Properties

Label 1600.2.j.b.1007.1
Level $1600$
Weight $2$
Character 1600.1007
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(143,1600)]
 
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, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1007.1
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1007
Dual form 1600.2.j.b.143.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.93185i q^{3} +(0.896575 + 0.896575i) q^{7} -0.732051 q^{9} +O(q^{10})\) \(q-1.93185i q^{3} +(0.896575 + 0.896575i) q^{7} -0.732051 q^{9} +(-4.09808 - 4.09808i) q^{11} -4.89898 q^{13} +(-0.707107 - 0.707107i) q^{17} +(3.09808 + 3.09808i) q^{19} +(1.73205 - 1.73205i) q^{21} +(-2.96713 + 2.96713i) q^{23} -4.38134i q^{27} +(-1.26795 + 1.26795i) q^{29} +4.19615i q^{31} +(-7.91688 + 7.91688i) q^{33} -10.9348 q^{37} +9.46410i q^{39} -6.46410i q^{41} -9.14162 q^{43} +(1.41421 - 1.41421i) q^{47} -5.39230i q^{49} +(-1.36603 + 1.36603i) q^{51} -9.89949i q^{53} +(5.98502 - 5.98502i) q^{57} +(-4.26795 + 4.26795i) q^{59} +(7.19615 + 7.19615i) q^{61} +(-0.656339 - 0.656339i) q^{63} -1.55291 q^{67} +(5.73205 + 5.73205i) q^{69} +12.9282 q^{71} +(3.91447 + 3.91447i) q^{73} -7.34847i q^{77} -8.19615 q^{79} -10.6603 q^{81} +4.65874i q^{83} +(2.44949 + 2.44949i) q^{87} -13.7321 q^{89} +(-4.39230 - 4.39230i) q^{91} +8.10634 q^{93} +(-3.10583 - 3.10583i) q^{97} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 12 q^{11} + 4 q^{19} - 24 q^{29} - 4 q^{51} - 48 q^{59} + 16 q^{61} + 32 q^{69} + 48 q^{71} - 24 q^{79} - 16 q^{81} - 96 q^{89} + 48 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.93185i 1.11536i −0.830058 0.557678i \(-0.811693\pi\)
0.830058 0.557678i \(-0.188307\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.896575 + 0.896575i 0.338874 + 0.338874i 0.855943 0.517070i \(-0.172977\pi\)
−0.517070 + 0.855943i \(0.672977\pi\)
\(8\) 0 0
\(9\) −0.732051 −0.244017
\(10\) 0 0
\(11\) −4.09808 4.09808i −1.23562 1.23562i −0.961775 0.273842i \(-0.911706\pi\)
−0.273842 0.961775i \(-0.588294\pi\)
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.707107 0.707107i −0.171499 0.171499i 0.616139 0.787638i \(-0.288696\pi\)
−0.787638 + 0.616139i \(0.788696\pi\)
\(18\) 0 0
\(19\) 3.09808 + 3.09808i 0.710747 + 0.710747i 0.966692 0.255944i \(-0.0823863\pi\)
−0.255944 + 0.966692i \(0.582386\pi\)
\(20\) 0 0
\(21\) 1.73205 1.73205i 0.377964 0.377964i
\(22\) 0 0
\(23\) −2.96713 + 2.96713i −0.618689 + 0.618689i −0.945195 0.326506i \(-0.894129\pi\)
0.326506 + 0.945195i \(0.394129\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.38134i 0.843190i
\(28\) 0 0
\(29\) −1.26795 + 1.26795i −0.235452 + 0.235452i −0.814964 0.579512i \(-0.803243\pi\)
0.579512 + 0.814964i \(0.303243\pi\)
\(30\) 0 0
\(31\) 4.19615i 0.753651i 0.926284 + 0.376826i \(0.122984\pi\)
−0.926284 + 0.376826i \(0.877016\pi\)
\(32\) 0 0
\(33\) −7.91688 + 7.91688i −1.37815 + 1.37815i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.9348 −1.79767 −0.898833 0.438292i \(-0.855584\pi\)
−0.898833 + 0.438292i \(0.855584\pi\)
\(38\) 0 0
\(39\) 9.46410i 1.51547i
\(40\) 0 0
\(41\) 6.46410i 1.00952i −0.863259 0.504762i \(-0.831580\pi\)
0.863259 0.504762i \(-0.168420\pi\)
\(42\) 0 0
\(43\) −9.14162 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41421 1.41421i 0.206284 0.206284i −0.596402 0.802686i \(-0.703403\pi\)
0.802686 + 0.596402i \(0.203403\pi\)
\(48\) 0 0
\(49\) 5.39230i 0.770329i
\(50\) 0 0
\(51\) −1.36603 + 1.36603i −0.191282 + 0.191282i
\(52\) 0 0
\(53\) 9.89949i 1.35980i −0.733305 0.679900i \(-0.762023\pi\)
0.733305 0.679900i \(-0.237977\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.98502 5.98502i 0.792736 0.792736i
\(58\) 0 0
\(59\) −4.26795 + 4.26795i −0.555640 + 0.555640i −0.928063 0.372423i \(-0.878527\pi\)
0.372423 + 0.928063i \(0.378527\pi\)
\(60\) 0 0
\(61\) 7.19615 + 7.19615i 0.921373 + 0.921373i 0.997127 0.0757537i \(-0.0241363\pi\)
−0.0757537 + 0.997127i \(0.524136\pi\)
\(62\) 0 0
\(63\) −0.656339 0.656339i −0.0826909 0.0826909i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.55291 −0.189719 −0.0948593 0.995491i \(-0.530240\pi\)
−0.0948593 + 0.995491i \(0.530240\pi\)
\(68\) 0 0
\(69\) 5.73205 + 5.73205i 0.690058 + 0.690058i
\(70\) 0 0
\(71\) 12.9282 1.53430 0.767148 0.641470i \(-0.221675\pi\)
0.767148 + 0.641470i \(0.221675\pi\)
\(72\) 0 0
\(73\) 3.91447 + 3.91447i 0.458154 + 0.458154i 0.898049 0.439895i \(-0.144984\pi\)
−0.439895 + 0.898049i \(0.644984\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.34847i 0.837436i
\(78\) 0 0
\(79\) −8.19615 −0.922139 −0.461070 0.887364i \(-0.652534\pi\)
−0.461070 + 0.887364i \(0.652534\pi\)
\(80\) 0 0
\(81\) −10.6603 −1.18447
