Properties

Label 1600.2.n.t.1343.2
Level $1600$
Weight $2$
Character 1600.1343
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1343,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1343.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1343
Dual form 1600.2.n.t.1407.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.22474 - 2.22474i) q^{3} +(2.00000 + 2.00000i) q^{7} -6.89898i q^{9} +O(q^{10})\) \(q+(2.22474 - 2.22474i) q^{3} +(2.00000 + 2.00000i) q^{7} -6.89898i q^{9} +3.44949i q^{11} +(4.44949 + 4.44949i) q^{13} +(0.775255 - 0.775255i) q^{17} +2.55051 q^{19} +8.89898 q^{21} +(0.449490 - 0.449490i) q^{23} +(-8.67423 - 8.67423i) q^{27} +2.89898i q^{29} -2.89898i q^{31} +(7.67423 + 7.67423i) q^{33} +(-6.00000 + 6.00000i) q^{37} +19.7980 q^{39} +5.00000 q^{41} +(2.89898 - 2.89898i) q^{43} +(-7.34847 - 7.34847i) q^{47} +1.00000i q^{49} -3.44949i q^{51} +(-6.44949 - 6.44949i) q^{53} +(5.67423 - 5.67423i) q^{57} -7.79796 q^{59} +0.898979 q^{61} +(13.7980 - 13.7980i) q^{63} +(4.22474 + 4.22474i) q^{67} -2.00000i q^{69} -2.00000i q^{71} +(-5.67423 - 5.67423i) q^{73} +(-6.89898 + 6.89898i) q^{77} -0.898979 q^{79} -17.8990 q^{81} +(4.67423 - 4.67423i) q^{83} +(6.44949 + 6.44949i) q^{87} +11.8990i q^{89} +17.7980i q^{91} +(-6.44949 - 6.44949i) q^{93} +(-4.89898 + 4.89898i) q^{97} +23.7980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{7} + 8 q^{13} + 8 q^{17} + 20 q^{19} + 16 q^{21} - 8 q^{23} - 20 q^{27} + 16 q^{33} - 24 q^{37} + 40 q^{39} + 20 q^{41} - 8 q^{43} - 16 q^{53} + 8 q^{57} + 8 q^{59} - 16 q^{61} + 16 q^{63} + 12 q^{67} - 8 q^{73} - 8 q^{77} + 16 q^{79} - 52 q^{81} + 4 q^{83} + 16 q^{87} - 16 q^{93} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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
\(3\) 2.22474 2.22474i 1.28446 1.28446i 0.346353 0.938104i \(-0.387420\pi\)
0.938104 0.346353i \(-0.112580\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 + 2.00000i 0.755929 + 0.755929i 0.975579 0.219650i \(-0.0704915\pi\)
−0.219650 + 0.975579i \(0.570491\pi\)
\(8\) 0 0
\(9\) 6.89898i 2.29966i
\(10\) 0 0
\(11\) 3.44949i 1.04006i 0.854148 + 0.520030i \(0.174079\pi\)
−0.854148 + 0.520030i \(0.825921\pi\)
\(12\) 0 0
\(13\) 4.44949 + 4.44949i 1.23407 + 1.23407i 0.962388 + 0.271678i \(0.0875787\pi\)
0.271678 + 0.962388i \(0.412421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.775255 0.775255i 0.188027 0.188027i −0.606816 0.794843i \(-0.707553\pi\)
0.794843 + 0.606816i \(0.207553\pi\)
\(18\) 0 0
\(19\) 2.55051 0.585127 0.292564 0.956246i \(-0.405492\pi\)
0.292564 + 0.956246i \(0.405492\pi\)
\(20\) 0 0
\(21\) 8.89898 1.94192
\(22\) 0 0
\(23\) 0.449490 0.449490i 0.0937251 0.0937251i −0.658690 0.752415i \(-0.728889\pi\)
0.752415 + 0.658690i \(0.228889\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −8.67423 8.67423i −1.66936 1.66936i
\(28\) 0 0
\(29\) 2.89898i 0.538327i 0.963095 + 0.269163i \(0.0867472\pi\)
−0.963095 + 0.269163i \(0.913253\pi\)
\(30\) 0 0
\(31\) 2.89898i 0.520672i −0.965518 0.260336i \(-0.916167\pi\)
0.965518 0.260336i \(-0.0838333\pi\)
\(32\) 0 0
\(33\) 7.67423 + 7.67423i 1.33591 + 1.33591i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 + 6.00000i −0.986394 + 0.986394i −0.999909 0.0135147i \(-0.995698\pi\)
0.0135147 + 0.999909i \(0.495698\pi\)
\(38\) 0 0
\(39\) 19.7980 3.17021
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 2.89898 2.89898i 0.442090 0.442090i −0.450624 0.892714i \(-0.648798\pi\)
0.892714 + 0.450624i \(0.148798\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.34847 7.34847i −1.07188 1.07188i −0.997208 0.0746766i \(-0.976208\pi\)
−0.0746766 0.997208i \(-0.523792\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 3.44949i 0.483025i
\(52\) 0 0
\(53\) −6.44949 6.44949i −0.885906 0.885906i 0.108221 0.994127i \(-0.465484\pi\)
−0.994127 + 0.108221i \(0.965484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.67423 5.67423i 0.751571 0.751571i
\(58\) 0 0
\(59\) −7.79796 −1.01521 −0.507604 0.861591i \(-0.669469\pi\)
−0.507604 + 0.861591i \(0.669469\pi\)
\(60\) 0 0
\(61\) 0.898979 0.115103 0.0575513 0.998343i \(-0.481671\pi\)
0.0575513 + 0.998343i \(0.481671\pi\)
\(62\) 0 0
\(63\) 13.7980 13.7980i 1.73838 1.73838i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.22474 + 4.22474i 0.516135 + 0.516135i 0.916400 0.400265i \(-0.131082\pi\)
−0.400265 + 0.916400i \(0.631082\pi\)
\(68\) 0 0
\(69\) 2.00000i 0.240772i
\(70\) 0 0
\(71\) 2.00000i 0.237356i −0.992933 0.118678i \(-0.962134\pi\)
0.992933 0.118678i \(-0.0378657\pi\)
\(72\) 0 0
\(73\) −5.67423 5.67423i −0.664119 0.664119i 0.292229 0.956348i \(-0.405603\pi\)
