0.435930\pi\)
−0.979811 + 0.199925i \(0.935930\pi\)
\(908\) 0 0
\(909\) −16.8630 + 16.8630i −0.559310 + 0.559310i
\(910\) 0 0
\(911\) −1.77171 −0.0586993 −0.0293497 0.999569i \(-0.509344\pi\)
−0.0293497 + 0.999569i \(0.509344\pi\)
\(912\) 0 0
\(913\) −1.67493 −0.0554322
\(914\) 0 0
\(915\) −7.12321 + 7.12321i −0.235486 + 0.235486i
\(916\) 0 0
\(917\) 45.4285 + 45.4285i 1.50018 + 1.50018i
\(918\) 0 0
\(919\) 46.2001i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(920\) 0 0
\(921\) 16.8804i 0.556229i
\(922\) 0 0
\(923\) 67.7963 + 67.7963i 2.23154 + 2.23154i
\(924\) 0 0
\(925\) −0.743961 + 0.743961i −0.0244613 + 0.0244613i
\(926\) 0 0
\(927\) −23.5995 −0.775110
\(928\) 0 0
\(929\) −35.5011 −1.16475 −0.582376 0.812920i \(-0.697877\pi\)
−0.582376 + 0.812920i \(0.697877\pi\)
\(930\) 0 0
\(931\) −4.04553 + 4.04553i −0.132587 + 0.132587i
\(932\) 0 0
\(933\) −8.07641 8.07641i −0.264410 0.264410i
\(934\) 0 0
\(935\) 2.47776i 0.0810313i
\(936\) 0 0
\(937\) 10.0385i 0.327945i −0.986465 0.163972i \(-0.947569\pi\)
0.986465 0.163972i \(-0.0524308\pi\)
\(938\) 0 0
\(939\) −5.40962 5.40962i −0.176536 0.176536i
\(940\) 0 0
\(941\) −42.3367 + 42.3367i −1.38014 + 1.38014i −0.535777 + 0.844359i \(0.679981\pi\)
−0.844359 + 0.535777i \(0.820019\pi\)
\(942\) 0 0
\(943\) 2.17481 0.0708215
\(944\) 0 0
\(945\) 20.3694 0.662616
\(946\) 0 0
\(947\) 36.3384 36.3384i 1.18084 1.18084i 0.201313 0.979527i \(-0.435479\pi\)
0.979527 0.201313i \(-0.0645209\pi\)
\(948\) 0 0
\(949\) −6.43246 6.43246i −0.208807 0.208807i
\(950\) 0 0
\(951\) 23.4729i 0.761161i
\(952\) 0 0
\(953\) 32.7338i 1.06035i −0.847888 0.530176i \(-0.822126\pi\)
0.847888 0.530176i \(-0.177874\pi\)
\(954\) 0 0
\(955\) 11.9750 + 11.9750i 0.387502 + 0.387502i
\(956\) 0 0
\(957\) −1.87322 + 1.87322i −0.0605527 + 0.0605527i
\(958\) 0 0
\(959\) 14.7952 0.477762
\(960\) 0 0
\(961\) −12.4244 −0.400789
\(962\) 0 0
\(963\) −4.65424 + 4.65424i −0.149981 + 0.149981i
\(964\) 0 0
\(965\) 11.7328 + 11.7328i 0.377693 + 0.377693i
\(966\) 0 0
\(967\) 49.7169i 1.59879i −0.600807 0.799394i \(-0.705154\pi\)
0.600807 0.799394i \(-0.294846\pi\)
\(968\) 0 0
\(969\) 1.71153i 0.0549823i
\(970\) 0 0
\(971\) 24.7937 + 24.7937i 0.795667 + 0.795667i 0.982409 0.186742i \(-0.0597928\pi\)
−0.186742 + 0.982409i \(0.559793\pi\)
\(972\) 0 0
\(973\) −21.0908 + 21.0908i −0.676139 + 0.676139i
\(974\) 0 0
\(975\) −7.08410 −0.226873
\(976\) 0 0
\(977\) −49.4546 −1.58219 −0.791096 0.611692i \(-0.790489\pi\)
−0.791096 + 0.611692i \(0.790489\pi\)
\(978\) 0 0
\(979\) −7.06579 + 7.06579i −0.225824 + 0.225824i