\(82\) 0 0
\(83\) 4.65874i 0.511363i 0.966761 + 0.255682i \(0.0822999\pi\)
−0.966761 + 0.255682i \(0.917700\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.44949 + 2.44949i 0.262613 + 0.262613i
\(88\) 0 0
\(89\) −13.7321 −1.45559 −0.727797 0.685792i \(-0.759456\pi\)
−0.727797 + 0.685792i \(0.759456\pi\)
\(90\) 0 0
\(91\) −4.39230 4.39230i −0.460439 0.460439i
\(92\) 0 0
\(93\) 8.10634 0.840589
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.10583 3.10583i −0.315349 0.315349i 0.531629 0.846978i \(-0.321580\pi\)
−0.846978 + 0.531629i \(0.821580\pi\)
\(98\) 0 0
\(99\) 3.00000 + 3.00000i 0.301511 + 0.301511i
\(100\) 0 0
\(101\) 3.92820 3.92820i 0.390871 0.390871i −0.484127 0.874998i \(-0.660863\pi\)
0.874998 + 0.484127i \(0.160863\pi\)
\(102\) 0 0
\(103\) −2.44949 + 2.44949i −0.241355 + 0.241355i −0.817411 0.576055i \(-0.804591\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.55291i 0.150126i −0.997179 0.0750629i \(-0.976084\pi\)
0.997179 0.0750629i \(-0.0239158\pi\)
\(108\) 0 0
\(109\) −11.3923 + 11.3923i −1.09118 + 1.09118i −0.0957826 + 0.995402i \(0.530535\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 21.1244i 2.00504i
\(112\) 0 0
\(113\) −9.33109 + 9.33109i −0.877795 + 0.877795i −0.993306 0.115511i \(-0.963149\pi\)
0.115511 + 0.993306i \(0.463149\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.58630 0.331554
\(118\) 0 0
\(119\) 1.26795i 0.116233i
\(120\) 0 0
\(121\) 22.5885i 2.05350i
\(122\) 0 0
\(123\) −12.4877 −1.12598
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.55291 1.55291i 0.137799 0.137799i −0.634843 0.772641i \(-0.718935\pi\)
0.772641 + 0.634843i \(0.218935\pi\)
\(128\) 0 0
\(129\) 17.6603i 1.55490i
\(130\) 0 0
\(131\) 9.92820 9.92820i 0.867431 0.867431i −0.124756 0.992187i \(-0.539815\pi\)
0.992187 + 0.124756i \(0.0398149\pi\)
\(132\) 0 0
\(133\) 5.55532i 0.481707i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.98262 + 1.98262i −0.169387 + 0.169387i −0.786710 0.617323i \(-0.788217\pi\)
0.617323 + 0.786710i \(0.288217\pi\)
\(138\) 0 0
\(139\) 8.29423 8.29423i 0.703507 0.703507i −0.261654 0.965162i \(-0.584268\pi\)
0.965162 + 0.261654i \(0.0842680\pi\)
\(140\) 0 0
\(141\) −2.73205 2.73205i −0.230080 0.230080i
\(142\) 0 0
\(143\) 20.0764 + 20.0764i 1.67887 + 1.67887i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.4171 −0.859191
\(148\) 0 0
\(149\) 9.12436 + 9.12436i 0.747496 + 0.747496i 0.974008 0.226512i \(-0.0727323\pi\)
−0.226512 + 0.974008i \(0.572732\pi\)
\(150\) 0 0
\(151\) 0.196152 0.0159627 0.00798133 0.999968i \(-0.497459\pi\)
0.00798133 + 0.999968i \(0.497459\pi\)
\(152\) 0 0
\(153\) 0.517638 + 0.517638i 0.0418486 + 0.0418486i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.656339i 0.0523815i 0.999657 + 0.0261908i \(0.00833773\pi\)
−0.999657 + 0.0261908i \(0.991662\pi\)
\(158\) 0 0
\(159\) −19.1244 −1.51666
\(160\) 0 0
\(161\) −5.32051 −0.419315
\(162\) 0 0
\(163\) 13.1440i 1.02952i −0.857334 0.514760i \(-0.827881\pi\)
0.857334 0.514760i \(-0.172119\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.4195 14.4195i −1.11582 1.11582i −0.992348 0.123469i \(-0.960598\pi\)
−0.123469 0.992348i \(-0.539402\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 0 0
\(171\) −2.26795 2.26795i −0.173434 0.173434i
\(172\) 0 0
\(173\) −8.76268 −0.666214 −0.333107 0.942889i \(-0.608097\pi\)
−0.333107 + 0.942889i \(0.608097\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.24504 + 8.24504i 0.619736 + 0.619736i
\(178\) 0 0
\(179\) −4.56218 4.56218i −0.340993 0.340993i 0.515747 0.856741i \(-0.327514\pi\)
−0.856741 + 0.515747i \(0.827514\pi\)
\(180\) 0 0
\(181\) 13.1962 13.1962i 0.980862 0.980862i −0.0189580 0.999820i \(-0.506035\pi\)
0.999820 + 0.0189580i \(0.00603488\pi\)
\(182\) 0 0
\(183\) 13.9019 13.9019i 1.02766 1.02766i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.79555i 0.423813i
\(188\) 0 0
\(189\) 3.92820 3.92820i 0.285735 0.285735i
\(190\) 0 0
\(191\) 0.339746i 0.0245832i −0.999924 0.0122916i \(-0.996087\pi\)
0.999924 0.0122916i \(-0.00391263\pi\)
\(192\) 0 0
\(193\) 5.22715 5.22715i 0.376258 0.376258i −0.493492 0.869750i \(-0.664280\pi\)
0.869750 + 0.493492i \(0.164280\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.9396 1.34939 0.674695 0.738097i \(-0.264275\pi\)
0.674695 + 0.738097i \(0.264275\pi\)
\(198\) 0 0
\(199\) 4.39230i 0.311362i −0.987807 0.155681i \(-0.950243\pi\)
0.987807 0.155681i \(-0.0497572\pi\)
\(200\) 0 0
\(201\) 3.00000i 0.211604i
\(202\) 0 0
\(203\) −2.27362 −0.159577
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.17209 2.17209i 0.150971 0.150971i
\(208\) 0 0
\(209\) 25.3923i 1.75642i
\(210\) 0 0