−0.956348 + 0.292229i \(0.905603\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.89898 + 6.89898i −0.786212 + 0.786212i
\(78\) 0 0
\(79\) −0.898979 −0.101143 −0.0505715 0.998720i \(-0.516104\pi\)
−0.0505715 + 0.998720i \(0.516104\pi\)
\(80\) 0 0
\(81\) −17.8990 −1.98878
\(82\) 0 0
\(83\) 4.67423 4.67423i 0.513064 0.513064i −0.402400 0.915464i \(-0.631824\pi\)
0.915464 + 0.402400i \(0.131824\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.44949 + 6.44949i 0.691458 + 0.691458i
\(88\) 0 0
\(89\) 11.8990i 1.26129i 0.776072 + 0.630645i \(0.217209\pi\)
−0.776072 + 0.630645i \(0.782791\pi\)
\(90\) 0 0
\(91\) 17.7980i 1.86573i
\(92\) 0 0
\(93\) −6.44949 6.44949i −0.668781 0.668781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 + 4.89898i −0.497416 + 0.497416i −0.910633 0.413217i \(-0.864405\pi\)
0.413217 + 0.910633i \(0.364405\pi\)
\(98\) 0 0
\(99\) 23.7980 2.39178
\(100\) 0 0
\(101\) −14.6969 −1.46240 −0.731200 0.682163i \(-0.761039\pi\)
−0.731200 + 0.682163i \(0.761039\pi\)
\(102\) 0 0
\(103\) 2.44949 2.44949i 0.241355 0.241355i −0.576055 0.817411i \(-0.695409\pi\)
0.817411 + 0.576055i \(0.195409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.57321 5.57321i −0.538783 0.538783i 0.384388 0.923171i \(-0.374412\pi\)
−0.923171 + 0.384388i \(0.874412\pi\)
\(108\) 0 0
\(109\) 3.10102i 0.297024i 0.988911 + 0.148512i \(0.0474483\pi\)
−0.988911 + 0.148512i \(0.952552\pi\)
\(110\) 0 0
\(111\) 26.6969i 2.53396i
\(112\) 0 0
\(113\) −11.6742 11.6742i −1.09822 1.09822i −0.994619 0.103601i \(-0.966964\pi\)
−0.103601 0.994619i \(-0.533036\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 30.6969 30.6969i 2.83793 2.83793i
\(118\) 0 0
\(119\) 3.10102 0.284270
\(120\) 0 0
\(121\) −0.898979 −0.0817254
\(122\) 0 0
\(123\) 11.1237 11.1237i 1.00299 1.00299i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.44949 8.44949i −0.749771 0.749771i 0.224665 0.974436i \(-0.427871\pi\)
−0.974436 + 0.224665i \(0.927871\pi\)
\(128\) 0 0
\(129\) 12.8990i 1.13569i
\(130\) 0 0
\(131\) 15.7980i 1.38027i −0.723679 0.690137i \(-0.757550\pi\)
0.723679 0.690137i \(-0.242450\pi\)
\(132\) 0 0
\(133\) 5.10102 + 5.10102i 0.442315 + 0.442315i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.12372 2.12372i 0.181442 0.181442i −0.610542 0.791984i \(-0.709048\pi\)
0.791984 + 0.610542i \(0.209048\pi\)
\(138\) 0 0
\(139\) 9.24745 0.784358 0.392179 0.919889i \(-0.371721\pi\)
0.392179 + 0.919889i \(0.371721\pi\)
\(140\) 0 0
\(141\) −32.6969 −2.75358
\(142\) 0 0
\(143\) −15.3485 + 15.3485i −1.28350 + 1.28350i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.22474 + 2.22474i 0.183494 + 0.183494i
\(148\) 0 0
\(149\) 18.8990i 1.54826i 0.633024 + 0.774132i \(0.281814\pi\)
−0.633024 + 0.774132i \(0.718186\pi\)
\(150\) 0 0
\(151\) 22.6969i 1.84705i −0.383537 0.923525i \(-0.625294\pi\)
0.383537 0.923525i \(-0.374706\pi\)
\(152\) 0 0
\(153\) −5.34847 5.34847i −0.432398 0.432398i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −8.24745 + 8.24745i −0.658218 + 0.658218i −0.954958 0.296740i \(-0.904100\pi\)
0.296740 + 0.954958i \(0.404100\pi\)
\(158\) 0 0
\(159\) −28.6969 −2.27582
\(160\) 0 0
\(161\) 1.79796 0.141699
\(162\) 0 0
\(163\) −7.77526 + 7.77526i −0.609005 + 0.609005i −0.942686 0.333681i \(-0.891709\pi\)
0.333681 + 0.942686i \(0.391709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.4495 10.4495i −0.808606 0.808606i 0.175817 0.984423i \(-0.443743\pi\)
−0.984423 + 0.175817i \(0.943743\pi\)
\(168\) 0 0
\(169\) 26.5959i 2.04584i
\(170\) 0 0
\(171\) 17.5959i 1.34559i
\(172\) 0 0
\(173\) 5.79796 + 5.79796i 0.440811 + 0.440811i 0.892284 0.451474i \(-0.149102\pi\)
−0.451474 + 0.892284i \(0.649102\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −17.3485 + 17.3485i −1.30399 + 1.30399i
\(178\) 0 0
\(179\) 15.2474 1.13965 0.569824 0.821767i \(-0.307011\pi\)
0.569824 + 0.821767i \(0.307011\pi\)
\(180\) 0 0
\(181\) 23.7980 1.76889 0.884444 0.466646i \(-0.154538\pi\)
0.884444 + 0.466646i \(0.154538\pi\)
\(182\) 0 0
\(183\) 2.00000 2.00000i 0.147844 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.67423 + 2.67423i 0.195559 + 0.195559i
\(188\) 0 0
\(189\) 34.6969i 2.52383i
\(190\) 0 0
\(191\) 13.1010i 0.947957i −0.880537 0.473978i \(-0.842818\pi\)
0.880537 0.473978i \(-0.157182\pi\)
\(192\) 0 0
\(193\) −3.87628 3.87628i −0.279020 0.279020i 0.553697 0.832718i \(-0.313216\pi\)
−0.832718 + 0.553697i \(0.813216\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.55051 + 1.55051i −0.110469 + 0.110469i −0.760181 0.649712i \(-0.774890\pi\)
0.649712 + 0.760181i \(0.274890\pi\)
\(198\) 0 0
\(199\) −9.79796 −0.694559 −0.347279 0.937762i \(-0.612894\pi\)
−0.347279 + 0.937762i \(0.612894\pi\)