\(980\) 0 0
\(981\) 6.36387 + 6.36387i 0.203183 + 0.203183i
\(982\) 0 0
\(983\) 23.9656i 0.764383i 0.924083 + 0.382191i \(0.124830\pi\)
−0.924083 + 0.382191i \(0.875170\pi\)
\(984\) 0 0
\(985\) 3.37137i 0.107421i
\(986\) 0 0
\(987\) 31.4056 + 31.4056i 0.999650 + 0.999650i
\(988\) 0 0
\(989\) 18.1124 18.1124i 0.575940 0.575940i
\(990\) 0 0
\(991\) 28.8345 0.915957 0.457978 0.888963i \(-0.348574\pi\)
0.457978 + 0.888963i \(0.348574\pi\)
\(992\) 0 0
\(993\) 27.9399 0.886645
\(994\) 0 0
\(995\) 7.17084 7.17084i 0.227331 0.227331i
\(996\) 0 0
\(997\) −6.03212 6.03212i −0.191039 0.191039i 0.605106 0.796145i \(-0.293131\pi\)
−0.796145 + 0.605106i \(0.793131\pi\)
\(998\) 0 0
\(999\) 5.31998i 0.168317i
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 640.2.l.b.481.6 16
4.3 odd 2 640.2.l.a.481.3 16
8.3 odd 2 320.2.l.a.241.6 16
8.5 even 2 80.2.l.a.21.1 16
16.3 odd 4 640.2.l.a.161.3 16
16.5 even 4 80.2.l.a.61.1 yes 16
16.11 odd 4 320.2.l.a.81.6 16
16.13 even 4 inner 640.2.l.b.161.6 16
24.5 odd 2 720.2.t.c.181.8 16
24.11 even 2 2880.2.t.c.2161.1 16
32.3 odd 8 5120.2.a.t.1.4 8
32.13 even 8 5120.2.a.s.1.4 8
32.19 odd 8 5120.2.a.u.1.5 8
32.29 even 8 5120.2.a.v.1.5 8
40.3 even 4 1600.2.q.g.49.6 16
40.13 odd 4 400.2.q.h.149.4 16
40.19 odd 2 1600.2.l.i.1201.3 16
40.27 even 4 1600.2.q.h.49.3 16
40.29 even 2 400.2.l.h.101.8 16
40.37 odd 4 400.2.q.g.149.5 16
48.5 odd 4 720.2.t.c.541.8 16
48.11 even 4 2880.2.t.c.721.4 16
80.27 even 4 1600.2.q.g.849.6 16
80.37 odd 4 400.2.q.h.349.4 16
80.43 even 4 1600.2.q.h.849.3 16
80.53 odd 4 400.2.q.g.349.5 16
80.59 odd 4 1600.2.l.i.401.3 16
80.69 even 4 400.2.l.h.301.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.1 16 8.5 even 2
80.2.l.a.61.1 yes 16 16.5 even 4
320.2.l.a.81.6 16 16.11 odd 4
320.2.l.a.241.6 16 8.3 odd 2
400.2.l.h.101.8 16 40.29 even 2
400.2.l.h.301.8 16 80.69 even 4
400.2.q.g.149.5 16 40.37 odd 4
400.2.q.g.349.5 16 80.53 odd 4
400.2.q.h.149.4 16 40.13 odd 4
400.2.q.h.349.4 16 80.37 odd 4
640.2.l.a.161.3 16 16.3 odd 4
640.2.l.a.481.3 16 4.3 odd 2
640.2.l.b.161.6 16 16.13 even 4 inner
640.2.l.b.481.6 16 1.1 even 1 trivial
720.2.t.c.181.8 16 24.5 odd 2
720.2.t.c.541.8 16 48.5 odd 4
1600.2.l.i.401.3 16 80.59 odd 4
1600.2.l.i.1201.3 16 40.19 odd 2
1600.2.q.g.49.6 16 40.3 even 4
1600.2.q.g.849.6 16 80.27 even 4
1600.2.q.h.49.3 16 40.27 even 4
1600.2.q.h.849.3 16 80.43 even 4
2880.2.t.c.721.4 16 48.11 even 4
2880.2.t.c.2161.1 16 24.11 even 2
5120.2.a.s.1.4 8 32.13 even 8
5120.2.a.t.1.4 8 32.3 odd 8
5120.2.a.u.1.5 8 32.19 odd 8
5120.2.a.v.1.5 8 32.29 even 8