\(211\) 4.29423 4.29423i 0.295627 0.295627i −0.543671 0.839298i \(-0.682966\pi\)
0.839298 + 0.543671i \(0.182966\pi\)
\(212\) 0 0
\(213\) 24.9754i 1.71128i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.76217 + 3.76217i −0.255393 + 0.255393i
\(218\) 0 0
\(219\) 7.56218 7.56218i 0.511005 0.511005i
\(220\) 0 0
\(221\) 3.46410 + 3.46410i 0.233021 + 0.233021i
\(222\) 0 0
\(223\) −2.44949 2.44949i −0.164030 0.164030i 0.620319 0.784349i \(-0.287003\pi\)
−0.784349 + 0.620319i \(0.787003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.4543 0.693876 0.346938 0.937888i \(-0.387221\pi\)
0.346938 + 0.937888i \(0.387221\pi\)
\(228\) 0 0
\(229\) −4.00000 4.00000i −0.264327 0.264327i 0.562482 0.826809i \(-0.309847\pi\)
−0.826809 + 0.562482i \(0.809847\pi\)
\(230\) 0 0
\(231\) −14.1962 −0.934038
\(232\) 0 0
\(233\) −5.65685 5.65685i −0.370593 0.370593i 0.497100 0.867693i \(-0.334398\pi\)
−0.867693 + 0.497100i \(0.834398\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 15.8338i 1.02851i
\(238\) 0 0
\(239\) 14.5359 0.940249 0.470125 0.882600i \(-0.344209\pi\)
0.470125 + 0.882600i \(0.344209\pi\)
\(240\) 0 0
\(241\) −23.1962 −1.49420 −0.747098 0.664714i \(-0.768553\pi\)
−0.747098 + 0.664714i \(0.768553\pi\)
\(242\) 0 0
\(243\) 7.45001i 0.477918i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.1774 15.1774i −0.965716 0.965716i
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −0.169873 0.169873i −0.0107223 0.0107223i 0.701725 0.712448i \(-0.252413\pi\)
−0.712448 + 0.701725i \(0.752413\pi\)
\(252\) 0 0
\(253\) 24.3190 1.52892
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.1769 + 10.1769i 0.634817 + 0.634817i 0.949272 0.314455i \(-0.101822\pi\)
−0.314455 + 0.949272i \(0.601822\pi\)
\(258\) 0 0
\(259\) −9.80385 9.80385i −0.609181 0.609181i
\(260\) 0 0
\(261\) 0.928203 0.928203i 0.0574543 0.0574543i
\(262\) 0 0
\(263\) 1.27551 1.27551i 0.0786515 0.0786515i −0.666687 0.745338i \(-0.732288\pi\)
0.745338 + 0.666687i \(0.232288\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 26.5283i 1.62350i
\(268\) 0 0
\(269\) 10.7321 10.7321i 0.654345 0.654345i −0.299691 0.954036i \(-0.596884\pi\)
0.954036 + 0.299691i \(0.0968837\pi\)
\(270\) 0 0
\(271\) 30.5885i 1.85812i −0.369934 0.929058i \(-0.620620\pi\)
0.369934 0.929058i \(-0.379380\pi\)
\(272\) 0 0
\(273\) −8.48528 + 8.48528i −0.513553 + 0.513553i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 3.07180i 0.183904i
\(280\) 0 0
\(281\) 8.53590i 0.509209i −0.967045 0.254605i \(-0.918055\pi\)
0.967045 0.254605i \(-0.0819453\pi\)
\(282\) 0 0
\(283\) −20.9730 −1.24671 −0.623357 0.781938i \(-0.714232\pi\)
−0.623357 + 0.781938i \(0.714232\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.79555 5.79555i 0.342101 0.342101i
\(288\) 0 0
\(289\) 16.0000i 0.941176i
\(290\) 0 0
\(291\) −6.00000 + 6.00000i −0.351726 + 0.351726i
\(292\) 0 0
\(293\) 9.89949i 0.578335i 0.957279 + 0.289167i \(0.0933784\pi\)
−0.957279 + 0.289167i \(0.906622\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −17.9551 + 17.9551i −1.04186 + 1.04186i
\(298\) 0 0
\(299\) 14.5359 14.5359i 0.840633 0.840633i
\(300\) 0 0
\(301\) −8.19615 8.19615i −0.472418 0.472418i
\(302\) 0 0
\(303\) −7.58871 7.58871i −0.435960 0.435960i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.27603 0.358192 0.179096 0.983832i \(-0.442683\pi\)
0.179096 + 0.983832i \(0.442683\pi\)
\(308\) 0 0
\(309\) 4.73205 + 4.73205i 0.269197 + 0.269197i
\(310\) 0 0
\(311\) −5.66025 −0.320964 −0.160482 0.987039i \(-0.551305\pi\)
−0.160482 + 0.987039i \(0.551305\pi\)
\(312\) 0 0
\(313\) 9.14162 + 9.14162i 0.516715 + 0.516715i 0.916576 0.399861i \(-0.130942\pi\)
−0.399861 + 0.916576i \(0.630942\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.0764i 1.12760i −0.825911 0.563801i \(-0.809338\pi\)
0.825911 0.563801i \(-0.190662\pi\)
\(318\) 0 0
\(319\) 10.3923 0.581857
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 4.38134i 0.243784i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 22.0082 + 22.0082i 1.21706 + 1.21706i
\(328\) 0 0
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) −6.49038 6.49038i −0.356744 0.356744i 0.505868 0.862611i \(-0.331172\pi\)
−0.862611 + 0.505868i \(0.831172\pi\)
\(332\) 0 0
\(333\) 8.00481 0.438661
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9.71003 9.71003i −0.528939 0.528939i 0.391317 0.920256i \(-0.372020\pi\)
−0.920256 + 0.391317i \(0.872020\pi\)
\(338\) 0 0
\(339\) 18.0263 + 18.0263i 0.979053 + 0.979053i
\(340\) 0 0
\(341\) 17.1962 17.1962i 0.931224 0.931224i
\(342\) 0 0
\(343\) 11.1106 11.1106i 0.599918 0.599918i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.38134i 0.235203i −0.993061 0.117601i \(-0.962479\pi\)