\(200\) 0 0
\(201\) 18.7980 1.32591
\(202\) 0 0
\(203\) −5.79796 + 5.79796i −0.406937 + 0.406937i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.10102 3.10102i −0.215536 0.215536i
\(208\) 0 0
\(209\) 8.79796i 0.608568i
\(210\) 0 0
\(211\) 13.2474i 0.911992i 0.889982 + 0.455996i \(0.150717\pi\)
−0.889982 + 0.455996i \(0.849283\pi\)
\(212\) 0 0
\(213\) −4.44949 4.44949i −0.304874 0.304874i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.79796 5.79796i 0.393591 0.393591i
\(218\) 0 0
\(219\) −25.2474 −1.70606
\(220\) 0 0
\(221\) 6.89898 0.464076
\(222\) 0 0
\(223\) 16.8990 16.8990i 1.13164 1.13164i 0.141735 0.989905i \(-0.454732\pi\)
0.989905 0.141735i \(-0.0452680\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.0000 + 10.0000i 0.663723 + 0.663723i 0.956256 0.292532i \(-0.0944979\pi\)
−0.292532 + 0.956256i \(0.594498\pi\)
\(228\) 0 0
\(229\) 13.7980i 0.911795i −0.890032 0.455897i \(-0.849318\pi\)
0.890032 0.455897i \(-0.150682\pi\)
\(230\) 0 0
\(231\) 30.6969i 2.01971i
\(232\) 0 0
\(233\) 16.8990 + 16.8990i 1.10709 + 1.10709i 0.993531 + 0.113558i \(0.0362246\pi\)
0.113558 + 0.993531i \(0.463775\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 + 2.00000i −0.129914 + 0.129914i
\(238\) 0 0
\(239\) 25.5959 1.65566 0.827831 0.560977i \(-0.189575\pi\)
0.827831 + 0.560977i \(0.189575\pi\)
\(240\) 0 0
\(241\) 5.89898 0.379987 0.189993 0.981785i \(-0.439153\pi\)
0.189993 + 0.981785i \(0.439153\pi\)
\(242\) 0 0
\(243\) −13.7980 + 13.7980i −0.885139 + 0.885139i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 11.3485 + 11.3485i 0.722086 + 0.722086i
\(248\) 0 0
\(249\) 20.7980i 1.31802i
\(250\) 0 0
\(251\) 17.4495i 1.10140i −0.834703 0.550701i \(-0.814360\pi\)
0.834703 0.550701i \(-0.185640\pi\)
\(252\) 0 0
\(253\) 1.55051 + 1.55051i 0.0974797 + 0.0974797i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.6969 + 18.6969i −1.16628 + 1.16628i −0.183209 + 0.983074i \(0.558649\pi\)
−0.983074 + 0.183209i \(0.941351\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 20.0000 1.23797
\(262\) 0 0
\(263\) −6.24745 + 6.24745i −0.385234 + 0.385234i −0.872984 0.487749i \(-0.837818\pi\)
0.487749 + 0.872984i \(0.337818\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 26.4722 + 26.4722i 1.62007 + 1.62007i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 26.8990i 1.63400i 0.576640 + 0.816998i \(0.304364\pi\)
−0.576640 + 0.816998i \(0.695636\pi\)
\(272\) 0 0
\(273\) 39.5959 + 39.5959i 2.39645 + 2.39645i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.55051 + 7.55051i −0.453666 + 0.453666i −0.896569 0.442903i \(-0.853949\pi\)
0.442903 + 0.896569i \(0.353949\pi\)
\(278\) 0 0
\(279\) −20.0000 −1.19737
\(280\) 0 0
\(281\) 31.7980 1.89691 0.948454 0.316916i \(-0.102647\pi\)
0.948454 + 0.316916i \(0.102647\pi\)
\(282\) 0 0
\(283\) −15.5732 + 15.5732i −0.925731 + 0.925731i −0.997427 0.0716951i \(-0.977159\pi\)
0.0716951 + 0.997427i \(0.477159\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 + 10.0000i 0.590281 + 0.590281i
\(288\) 0 0
\(289\) 15.7980i 0.929292i
\(290\) 0 0
\(291\) 21.7980i 1.27782i
\(292\) 0 0
\(293\) −6.89898 6.89898i −0.403043 0.403043i 0.476261 0.879304i \(-0.341992\pi\)
−0.879304 + 0.476261i \(0.841992\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 29.9217 29.9217i 1.73623 1.73623i
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 11.5959 0.668378
\(302\) 0 0
\(303\) −32.6969 + 32.6969i −1.87839 + 1.87839i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.426786 0.426786i −0.0243580 0.0243580i 0.694823 0.719181i \(-0.255483\pi\)
−0.719181 + 0.694823i \(0.755483\pi\)
\(308\) 0 0
\(309\) 10.8990i 0.620021i
\(310\) 0 0
\(311\) 15.1010i 0.856300i −0.903708 0.428150i \(-0.859165\pi\)
0.903708 0.428150i \(-0.140835\pi\)
\(312\) 0 0
\(313\) −16.8990 16.8990i −0.955187 0.955187i 0.0438513 0.999038i \(-0.486037\pi\)
−0.999038 + 0.0438513i \(0.986037\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.69694 + 6.69694i −0.376138 + 0.376138i −0.869707 0.493569i \(-0.835692\pi\)
0.493569 + 0.869707i \(0.335692\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) −24.7980 −1.38409
\(322\) 0 0
\(323\) 1.97730 1.97730i 0.110020 0.110020i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.89898 + 6.89898i 0.381514 + 0.381514i
\(328\) 0 0
\(329\) 29.3939i 1.62054i
\(330\) 0 0
\(331\) 10.3485i 0.568803i 0.958705 + 0.284402i \(0.0917949\pi\)
−0.958705 + 0.284402i \(0.908205\pi\)
\(332\) 0 0
\(333\) 41.3939 + 41.3939i 2.26837 + 2.26837i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.42679 7.42679i 0.404563 0.404563i −0.475275 0.879837i \(-0.657651\pi\)