0.993061 0.117601i \(-0.0375205\pi\)
\(348\) 0 0
\(349\) −7.58846 + 7.58846i −0.406201 + 0.406201i −0.880411 0.474211i \(-0.842734\pi\)
0.474211 + 0.880411i \(0.342734\pi\)
\(350\) 0 0
\(351\) 21.4641i 1.14567i
\(352\) 0 0
\(353\) 24.5964 24.5964i 1.30914 1.30914i 0.387097 0.922039i \(-0.373478\pi\)
0.922039 0.387097i \(-0.126522\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.44949 −0.129641
\(358\) 0 0
\(359\) 6.58846i 0.347725i 0.984770 + 0.173863i \(0.0556249\pi\)
−0.984770 + 0.173863i \(0.944375\pi\)
\(360\) 0 0
\(361\) 0.196152i 0.0103238i
\(362\) 0 0
\(363\) 43.6375 2.29038
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −22.5259 + 22.5259i −1.17584 + 1.17584i −0.195048 + 0.980794i \(0.562486\pi\)
−0.980794 + 0.195048i \(0.937514\pi\)
\(368\) 0 0
\(369\) 4.73205i 0.246341i
\(370\) 0 0
\(371\) 8.87564 8.87564i 0.460800 0.460800i
\(372\) 0 0
\(373\) 1.13681i 0.0588619i 0.999567 + 0.0294310i \(0.00936952\pi\)
−0.999567 + 0.0294310i \(0.990630\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.21166 6.21166i 0.319917 0.319917i
\(378\) 0 0
\(379\) −5.29423 + 5.29423i −0.271946 + 0.271946i −0.829883 0.557937i \(-0.811593\pi\)
0.557937 + 0.829883i \(0.311593\pi\)
\(380\) 0 0
\(381\) −3.00000 3.00000i −0.153695 0.153695i
\(382\) 0 0
\(383\) 7.20977 + 7.20977i 0.368402 + 0.368402i 0.866894 0.498492i \(-0.166113\pi\)
−0.498492 + 0.866894i \(0.666113\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.69213 0.340180
\(388\) 0 0
\(389\) −26.3205 26.3205i −1.33450 1.33450i −0.901293 0.433209i \(-0.857381\pi\)
−0.433209 0.901293i \(-0.642619\pi\)
\(390\) 0 0
\(391\) 4.19615 0.212209
\(392\) 0 0
\(393\) −19.1798 19.1798i −0.967494 0.967494i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.3843i 0.671737i 0.941909 + 0.335868i \(0.109030\pi\)
−0.941909 + 0.335868i \(0.890970\pi\)
\(398\) 0 0
\(399\) 10.7321 0.537275
\(400\) 0 0
\(401\) 12.1244 0.605461 0.302731 0.953076i \(-0.402102\pi\)
0.302731 + 0.953076i \(0.402102\pi\)
\(402\) 0 0
\(403\) 20.5569i 1.02401i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.8115 + 44.8115i 2.22122 + 2.22122i
\(408\) 0 0
\(409\) −4.60770 −0.227836 −0.113918 0.993490i \(-0.536340\pi\)
−0.113918 + 0.993490i \(0.536340\pi\)
\(410\) 0 0
\(411\) 3.83013 + 3.83013i 0.188926 + 0.188926i
\(412\) 0 0
\(413\) −7.65308 −0.376583
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.0232 16.0232i −0.784660 0.784660i
\(418\) 0 0
\(419\) −16.5622 16.5622i −0.809115 0.809115i 0.175385 0.984500i \(-0.443883\pi\)
−0.984500 + 0.175385i \(0.943883\pi\)
\(420\) 0 0
\(421\) −26.3923 + 26.3923i −1.28628 + 1.28628i −0.349254 + 0.937028i \(0.613565\pi\)
−0.937028 + 0.349254i \(0.886435\pi\)
\(422\) 0 0
\(423\) −1.03528 + 1.03528i −0.0503369 + 0.0503369i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.9038i 0.624458i
\(428\) 0 0
\(429\) 38.7846 38.7846i 1.87254 1.87254i
\(430\) 0 0
\(431\) 6.58846i 0.317355i 0.987330 + 0.158677i \(0.0507230\pi\)
−0.987330 + 0.158677i \(0.949277\pi\)
\(432\) 0 0
\(433\) 4.57081 4.57081i 0.219659 0.219659i −0.588696 0.808355i \(-0.700358\pi\)
0.808355 + 0.588696i \(0.200358\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.3848 −0.879463
\(438\) 0 0
\(439\) 40.3923i 1.92782i −0.266229 0.963910i \(-0.585778\pi\)
0.266229 0.963910i \(-0.414222\pi\)
\(440\) 0 0
\(441\) 3.94744i 0.187973i
\(442\) 0 0
\(443\) 16.5545 0.786526 0.393263 0.919426i \(-0.371346\pi\)
0.393263 + 0.919426i \(0.371346\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 17.6269 17.6269i 0.833724 0.833724i
\(448\) 0 0
\(449\) 6.12436i 0.289026i −0.989503 0.144513i \(-0.953838\pi\)
0.989503 0.144513i \(-0.0461616\pi\)
\(450\) 0 0
\(451\) −26.4904 + 26.4904i −1.24738 + 1.24738i
\(452\) 0 0
\(453\) 0.378937i 0.0178040i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.1953 18.1953i 0.851141 0.851141i −0.139133 0.990274i \(-0.544432\pi\)
0.990274 + 0.139133i \(0.0444316\pi\)
\(458\) 0 0
\(459\) −3.09808 + 3.09808i −0.144606 + 0.144606i
\(460\) 0 0
\(461\) −13.2679 13.2679i −0.617950 0.617950i 0.327055 0.945005i \(-0.393944\pi\)
−0.945005 + 0.327055i \(0.893944\pi\)
\(462\) 0 0
\(463\) 8.00481 + 8.00481i 0.372015 + 0.372015i 0.868211 0.496196i \(-0.165270\pi\)
−0.496196 + 0.868211i \(0.665270\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.41421 0.0654420 0.0327210 0.999465i \(-0.489583\pi\)
0.0327210 + 0.999465i \(0.489583\pi\)
\(468\) 0 0
\(469\) −1.39230 1.39230i −0.0642907 0.0642907i
\(470\) 0 0
\(471\) 1.26795 0.0584240
\(472\) 0 0
\(473\) 37.4631 + 37.4631i 1.72255 + 1.72255i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.24693i 0.331814i
\(478\) 0 0
\(479\) −24.3397 −1.11211 −0.556056 0.831145i \(-0.687686\pi\)
−0.556056 + 0.831145i \(0.687686\pi\)
\(480\) 0 0
\(481\) 53.5692 2.44255
\(482\) 0 0
\(483\) 10.2784i 0.467685i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −23.4225 23.4225i −1.06137 1.06137i −0.997989 0.0633836i \(-0.979811\pi\)
−0.0633836 0.997989i \(-0.520189\pi\)
\(488\) 0 0
\(489\) −25.3923 −1.14828
\(490\) 0 0
\(491\) 14.3205 + 14.3205i 0.646275 + 0.646275i 0.952091 0.305815i \(-0.0989290\pi\)
−0.305815 + 0.952091i \(0.598929\pi\)
\(492\) 0 0
\(493\) 1.79315 0.0807595
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.5911 + 11.5911i 0.519932 + 0.519932i
\(498\) 0 0
\(499\) 23.0000 + 23.0000i 1.02962 + 1.02962i 0.999548 + 0.0300737i \(0.00957421\pi\)
0.0300737 + 0.999548i \(0.490426\pi\)
\(500\) 0 0
\(501\) −27.8564 + 27.8564i −1.24453 + 1.24453i
\(502\) 0 0
\(503\) 7.62587 7.62587i 0.340021 0.340021i −0.516354 0.856375i \(-0.672711\pi\)
0.856375 + 0.516354i \(0.172711\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 21.2504i 0.943762i
\(508\) 0 0
\(509\) −15.0000 + 15.0000i −0.664863 + 0.664863i −0.956522 0.291659i \(-0.905793\pi\)
0.291659 + 0.956522i \(0.405793\pi\)
\(510\) 0 0
\(511\) 7.01924i 0.310513i
\(512\) 0 0
\(513\) 13.5737 13.5737i 0.599295 0.599295i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.5911 −0.509776
\(518\) 0 0
\(519\) 16.9282i 0.743066i
\(520\) 0 0
\(521\) 45.2487i 1.98238i −0.132440 0.991191i \(-0.542281\pi\)
0.132440 0.991191i \(-0.457719\pi\)
\(522\) 0 0
\(523\) 26.0478 1.13899 0.569496 0.821994i \(-0.307139\pi\)
0.569496 + 0.821994i \(0.307139\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.96713 2.96713i 0.129250 0.129250i
\(528\) 0 0
\(529\) 5.39230i 0.234448i
\(530\) 0 0
\(531\) 3.12436 3.12436i 0.135585 0.135585i
\(532\) 0 0
\(533\) 31.6675i 1.37167i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.81345 + 8.81345i −0.380328 + 0.380328i
\(538\) 0 0
\(539\) −22.0981 + 22.0981i −0.951832 + 0.951832i
\(540\) 0 0
\(541\) 30.1962 + 30.1962i 1.29823 + 1.29823i 0.929558 + 0.368676i \(0.120189\pi\)
0.368676 + 0.929558i \(0.379811\pi\)
\(542\) 0 0
\(543\) −25.4930 25.4930i −1.09401 1.09401i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.2808 0.610604 0.305302 0.952256i \(-0.401243\pi\)
0.305302 + 0.952256i \(0.401243\pi\)
\(548\) 0 0
\(549\) −5.26795 5.26795i −0.224831 0.224831i
\(550\) 0 0
\(551\) −7.85641 −0.334694
\(552\) 0 0
\(553\) −7.34847 7.34847i −0.312489 0.312489i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.38323i 0.143352i 0.997428 + 0.0716760i \(0.0228348\pi\)
−0.997428 + 0.0716760i \(0.977165\pi\)
\(558\) 0 0
\(559\) 44.7846 1.89419
\(560\) 0 0
\(561\) 11.1962 0.472702
\(562\) 0 0
\(563\) 38.4612i 1.62094i 0.585777 + 0.810472i \(0.300790\pi\)
−0.585777 + 0.810472i \(0.699210\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −9.55772 9.55772i −0.401387 0.401387i
\(568\) 0 0
\(569\) 14.0718 0.589920 0.294960 0.955510i \(-0.404694\pi\)
0.294960 + 0.955510i \(0.404694\pi\)
\(570\) 0 0
\(571\) −0.607695 0.607695i −0.0254313 0.0254313i 0.694277 0.719708i \(-0.255724\pi\)
−0.719708 + 0.694277i \(0.755724\pi\)
\(572\) 0 0
\(573\) −0.656339 −0.0274189
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.4722 + 13.4722i 0.560855 + 0.560855i 0.929550 0.368695i \(-0.120195\pi\)
−0.368695 + 0.929550i \(0.620195\pi\)
\(578\) 0 0
\(579\) −10.0981 10.0981i −0.419662 0.419662i
\(580\) 0 0
\(581\) −4.17691 + 4.17691i −0.173288 + 0.173288i
\(582\) 0 0
\(583\) −40.5689 + 40.5689i −1.68019 + 1.68019i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.9483i 1.89649i −0.317538 0.948245i \(-0.602856\pi\)
0.317538 0.948245i \(-0.397144\pi\)
\(588\) 0 0
\(589\) −13.0000 + 13.0000i −0.535656 + 0.535656i
\(590\) 0 0
\(591\) 36.5885i 1.50505i
\(592\) 0 0
\(593\) −11.3001 + 11.3001i −0.464040 + 0.464040i −0.899977 0.435937i \(-0.856417\pi\)
0.435937 + 0.899977i \(0.356417\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.48528 −0.347279
\(598\) 0 0
\(599\) 26.1962i 1.07035i 0.844743 + 0.535173i \(0.179754\pi\)
−0.844743 + 0.535173i \(0.820246\pi\)
\(600\) 0 0
\(601\) 23.1962i 0.946191i 0.881011 + 0.473095i \(0.156863\pi\)
−0.881011 + 0.473095i \(0.843137\pi\)
\(602\) 0 0
\(603\) 1.13681 0.0462946
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −24.3190 + 24.3190i −0.987079 + 0.987079i −0.999918 0.0128385i \(-0.995913\pi\)
0.0128385 + 0.999918i \(0.495913\pi\)
\(608\) 0 0
\(609\) 4.39230i 0.177985i
\(610\) 0 0
\(611\) −6.92820 + 6.92820i −0.280285 + 0.280285i
\(612\) 0 0
\(613\) 13.5601i 0.547688i −0.961774 0.273844i \(-0.911705\pi\)
0.961774 0.273844i \(-0.0882953\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.9048 22.9048i 0.922113 0.922113i −0.0750653 0.997179i \(-0.523917\pi\)