0.879837 + 0.475275i \(0.157651\pi\)
\(338\) 0 0
\(339\) −51.9444 −2.82123
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) 12.0000 12.0000i 0.647939 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.6742 20.6742i −1.10985 1.10985i −0.993169 0.116682i \(-0.962774\pi\)
−0.116682 0.993169i \(-0.537226\pi\)
\(348\) 0 0
\(349\) 10.6969i 0.572594i 0.958141 + 0.286297i \(0.0924244\pi\)
−0.958141 + 0.286297i \(0.907576\pi\)
\(350\) 0 0
\(351\) 77.1918i 4.12020i
\(352\) 0 0
\(353\) 7.10102 + 7.10102i 0.377949 + 0.377949i 0.870362 0.492413i \(-0.163885\pi\)
−0.492413 + 0.870362i \(0.663885\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 6.89898 6.89898i 0.365133 0.365133i
\(358\) 0 0
\(359\) −19.1010 −1.00811 −0.504057 0.863671i \(-0.668160\pi\)
−0.504057 + 0.863671i \(0.668160\pi\)
\(360\) 0 0
\(361\) −12.4949 −0.657626
\(362\) 0 0
\(363\) −2.00000 + 2.00000i −0.104973 + 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 18.6969 + 18.6969i 0.975972 + 0.975972i 0.999718 0.0237458i \(-0.00755923\pi\)
−0.0237458 + 0.999718i \(0.507559\pi\)
\(368\) 0 0
\(369\) 34.4949i 1.79573i
\(370\) 0 0
\(371\) 25.7980i 1.33936i
\(372\) 0 0
\(373\) 2.44949 + 2.44949i 0.126830 + 0.126830i 0.767672 0.640843i \(-0.221415\pi\)
−0.640843 + 0.767672i \(0.721415\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.8990 + 12.8990i −0.664331 + 0.664331i
\(378\) 0 0
\(379\) 33.0454 1.69743 0.848714 0.528852i \(-0.177377\pi\)
0.848714 + 0.528852i \(0.177377\pi\)
\(380\) 0 0
\(381\) −37.5959 −1.92610
\(382\) 0 0
\(383\) −11.7980 + 11.7980i −0.602848 + 0.602848i −0.941067 0.338220i \(-0.890175\pi\)
0.338220 + 0.941067i \(0.390175\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.0000 20.0000i −1.01666 1.01666i
\(388\) 0 0
\(389\) 11.7980i 0.598180i 0.954225 + 0.299090i \(0.0966831\pi\)
−0.954225 + 0.299090i \(0.903317\pi\)
\(390\) 0 0
\(391\) 0.696938i 0.0352457i
\(392\) 0 0
\(393\) −35.1464 35.1464i −1.77290 1.77290i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.79796 5.79796i 0.290991 0.290991i −0.546481 0.837472i \(-0.684033\pi\)
0.837472 + 0.546481i \(0.184033\pi\)
\(398\) 0 0
\(399\) 22.6969 1.13627
\(400\) 0 0
\(401\) −4.10102 −0.204795 −0.102398 0.994744i \(-0.532651\pi\)
−0.102398 + 0.994744i \(0.532651\pi\)
\(402\) 0 0
\(403\) 12.8990 12.8990i 0.642544 0.642544i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.6969 20.6969i −1.02591 1.02591i
\(408\) 0 0
\(409\) 10.7980i 0.533925i −0.963707 0.266962i \(-0.913980\pi\)
0.963707 0.266962i \(-0.0860199\pi\)
\(410\) 0 0
\(411\) 9.44949i 0.466109i
\(412\) 0 0
\(413\) −15.5959 15.5959i −0.767425 0.767425i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.5732 20.5732i 1.00747 1.00747i
\(418\) 0 0
\(419\) −22.1464 −1.08192 −0.540962 0.841047i \(-0.681940\pi\)
−0.540962 + 0.841047i \(0.681940\pi\)
\(420\) 0 0
\(421\) 27.5959 1.34494 0.672471 0.740123i \(-0.265233\pi\)
0.672471 + 0.740123i \(0.265233\pi\)
\(422\) 0 0
\(423\) −50.6969 + 50.6969i −2.46497 + 2.46497i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.79796 + 1.79796i 0.0870093 + 0.0870093i
\(428\) 0 0
\(429\) 68.2929i 3.29721i
\(430\) 0 0
\(431\) 12.8990i 0.621322i 0.950521 + 0.310661i \(0.100550\pi\)
−0.950521 + 0.310661i \(0.899450\pi\)
\(432\) 0 0
\(433\) −6.32577 6.32577i −0.303997 0.303997i 0.538578 0.842575i \(-0.318962\pi\)
−0.842575 + 0.538578i \(0.818962\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.14643 1.14643i 0.0548411 0.0548411i
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 6.89898 0.328523
\(442\) 0 0
\(443\) −16.9217 + 16.9217i −0.803973 + 0.803973i −0.983714 0.179741i \(-0.942474\pi\)
0.179741 + 0.983714i \(0.442474\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.0454 + 42.0454i 1.98868 + 1.98868i
\(448\) 0 0
\(449\) 5.89898i 0.278390i −0.990265 0.139195i \(-0.955549\pi\)
0.990265 0.139195i \(-0.0444515\pi\)
\(450\) 0 0
\(451\) 17.2474i 0.812151i
\(452\) 0 0
\(453\) −50.4949 50.4949i −2.37246 2.37246i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.0227 + 23.0227i −1.07696 + 1.07696i −0.0801759 + 0.996781i \(0.525548\pi\)
−0.996781 + 0.0801759i \(0.974452\pi\)
\(458\) 0 0
\(459\) −13.4495 −0.627768
\(460\) 0 0
\(461\) −2.20204 −0.102559 −0.0512796 0.998684i \(-0.516330\pi\)
−0.0512796 + 0.998684i \(0.516330\pi\)
\(462\) 0 0
\(463\) −2.69694 + 2.69694i −0.125337 + 0.125337i −0.766993 0.641656i \(-0.778248\pi\)
0.641656 + 0.766993i \(0.278248\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.4949 26.4949i −1.22604 1.22604i −0.965450 0.260587i \(-0.916084\pi\)
−0.260587 0.965450i \(-0.583916\pi\)
\(468\) 0 0
\(469\) 16.8990i 0.780322i
\(470\) 0 0
\(471\) 36.6969i 1.69091i