0.997179 + 0.0750653i \(0.0239165\pi\)
\(618\) 0 0
\(619\) −15.7846 + 15.7846i −0.634437 + 0.634437i −0.949178 0.314741i \(-0.898082\pi\)
0.314741 + 0.949178i \(0.398082\pi\)
\(620\) 0 0
\(621\) 13.0000 + 13.0000i 0.521672 + 0.521672i
\(622\) 0 0
\(623\) −12.3118 12.3118i −0.493263 0.493263i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −49.0542 −1.95903
\(628\) 0 0
\(629\) 7.73205 + 7.73205i 0.308297 + 0.308297i
\(630\) 0 0
\(631\) 18.5885 0.739995 0.369997 0.929033i \(-0.379359\pi\)
0.369997 + 0.929033i \(0.379359\pi\)
\(632\) 0 0
\(633\) −8.29581 8.29581i −0.329729 0.329729i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.4168i 1.04667i
\(638\) 0 0
\(639\) −9.46410 −0.374394
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) 0 0
\(643\) 32.3238i 1.27473i −0.770563 0.637364i \(-0.780025\pi\)
0.770563 0.637364i \(-0.219975\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.96524 + 3.96524i 0.155890 + 0.155890i 0.780743 0.624853i \(-0.214841\pi\)
−0.624853 + 0.780743i \(0.714841\pi\)
\(648\) 0 0
\(649\) 34.9808 1.37312
\(650\) 0 0
\(651\) 7.26795 + 7.26795i 0.284853 + 0.284853i
\(652\) 0 0
\(653\) −7.62587 −0.298423 −0.149212 0.988805i \(-0.547674\pi\)
−0.149212 + 0.988805i \(0.547674\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.86559 2.86559i −0.111797 0.111797i
\(658\) 0 0
\(659\) −26.4904 26.4904i −1.03192 1.03192i −0.999474 0.0324452i \(-0.989671\pi\)
−0.0324452 0.999474i \(-0.510329\pi\)
\(660\) 0 0
\(661\) −8.00000 + 8.00000i −0.311164 + 0.311164i −0.845360 0.534196i \(-0.820614\pi\)
0.534196 + 0.845360i \(0.320614\pi\)
\(662\) 0 0
\(663\) 6.69213 6.69213i 0.259901 0.259901i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.52433i 0.291343i
\(668\) 0 0
\(669\) −4.73205 + 4.73205i −0.182952 + 0.182952i
\(670\) 0 0
\(671\) 58.9808i 2.27693i
\(672\) 0 0
\(673\) 7.17260 7.17260i 0.276484 0.276484i −0.555220 0.831704i \(-0.687366\pi\)
0.831704 + 0.555220i \(0.187366\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.1127 1.19576 0.597879 0.801586i \(-0.296010\pi\)
0.597879 + 0.801586i \(0.296010\pi\)
\(678\) 0 0
\(679\) 5.56922i 0.213727i
\(680\) 0 0
\(681\) 20.1962i 0.773918i
\(682\) 0 0
\(683\) 26.7314 1.02285 0.511423 0.859329i \(-0.329118\pi\)
0.511423 + 0.859329i \(0.329118\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −7.72741 + 7.72741i −0.294819 + 0.294819i
\(688\) 0 0
\(689\) 48.4974i 1.84760i
\(690\) 0 0
\(691\) −25.8827 + 25.8827i −0.984624 + 0.984624i −0.999884 0.0152598i \(-0.995142\pi\)
0.0152598 + 0.999884i \(0.495142\pi\)
\(692\) 0 0
\(693\) 5.37945i 0.204349i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.57081 + 4.57081i −0.173132 + 0.173132i
\(698\) 0 0
\(699\) −10.9282 + 10.9282i −0.413343 + 0.413343i
\(700\) 0 0
\(701\) −12.5885 12.5885i −0.475459 0.475459i 0.428217 0.903676i \(-0.359142\pi\)
−0.903676 + 0.428217i \(0.859142\pi\)
\(702\) 0 0
\(703\) −33.8768 33.8768i −1.27769 1.27769i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.04386 0.264912
\(708\) 0 0
\(709\) −2.00000 2.00000i −0.0751116 0.0751116i 0.668553 0.743665i \(-0.266914\pi\)
−0.743665 + 0.668553i \(0.766914\pi\)
\(710\) 0 0
\(711\) 6.00000 0.225018
\(712\) 0 0
\(713\) −12.4505 12.4505i −0.466276 0.466276i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 28.0812i 1.04871i
\(718\) 0 0
\(719\) 4.73205 0.176476 0.0882379 0.996099i \(-0.471876\pi\)
0.0882379 + 0.996099i \(0.471876\pi\)
\(720\) 0 0
\(721\) −4.39230 −0.163578
\(722\) 0 0
\(723\) 44.8115i 1.66656i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.34847 + 7.34847i 0.272540 + 0.272540i 0.830122 0.557582i \(-0.188271\pi\)
−0.557582 + 0.830122i \(0.688271\pi\)
\(728\) 0 0
\(729\) −17.5885 −0.651424
\(730\) 0 0
\(731\) 6.46410 + 6.46410i 0.239083 + 0.239083i
\(732\) 0 0
\(733\) −47.6771 −1.76099 −0.880497 0.474051i \(-0.842791\pi\)
−0.880497 + 0.474051i \(0.842791\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.36396 + 6.36396i 0.234420 + 0.234420i
\(738\) 0 0
\(739\) −0.607695 0.607695i −0.0223544 0.0223544i 0.695841 0.718196i \(-0.255032\pi\)
−0.718196 + 0.695841i \(0.755032\pi\)
\(740\) 0 0
\(741\) −29.3205 + 29.3205i −1.07712 + 1.07712i
\(742\) 0 0
\(743\) −26.4267 + 26.4267i −0.969503 + 0.969503i −0.999549 0.0300451i \(-0.990435\pi\)
0.0300451 + 0.999549i \(0.490435\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.41044i 0.124781i
\(748\) 0 0
\(749\) 1.39230 1.39230i 0.0508737 0.0508737i
\(750\) 0 0
\(751\) 23.6077i 0.861457i 0.902482 + 0.430729i \(0.141743\pi\)
−0.902482 + 0.430729i \(0.858257\pi\)
\(752\) 0 0
\(753\) −0.328169 + 0.328169i −0.0119592 + 0.0119592i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.37945 0.195520 0.0977598 0.995210i \(-0.468832\pi\)
0.0977598 + 0.995210i \(0.468832\pi\)
\(758\) 0 0
\(759\) 46.9808i 1.70529i
\(760\) 0 0
\(761\) 25.3923i 0.920470i −0.887797 0.460235i \(-0.847765\pi\)
0.887797 0.460235i \(-0.152235\pi\)
\(762\) 0 0
\(763\) −20.4281 −0.739548
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.9086 20.9086i 0.754966 0.754966i
\(768\) 0 0
\(769\) 24.8038i 0.894450i 0.894422 + 0.447225i \(0.147588\pi\)
−0.894422 + 0.447225i \(0.852412\pi\)
\(770\) 0 0
\(771\) 19.6603 19.6603i 0.708047 0.708047i
\(772\) 0 0
\(773\) 40.9850i 1.47413i 0.675823 + 0.737064i \(0.263788\pi\)
−0.675823 + 0.737064i \(0.736212\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −18.9396 + 18.9396i −0.679454 + 0.679454i
\(778\) 0 0
\(779\) 20.0263 20.0263i 0.717516 0.717516i
\(780\) 0 0
\(781\) −52.9808 52.9808i −1.89580 1.89580i
\(782\) 0 0
\(783\) 5.55532 + 5.55532i 0.198531 + 0.198531i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.1026 0.360118 0.180059 0.983656i \(-0.442371\pi\)
0.180059 + 0.983656i \(0.442371\pi\)
\(788\) 0 0
\(789\) −2.46410 2.46410i −0.0877243 0.0877243i
\(790\) 0 0
\(791\) −16.7321 −0.594923
\(792\) 0 0
\(793\) −35.2538 35.2538i −1.25190 1.25190i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.0454i 0.780888i −0.920627 0.390444i \(-0.872321\pi\)
0.920627 0.390444i \(-0.127679\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 0 0
\(801\) 10.0526 0.355190
\(802\) 0 0
\(803\) 32.0836i 1.13221i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20.7327 20.7327i −0.729827 0.729827i
\(808\) 0 0
\(809\) 6.24871 0.219693 0.109846 0.993949i \(-0.464964\pi\)
0.109846 + 0.993949i \(0.464964\pi\)
\(810\) 0 0
\(811\) 3.78461 + 3.78461i 0.132896 + 0.132896i 0.770426 0.637530i \(-0.220044\pi\)
−0.637530 + 0.770426i \(0.720044\pi\)
\(812\) 0 0
\(813\) −59.0924 −2.07246
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −28.3214 28.3214i −0.990842 0.990842i
\(818\) 0 0
\(819\) 3.21539 + 3.21539i 0.112355 + 0.112355i
\(820\) 0 0
\(821\) 18.4641 18.4641i 0.644402 0.644402i −0.307233 0.951634i \(-0.599403\pi\)
0.951634 + 0.307233i \(0.0994030\pi\)
\(822\) 0 0
\(823\) −11.5911 + 11.5911i −0.404041 + 0.404041i −0.879654 0.475614i \(-0.842226\pi\)
0.475614 + 0.879654i \(0.342226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.4524i 0.398239i −0.979975 0.199120i \(-0.936192\pi\)
0.979975 0.199120i \(-0.0638082\pi\)
\(828\) 0 0
\(829\) −14.5885 + 14.5885i −0.506678 + 0.506678i −0.913505 0.406827i \(-0.866635\pi\)
0.406827 + 0.913505i \(0.366635\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.81294 + 3.81294i −0.132110 + 0.132110i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 18.3848 0.635471
\(838\) 0 0
\(839\) 27.4641i 0.948166i 0.880480 + 0.474083i \(0.157220\pi\)
−0.880480 + 0.474083i \(0.842780\pi\)
\(840\) 0 0
\(841\) 25.7846i 0.889124i
\(842\) 0 0
\(843\) −16.4901 −0.567949
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −20.2523 + 20.2523i −0.695876 + 0.695876i
\(848\) 0 0
\(849\) 40.5167i 1.39053i
\(850\) 0 0
\(851\) 32.4449 32.4449i 1.11220 1.11220i
\(852\) 0 0
\(853\) 21.6937i 0.742777i 0.928477 + 0.371389i \(0.121118\pi\)
−0.928477 + 0.371389i \(0.878882\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.53365 + 4.53365i −0.154866 + 0.154866i −0.780287 0.625421i \(-0.784927\pi\)
0.625421 + 0.780287i \(0.284927\pi\)
\(858\) 0 0
\(859\) −27.9019 + 27.9019i −0.952001 + 0.952001i −0.998900 0.0468983i \(-0.985066\pi\)
0.0468983 + 0.998900i \(0.485066\pi\)
\(860\) 0 0
\(861\) −11.1962 11.1962i −0.381564 0.381564i
\(862\) 0 0
\(863\) −11.8685 11.8685i −0.404009 0.404009i 0.475634 0.879643i \(-0.342219\pi\)
−0.879643 + 0.475634i \(0.842219\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −30.9096 −1.04975
\(868\) 0 0
\(869\) 33.5885 + 33.5885i 1.13941 + 1.13941i
\(870\) 0 0
\(871\) 7.60770 0.257777
\(872\) 0 0
\(873\) 2.27362 + 2.27362i 0.0769505 + 0.0769505i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.480473i 0.0162244i −0.999967 0.00811222i \(-0.997418\pi\)
0.999967 0.00811222i \(-0.00258223\pi\)
\(878\) 0 0
\(879\) 19.1244 0.645049
\(880\) 0 0
\(881\) 13.8564 0.466834 0.233417 0.972377i \(-0.425009\pi\)
0.233417 + 0.972377i \(0.425009\pi\)
\(882\) 0 0
\(883\) 33.8768i 1.14004i 0.821630 + 0.570022i \(0.193065\pi\)
−0.821630 + 0.570022i \(0.806935\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.6932 16.6932i −0.560502 0.560502i 0.368948 0.929450i \(-0.379718\pi\)
−0.929450 + 0.368948i \(0.879718\pi\)
\(888\) 0 0
\(889\) 2.78461 0.0933928
\(890\) 0 0
\(891\) 43.6865 + 43.6865i 1.46355 + 1.46355i