\(472\) 0 0
\(473\) 10.0000 + 10.0000i 0.459800 + 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −44.4949 + 44.4949i −2.03728 + 2.03728i
\(478\) 0 0
\(479\) 2.89898 0.132458 0.0662289 0.997804i \(-0.478903\pi\)
0.0662289 + 0.997804i \(0.478903\pi\)
\(480\) 0 0
\(481\) −53.3939 −2.43455
\(482\) 0 0
\(483\) 4.00000 4.00000i 0.182006 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.4495 + 12.4495i 0.564140 + 0.564140i 0.930481 0.366341i \(-0.119389\pi\)
−0.366341 + 0.930481i \(0.619389\pi\)
\(488\) 0 0
\(489\) 34.5959i 1.56448i
\(490\) 0 0
\(491\) 1.59592i 0.0720228i 0.999351 + 0.0360114i \(0.0114653\pi\)
−0.999351 + 0.0360114i \(0.988535\pi\)
\(492\) 0 0
\(493\) 2.24745 + 2.24745i 0.101220 + 0.101220i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.00000 4.00000i 0.179425 0.179425i
\(498\) 0 0
\(499\) −7.79796 −0.349085 −0.174542 0.984650i \(-0.555845\pi\)
−0.174542 + 0.984650i \(0.555845\pi\)
\(500\) 0 0
\(501\) −46.4949 −2.07724
\(502\) 0 0
\(503\) 13.7980 13.7980i 0.615221 0.615221i −0.329081 0.944302i \(-0.606739\pi\)
0.944302 + 0.329081i \(0.106739\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 59.1691 + 59.1691i 2.62779 + 2.62779i
\(508\) 0 0
\(509\) 20.2020i 0.895440i −0.894174 0.447720i \(-0.852236\pi\)
0.894174 0.447720i \(-0.147764\pi\)
\(510\) 0 0
\(511\) 22.6969i 1.00405i
\(512\) 0 0
\(513\) −22.1237 22.1237i −0.976786 0.976786i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 25.3485 25.3485i 1.11482 1.11482i
\(518\) 0 0
\(519\) 25.7980 1.13240
\(520\) 0 0
\(521\) −19.0000 −0.832405 −0.416203 0.909272i \(-0.636639\pi\)
−0.416203 + 0.909272i \(0.636639\pi\)
\(522\) 0 0
\(523\) −8.22474 + 8.22474i −0.359643 + 0.359643i −0.863681 0.504038i \(-0.831847\pi\)
0.504038 + 0.863681i \(0.331847\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.24745 2.24745i −0.0979004 0.0979004i
\(528\) 0 0
\(529\) 22.5959i 0.982431i
\(530\) 0 0
\(531\) 53.7980i 2.33463i
\(532\) 0 0
\(533\) 22.2474 + 22.2474i 0.963644 + 0.963644i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 33.9217 33.9217i 1.46383 1.46383i
\(538\) 0 0
\(539\) −3.44949 −0.148580
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 52.9444 52.9444i 2.27206 2.27206i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.32577 1.32577i −0.0566856 0.0566856i 0.678196 0.734881i \(-0.262762\pi\)
−0.734881 + 0.678196i \(0.762762\pi\)
\(548\) 0 0
\(549\) 6.20204i 0.264697i
\(550\) 0 0
\(551\) 7.39388i 0.314990i
\(552\) 0 0
\(553\) −1.79796 1.79796i −0.0764570 0.0764570i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.44949 8.44949i 0.358016 0.358016i −0.505065 0.863081i \(-0.668531\pi\)
0.863081 + 0.505065i \(0.168531\pi\)
\(558\) 0 0
\(559\) 25.7980 1.09114
\(560\) 0 0
\(561\) 11.8990 0.502375
\(562\) 0 0
\(563\) −10.8990 + 10.8990i −0.459337 + 0.459337i −0.898438 0.439101i \(-0.855297\pi\)
0.439101 + 0.898438i \(0.355297\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −35.7980 35.7980i −1.50337 1.50337i
\(568\) 0 0
\(569\) 44.5959i 1.86956i 0.355230 + 0.934779i \(0.384403\pi\)
−0.355230 + 0.934779i \(0.615597\pi\)
\(570\) 0 0
\(571\) 14.0000i 0.585882i 0.956131 + 0.292941i \(0.0946339\pi\)
−0.956131 + 0.292941i \(0.905366\pi\)
\(572\) 0 0
\(573\) −29.1464 29.1464i −1.21761 1.21761i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −9.22474 + 9.22474i −0.384031 + 0.384031i −0.872552 0.488521i \(-0.837537\pi\)
0.488521 + 0.872552i \(0.337537\pi\)
\(578\) 0 0
\(579\) −17.2474 −0.716780
\(580\) 0 0
\(581\) 18.6969 0.775680
\(582\) 0 0
\(583\) 22.2474 22.2474i 0.921395 0.921395i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.5732 15.5732i −0.642775 0.642775i 0.308461 0.951237i \(-0.400186\pi\)
−0.951237 + 0.308461i \(0.900186\pi\)
\(588\) 0 0
\(589\) 7.39388i 0.304659i
\(590\) 0 0
\(591\) 6.89898i 0.283786i
\(592\) 0 0
\(593\) 5.22474 + 5.22474i 0.214555 + 0.214555i 0.806199 0.591644i \(-0.201521\pi\)
−0.591644 + 0.806199i \(0.701521\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −21.7980 + 21.7980i −0.892131 + 0.892131i
\(598\) 0 0
\(599\) 1.30306 0.0532417 0.0266208 0.999646i \(-0.491525\pi\)
0.0266208 + 0.999646i \(0.491525\pi\)
\(600\) 0 0
\(601\) −13.6969 −0.558710 −0.279355 0.960188i \(-0.590121\pi\)
−0.279355 + 0.960188i \(0.590121\pi\)
\(602\) 0 0
\(603\) 29.1464 29.1464i 1.18693 1.18693i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.79796 + 5.79796i 0.235332 + 0.235332i 0.814914 0.579582i \(-0.196784\pi\)
−0.579582 + 0.814914i \(0.696784\pi\)
\(608\) 0 0
\(609\) 25.7980i 1.04539i
\(610\) 0 0
\(611\) 65.3939i 2.64555i
\(612\) 0 0
\(613\) −5.14643 5.14643i −0.207862 0.207862i 0.595496 0.803358i \(-0.296956\pi\)
−0.803358 + 0.595496i \(0.796956\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.5959 23.5959i 0.949936 0.949936i −0.0488693 0.998805i \(-0.515562\pi\)
0.998805 + 0.0488693i \(0.0155618\pi\)
\(618\) 0 0
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 0 0
\(621\) −7.79796 −0.312921
\(622\) 0 0
\(623\) −23.7980 + 23.7980i −0.953445 + 0.953445i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 19.5732 + 19.5732i 0.781679 + 0.781679i
\(628\) 0 0
\(629\) 9.30306i 0.370937i
\(630\) 0 0
\(631\) 4.69694i 0.186982i −0.995620 0.0934911i \(-0.970197\pi\)
0.995620 0.0934911i \(-0.0298027\pi\)
\(632\) 0 0
\(633\) 29.4722 + 29.4722i 1.17141 + 1.17141i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.44949 + 4.44949i −0.176295 + 0.176295i
\(638\) 0 0
\(639\) −13.7980 −0.545839
\(640\) 0 0
\(641\) 15.7980 0.623982 0.311991 0.950085i \(-0.399004\pi\)
0.311991 + 0.950085i \(0.399004\pi\)
\(642\) 0 0
\(643\) 0.696938 0.696938i 0.0274846 0.0274846i −0.693231 0.720716i \(-0.743813\pi\)
0.720716 + 0.693231i \(0.243813\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.5959 27.5959i −1.08491 1.08491i −0.996044 0.0888637i \(-0.971676\pi\)
−0.0888637 0.996044i \(-0.528324\pi\)
\(648\) 0 0
\(649\) 26.8990i 1.05588i
\(650\) 0 0
\(651\) 25.7980i 1.01110i
\(652\) 0 0
\(653\) −26.0000 26.0000i −1.01746 1.01746i −0.999845 0.0176138i \(-0.994393\pi\)
−0.0176138 0.999845i \(-0.505607\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −39.1464 + 39.1464i −1.52725 + 1.52725i
\(658\) 0 0
\(659\) 4.14643 0.161522 0.0807610 0.996733i \(-0.474265\pi\)
0.0807610 + 0.996733i \(0.474265\pi\)
\(660\) 0 0
\(661\) −40.2929 −1.56721 −0.783605 0.621259i \(-0.786621\pi\)
−0.783605 + 0.621259i \(0.786621\pi\)
\(662\) 0 0
\(663\) 15.3485 15.3485i 0.596085 0.596085i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.30306 + 1.30306i 0.0504547 + 0.0504547i
\(668\) 0 0
\(669\) 75.1918i 2.90708i
\(670\) 0 0
\(671\) 3.10102i 0.119714i
\(672\) 0 0
\(673\) 13.7980 + 13.7980i 0.531872 + 0.531872i 0.921129 0.389257i \(-0.127268\pi\)
−0.389257 + 0.921129i \(0.627268\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.0000 28.0000i 1.07613 1.07613i 0.0792746 0.996853i \(-0.474740\pi\)
0.996853 0.0792746i \(-0.0252604\pi\)
\(678\) 0 0
\(679\) −19.5959 −0.752022
\(680\) 0 0
\(681\) 44.4949 1.70505
\(682\) 0 0
\(683\) −12.2247 + 12.2247i −0.467767 + 0.467767i −0.901190 0.433424i \(-0.857305\pi\)
0.433424 + 0.901190i \(0.357305\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −30.6969 30.6969i −1.17116 1.17116i
\(688\) 0 0
\(689\) 57.3939i 2.18653i
\(690\) 0 0
\(691\) 39.9444i 1.51956i −0.650183 0.759778i \(-0.725308\pi\)
0.650183 0.759778i \(-0.274692\pi\)
\(692\) 0 0
\(693\) 47.5959 + 47.5959i 1.80802 + 1.80802i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.87628 3.87628i 0.146824 0.146824i
\(698\) 0 0
\(699\) 75.1918 2.84402
\(700\) 0 0
\(701\) 13.1010 0.494819 0.247409 0.968911i \(-0.420421\pi\)
0.247409 + 0.968911i \(0.420421\pi\)
\(702\) 0 0
\(703\) −15.3031 + 15.3031i −0.577166 + 0.577166i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.3939 29.3939i −1.10547 1.10547i
\(708\) 0 0
\(709\) 12.0000i 0.450669i −0.974281 0.225335i \(-0.927652\pi\)
0.974281 0.225335i \(-0.0723476\pi\)
\(710\) 0 0
\(711\) 6.20204i 0.232595i
\(712\) 0 0
\(713\) −1.30306 1.30306i −0.0488000 0.0488000i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 56.9444 56.9444i 2.12663 2.12663i
\(718\) 0 0
\(719\) 31.1010 1.15987 0.579936 0.814662i \(-0.303077\pi\)
0.579936 + 0.814662i \(0.303077\pi\)
\(720\) 0 0
\(721\) 9.79796 0.364895
\(722\) 0 0
\(723\) 13.1237 13.1237i 0.488077 0.488077i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.1464 + 17.1464i 0.635926 + 0.635926i 0.949548 0.313622i \(-0.101542\pi\)
−0.313622 + 0.949548i \(0.601542\pi\)
\(728\) 0 0
\(729\) 7.69694i 0.285072i
\(730\) 0 0
\(731\) 4.49490i 0.166250i
\(732\) 0 0
\(733\) 15.5505 + 15.5505i 0.574371 + 0.574371i 0.933347 0.358976i \(-0.116874\pi\)
−0.358976 + 0.933347i \(0.616874\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.5732 + 14.5732i −0.536811 + 0.536811i
\(738\) 0 0
\(739\) 33.5959 1.23585 0.617923 0.786239i \(-0.287974\pi\)
0.617923 + 0.786239i \(0.287974\pi\)
\(740\) 0 0
\(741\) 50.4949 1.85498
\(742\) 0 0
\(743\) −5.34847 + 5.34847i −0.196216 + 0.196216i −0.798376 0.602160i \(-0.794307\pi\)
0.602160 + 0.798376i \(0.294307\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −32.2474 32.2474i −1.17987 1.17987i
\(748\) 0 0
\(749\) 22.2929i 0.814563i
\(750\) 0 0
\(751\) 41.1918i 1.50311i 0.659670 + 0.751556i \(0.270696\pi\)
−0.659670 + 0.751556i \(0.729304\pi\)