\(892\) 0 0
\(893\) 8.76268 0.293232
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −28.0812 28.0812i −0.937604 0.937604i
\(898\) 0 0
\(899\) −5.32051 5.32051i −0.177449 0.177449i
\(900\) 0 0
\(901\) −7.00000 + 7.00000i −0.233204 + 0.233204i
\(902\) 0 0
\(903\) −15.8338 + 15.8338i −0.526914 + 0.526914i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 52.8807i 1.75587i 0.478775 + 0.877937i \(0.341081\pi\)
−0.478775 + 0.877937i \(0.658919\pi\)
\(908\) 0 0
\(909\) −2.87564 + 2.87564i −0.0953791 + 0.0953791i
\(910\) 0 0
\(911\) 24.2487i 0.803396i −0.915772 0.401698i \(-0.868420\pi\)
0.915772 0.401698i \(-0.131580\pi\)
\(912\) 0 0
\(913\) 19.0919 19.0919i 0.631849 0.631849i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.8028 0.587899
\(918\) 0 0
\(919\) 4.19615i 0.138418i −0.997602 0.0692091i \(-0.977952\pi\)
0.997602 0.0692091i \(-0.0220476\pi\)
\(920\) 0 0
\(921\) 12.1244i 0.399511i
\(922\) 0 0
\(923\) −63.3350 −2.08470
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.79315 1.79315i 0.0588948 0.0588948i
\(928\) 0 0
\(929\) 36.0000i 1.18112i 0.806993 + 0.590561i \(0.201093\pi\)
−0.806993 + 0.590561i \(0.798907\pi\)
\(930\) 0 0
\(931\) 16.7058 16.7058i 0.547510 0.547510i
\(932\) 0 0
\(933\) 10.9348i 0.357988i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.7482 19.7482i 0.645146 0.645146i −0.306670 0.951816i \(-0.599215\pi\)
0.951816 + 0.306670i \(0.0992147\pi\)
\(938\) 0 0
\(939\) 17.6603 17.6603i 0.576321 0.576321i
\(940\) 0 0
\(941\) 26.7846 + 26.7846i 0.873153 + 0.873153i 0.992815 0.119661i \(-0.0381809\pi\)
−0.119661 + 0.992815i \(0.538181\pi\)
\(942\) 0 0
\(943\) 19.1798 + 19.1798i 0.624581 + 0.624581i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.8338 0.514528 0.257264 0.966341i \(-0.417179\pi\)
0.257264 + 0.966341i \(0.417179\pi\)
\(948\) 0 0
\(949\) −19.1769 19.1769i −0.622509 0.622509i
\(950\) 0 0
\(951\) −38.7846 −1.25768
\(952\) 0 0
\(953\) −1.12321 1.12321i −0.0363843 0.0363843i 0.688681 0.725065i \(-0.258190\pi\)
−0.725065 + 0.688681i \(0.758190\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 20.0764i 0.648978i
\(958\) 0 0
\(959\) −3.55514 −0.114801
\(960\) 0 0
\(961\) 13.3923 0.432010
\(962\) 0 0
\(963\) 1.13681i 0.0366333i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −27.9053 27.9053i −0.897375 0.897375i 0.0978283 0.995203i \(-0.468810\pi\)
−0.995203 + 0.0978283i \(0.968810\pi\)
\(968\) 0 0
\(969\) −8.46410 −0.271906
\(970\) 0 0
\(971\) −0.633975 0.633975i −0.0203452 0.0203452i 0.696861 0.717206i \(-0.254579\pi\)
−0.717206 + 0.696861i \(0.754579\pi\)
\(972\) 0 0
\(973\) 14.8728 0.476800
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 10.3292 + 10.3292i 0.330460 + 0.330460i 0.852761 0.522301i \(-0.174926\pi\)
−0.522301 + 0.852761i \(0.674926\pi\)
\(978\) 0 0
\(979\) 56.2750 + 56.2750i 1.79856 + 1.79856i
\(980\) 0 0
\(981\) 8.33975 8.33975i 0.266268 0.266268i
\(982\) 0 0
\(983\) 7.48717 7.48717i 0.238804 0.238804i −0.577551 0.816355i \(-0.695992\pi\)
0.816355 + 0.577551i \(0.195992\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.89898i 0.155936i
\(988\) 0 0
\(989\) 27.1244 27.1244i 0.862504 0.862504i
\(990\) 0 0
\(991\) 18.5885i 0.590482i −0.955423 0.295241i \(-0.904600\pi\)
0.955423 0.295241i \(-0.0953999\pi\)
\(992\) 0 0
\(993\) −12.5385 + 12.5385i −0.397896 + 0.397896i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −54.3692 −1.72189 −0.860946 0.508697i \(-0.830127\pi\)
−0.860946 + 0.508697i \(0.830127\pi\)
\(998\) 0 0
\(999\) 47.9090i 1.51577i
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.j.b.1007.1 8
4.3 odd 2 400.2.j.b.307.4 yes 8
5.2 odd 4 1600.2.s.b.943.1 8
5.3 odd 4 1600.2.s.b.943.4 8
5.4 even 2 inner 1600.2.j.b.1007.4 8
16.5 even 4 400.2.s.b.107.3 yes 8
16.11 odd 4 1600.2.s.b.207.4 8
20.3 even 4 400.2.s.b.243.1 yes 8
20.7 even 4 400.2.s.b.243.4 yes 8
20.19 odd 2 400.2.j.b.307.1 yes 8
80.27 even 4 inner 1600.2.j.b.143.1 8
80.37 odd 4 400.2.j.b.43.2 8
80.43 even 4 inner 1600.2.j.b.143.4 8
80.53 odd 4 400.2.j.b.43.3 yes 8
80.59 odd 4 1600.2.s.b.207.1 8
80.69 even 4 400.2.s.b.107.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.b.43.2 8 80.37 odd 4
400.2.j.b.43.3 yes 8 80.53 odd 4
400.2.j.b.307.1 yes 8 20.19 odd 2
400.2.j.b.307.4 yes 8 4.3 odd 2
400.2.s.b.107.2 yes 8 80.69 even 4
400.2.s.b.107.3 yes 8 16.5 even 4
400.2.s.b.243.1 yes 8 20.3 even 4
400.2.s.b.243.4 yes 8 20.7 even 4
1600.2.j.b.143.1 8 80.27 even 4 inner
1600.2.j.b.143.4 8 80.43 even 4 inner
1600.2.j.b.1007.1 8 1.1 even 1 trivial
1600.2.j.b.1007.4 8 5.4 even 2 inner
1600.2.s.b.207.1 8 80.59 odd 4
1600.2.s.b.207.4 8 16.11 odd 4
1600.2.s.b.943.1 8 5.2 odd 4
1600.2.s.b.943.4 8 5.3 odd 4