\(752\) 0 0
\(753\) −38.8207 38.8207i −1.41470 1.41470i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −14.6969 + 14.6969i −0.534169 + 0.534169i −0.921810 0.387641i \(-0.873290\pi\)
0.387641 + 0.921810i \(0.373290\pi\)
\(758\) 0 0
\(759\) 6.89898 0.250417
\(760\) 0 0
\(761\) −17.0000 −0.616250 −0.308125 0.951346i \(-0.599701\pi\)
−0.308125 + 0.951346i \(0.599701\pi\)
\(762\) 0 0
\(763\) −6.20204 + 6.20204i −0.224529 + 0.224529i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −34.6969 34.6969i −1.25283 1.25283i
\(768\) 0 0
\(769\) 25.6969i 0.926655i −0.886187 0.463328i \(-0.846655\pi\)
0.886187 0.463328i \(-0.153345\pi\)
\(770\) 0 0
\(771\) 83.1918i 2.99608i
\(772\) 0 0
\(773\) 28.9444 + 28.9444i 1.04106 + 1.04106i 0.999120 + 0.0419370i \(0.0133529\pi\)
0.0419370 + 0.999120i \(0.486647\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −53.3939 + 53.3939i −1.91549 + 1.91549i
\(778\) 0 0
\(779\) 12.7526 0.456908
\(780\) 0 0
\(781\) 6.89898 0.246865
\(782\) 0 0
\(783\) 25.1464 25.1464i 0.898660 0.898660i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 6.89898 + 6.89898i 0.245922 + 0.245922i 0.819295 0.573373i \(-0.194365\pi\)
−0.573373 + 0.819295i \(0.694365\pi\)
\(788\) 0 0
\(789\) 27.7980i 0.989634i
\(790\) 0 0
\(791\) 46.6969i 1.66035i
\(792\) 0 0
\(793\) 4.00000 + 4.00000i 0.142044 + 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.3485 + 11.3485i −0.401983 + 0.401983i −0.878931 0.476948i \(-0.841743\pi\)
0.476948 + 0.878931i \(0.341743\pi\)
\(798\) 0 0
\(799\) −11.3939 −0.403086
\(800\) 0 0
\(801\) 82.0908 2.90054
\(802\) 0 0
\(803\) 19.5732 19.5732i 0.690724 0.690724i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.202041i 0.00710338i −0.999994 0.00355169i \(-0.998869\pi\)
0.999994 0.00355169i \(-0.00113054\pi\)
\(810\) 0 0
\(811\) 26.0000i 0.912983i 0.889728 + 0.456492i \(0.150894\pi\)
−0.889728 + 0.456492i \(0.849106\pi\)
\(812\) 0 0
\(813\) 59.8434 + 59.8434i 2.09880 + 2.09880i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.39388 7.39388i 0.258679 0.258679i
\(818\) 0 0
\(819\) 122.788 4.29055
\(820\) 0 0
\(821\) −15.7980 −0.551353 −0.275676 0.961251i \(-0.588902\pi\)
−0.275676 + 0.961251i \(0.588902\pi\)
\(822\) 0 0
\(823\) 10.6969 10.6969i 0.372872 0.372872i −0.495650 0.868522i \(-0.665070\pi\)
0.868522 + 0.495650i \(0.165070\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.07832 + 3.07832i 0.107044 + 0.107044i 0.758600 0.651557i \(-0.225884\pi\)
−0.651557 + 0.758600i \(0.725884\pi\)
\(828\) 0 0
\(829\) 37.1010i 1.28857i −0.764785 0.644286i \(-0.777155\pi\)
0.764785 0.644286i \(-0.222845\pi\)
\(830\) 0 0
\(831\) 33.5959i 1.16543i
\(832\) 0 0
\(833\) 0.775255 + 0.775255i 0.0268610 + 0.0268610i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −25.1464 + 25.1464i −0.869188 + 0.869188i
\(838\) 0 0
\(839\) −39.1918 −1.35305 −0.676526 0.736419i \(-0.736515\pi\)
−0.676526 + 0.736419i \(0.736515\pi\)
\(840\) 0 0
\(841\) 20.5959 0.710204
\(842\) 0 0
\(843\) 70.7423 70.7423i 2.43650 2.43650i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.79796 1.79796i −0.0617786 0.0617786i
\(848\) 0 0
\(849\) 69.2929i 2.37812i
\(850\) 0 0
\(851\) 5.39388i 0.184900i
\(852\) 0 0
\(853\) −4.65153 4.65153i −0.159265 0.159265i 0.622976 0.782241i \(-0.285924\pi\)
−0.782241 + 0.622976i \(0.785924\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.42679 + 7.42679i −0.253694 + 0.253694i −0.822483 0.568789i \(-0.807412\pi\)
0.568789 + 0.822483i \(0.307412\pi\)
\(858\) 0 0
\(859\) −42.3485 −1.44491 −0.722456 0.691417i \(-0.756987\pi\)
−0.722456 + 0.691417i \(0.756987\pi\)
\(860\) 0 0
\(861\) 44.4949 1.51638
\(862\) 0 0
\(863\) 5.55051 5.55051i 0.188942 0.188942i −0.606297 0.795238i \(-0.707346\pi\)
0.795238 + 0.606297i \(0.207346\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 35.1464 + 35.1464i 1.19364 + 1.19364i
\(868\) 0 0
\(869\) 3.10102i 0.105195i
\(870\) 0 0
\(871\) 37.5959i 1.27389i
\(872\) 0 0
\(873\) 33.7980 + 33.7980i 1.14389 + 1.14389i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.6969 10.6969i 0.361210 0.361210i −0.503048 0.864258i \(-0.667788\pi\)
0.864258 + 0.503048i \(0.167788\pi\)
\(878\) 0 0
\(879\) −30.6969 −1.03538
\(880\) 0 0
\(881\) 16.2020 0.545861 0.272930 0.962034i \(-0.412007\pi\)
0.272930 + 0.962034i \(0.412007\pi\)
\(882\) 0 0
\(883\) 21.8207 21.8207i 0.734324 0.734324i −0.237149 0.971473i \(-0.576213\pi\)
0.971473 + 0.237149i \(0.0762131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 40.0000 + 40.0000i 1.34307 + 1.34307i 0.892988 + 0.450081i \(0.148605\pi\)
0.450081 + 0.892988i \(0.351395\pi\)
\(888\) 0 0
\(889\) 33.7980i 1.13355i
\(890\) 0 0
\(891\) 61.7423i 2.06845i
\(892\) 0 0
\(893\) −18.7423 18.7423i −0.627189 0.627189i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.89898 8.89898i 0.297128 0.297128i
\(898\) 0 0
\(899\) 8.40408 0.280292
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) 25.7980 25.7980i 0.858502 0.858502i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 10.8990 + 10.8990i 0.361895 + 0.361895i 0.864510 0.502615i \(-0.167629\pi\)
−0.502615 + 0.864510i \(0.667629\pi\)
\(908\) 0 0
\(909\) 101.394i 3.36302i
\(910\) 0 0
\(911\) 29.7980i 0.987250i −0.869675 0.493625i \(-0.835671\pi\)
0.869675 0.493625i \(-0.164329\pi\)
\(912\) 0 0
\(913\) 16.1237 + 16.1237i 0.533617 + 0.533617i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 31.5959 31.5959i 1.04339 1.04339i
\(918\) 0 0
\(919\) 56.6969 1.87026 0.935130 0.354306i \(-0.115283\pi\)
0.935130 + 0.354306i \(0.115283\pi\)
\(920\) 0 0
\(921\) −1.89898 −0.0625735
\(922\) 0 0
\(923\) 8.89898 8.89898i 0.292913 0.292913i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −16.8990 16.8990i −0.555035 0.555035i
\(928\) 0 0
\(929\) 6.40408i 0.210111i −0.994466 0.105056i \(-0.966498\pi\)
0.994466 0.105056i \(-0.0335020\pi\)
\(930\) 0 0
\(931\) 2.55051i 0.0835896i
\(932\) 0 0
\(933\) −33.5959 33.5959i −1.09988 1.09988i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14.3258 14.3258i 0.468002 0.468002i −0.433264 0.901267i \(-0.642638\pi\)
0.901267 + 0.433264i \(0.142638\pi\)
\(938\) 0 0
\(939\) −75.1918 −2.45379
\(940\) 0 0
\(941\) −30.4949 −0.994105 −0.497053 0.867720i \(-0.665584\pi\)
−0.497053 + 0.867720i \(0.665584\pi\)
\(942\) 0 0
\(943\) 2.24745 2.24745i 0.0731870 0.0731870i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8990 + 26.8990i 0.874099 + 0.874099i 0.992916 0.118817i \(-0.0379101\pi\)
−0.118817 + 0.992916i \(0.537910\pi\)
\(948\) 0 0
\(949\) 50.4949i 1.63913i
\(950\) 0 0
\(951\) 29.7980i 0.966265i
\(952\) 0 0
\(953\) −17.8763 17.8763i −0.579069 0.579069i 0.355577 0.934647i \(-0.384284\pi\)
−0.934647 + 0.355577i \(0.884284\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −22.2474 + 22.2474i −0.719158 + 0.719158i
\(958\) 0 0
\(959\) 8.49490 0.274315
\(960\) 0 0
\(961\) 22.5959 0.728901
\(962\) 0 0
\(963\) −38.4495 + 38.4495i −1.23902 + 1.23902i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.8990 16.8990i −0.543435 0.543435i 0.381099 0.924534i \(-0.375546\pi\)
−0.924534 + 0.381099i \(0.875546\pi\)
\(968\) 0 0
\(969\) 8.79796i 0.282631i
\(970\) 0 0
\(971\) 32.1464i 1.03163i 0.856701 + 0.515814i \(0.172510\pi\)
−0.856701 + 0.515814i \(0.827490\pi\)
\(972\) 0 0
\(973\) 18.4949 + 18.4949i 0.592919 + 0.592919i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.6742 33.6742i 1.07733 1.07733i 0.0805866 0.996748i \(-0.474321\pi\)
0.996748 0.0805866i \(-0.0256794\pi\)
\(978\) 0 0
\(979\) −41.0454 −1.31182
\(980\) 0 0
\(981\) 21.3939 0.683054
\(982\) 0 0
\(983\) 2.89898 2.89898i 0.0924631 0.0924631i −0.659362 0.751825i \(-0.729174\pi\)
0.751825 + 0.659362i \(0.229174\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −65.3939 65.3939i −2.08151 2.08151i
\(988\) 0 0
\(989\) 2.60612i 0.0828699i
\(990\) 0 0
\(991\) 32.8990i 1.04507i 0.852618 + 0.522535i \(0.175014\pi\)
−0.852618 + 0.522535i \(0.824986\pi\)
\(992\) 0 0
\(993\) 23.0227 + 23.0227i 0.730603 + 0.730603i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −17.7980 + 17.7980i −0.563667 + 0.563667i −0.930347 0.366680i \(-0.880494\pi\)
0.366680 + 0.930347i \(0.380494\pi\)
\(998\) 0 0
\(999\) 104.091 3.29329
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 1600.2.n.t.1343.2 4
4.3 odd 2 1600.2.n.p.1343.1 4
5.2 odd 4 1600.2.n.p.1407.1 4
5.3 odd 4 1600.2.n.s.1407.2 4
5.4 even 2 1600.2.n.o.1343.1 4
8.3 odd 2 800.2.n.m.543.2 yes 4
8.5 even 2 800.2.n.k.543.1 4
20.3 even 4 1600.2.n.o.1407.1 4
20.7 even 4 inner 1600.2.n.t.1407.2 4
20.19 odd 2 1600.2.n.s.1343.2 4
40.3 even 4 800.2.n.n.607.2 yes 4
40.13 odd 4 800.2.n.l.607.1 yes 4
40.19 odd 2 800.2.n.l.543.1 yes 4
40.27 even 4 800.2.n.k.607.1 yes 4
40.29 even 2 800.2.n.n.543.2 yes 4
40.37 odd 4 800.2.n.m.607.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.n.k.543.1 4 8.5 even 2
800.2.n.k.607.1 yes 4 40.27 even 4
800.2.n.l.543.1 yes 4 40.19 odd 2
800.2.n.l.607.1 yes 4 40.13 odd 4
800.2.n.m.543.2 yes 4 8.3 odd 2
800.2.n.m.607.2 yes 4 40.37 odd 4
800.2.n.n.543.2 yes 4 40.29 even 2
800.2.n.n.607.2 yes 4 40.3 even 4
1600.2.n.o.1343.1 4 5.4 even 2
1600.2.n.o.1407.1 4 20.3 even 4
1600.2.n.p.1343.1 4 4.3 odd 2
1600.2.n.p.1407.1 4 5.2 odd 4
1600.2.n.s.1343.2 4 20.19 odd 2
1600.2.n.s.1407.2 4 5.3 odd 4
1600.2.n.t.1343.2 4 1.1 even 1 trivial
1600.2.n.t.1407.2 4 20.7 even 4 inner