Properties

Label 1521.2.b.l.1351.5
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,2,Mod(1351,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), 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: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.5
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.l.1351.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.801938i q^{2} +1.35690 q^{4} +0.246980i q^{5} -2.35690i q^{7} +2.69202i q^{8} +O(q^{10})\) \(q+0.801938i q^{2} +1.35690 q^{4} +0.246980i q^{5} -2.35690i q^{7} +2.69202i q^{8} -0.198062 q^{10} +4.24698i q^{11} +1.89008 q^{14} +0.554958 q^{16} +2.15883 q^{17} +0.0881460i q^{19} +0.335126i q^{20} -3.40581 q^{22} +1.49396 q^{23} +4.93900 q^{25} -3.19806i q^{28} -4.63102 q^{29} +6.63102i q^{31} +5.82908i q^{32} +1.73125i q^{34} +0.582105 q^{35} +5.69202i q^{37} -0.0706876 q^{38} -0.664874 q^{40} -11.5918i q^{41} +0.295897 q^{43} +5.76271i q^{44} +1.19806i q^{46} +7.35690i q^{47} +1.44504 q^{49} +3.96077i q^{50} +10.3937 q^{53} -1.04892 q^{55} +6.34481 q^{56} -3.71379i q^{58} +6.78017i q^{59} +3.47219 q^{61} -5.31767 q^{62} -3.56465 q^{64} -7.67994i q^{67} +2.92931 q^{68} +0.466812i q^{70} -8.66487i q^{71} +6.73556i q^{73} -4.56465 q^{74} +0.119605i q^{76} +10.0097 q^{77} +9.97046 q^{79} +0.137063i q^{80} +9.29590 q^{82} +1.60925i q^{83} +0.533188i q^{85} +0.237291i q^{86} -11.4330 q^{88} +2.88471i q^{89} +2.02715 q^{92} -5.89977 q^{94} -0.0217703 q^{95} +8.05861i q^{97} +1.15883i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62} + 22 q^{64} + 42 q^{68} + 16 q^{74} + 16 q^{77} - 10 q^{79} + 28 q^{82} - 30 q^{88} + 10 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.801938i 0.567056i 0.958964 + 0.283528i \(0.0915048\pi\)
−0.958964 + 0.283528i \(0.908495\pi\)
\(3\) 0 0
\(4\) 1.35690 0.678448
\(5\) 0.246980i 0.110453i 0.998474 + 0.0552263i \(0.0175880\pi\)
−0.998474 + 0.0552263i \(0.982412\pi\)
\(6\) 0 0
\(7\) − 2.35690i − 0.890823i −0.895326 0.445411i \(-0.853057\pi\)
0.895326 0.445411i \(-0.146943\pi\)
\(8\) 2.69202i 0.951773i
\(9\) 0 0
\(10\) −0.198062 −0.0626328
\(11\) 4.24698i 1.28051i 0.768161 + 0.640256i \(0.221172\pi\)
−0.768161 + 0.640256i \(0.778828\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.89008 0.505146
\(15\) 0 0
\(16\) 0.554958 0.138740
\(17\) 2.15883 0.523594 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) 0 0
\(19\) 0.0881460i 0.0202221i 0.999949 + 0.0101110i \(0.00321850\pi\)
−0.999949 + 0.0101110i \(0.996782\pi\)
\(20\) 0.335126i 0.0749364i
\(21\) 0 0
\(22\) −3.40581 −0.726122
\(23\) 1.49396 0.311512 0.155756 0.987796i \(-0.450219\pi\)
0.155756 + 0.987796i \(0.450219\pi\)
\(24\) 0 0
\(25\) 4.93900 0.987800
\(26\) 0 0
\(27\) 0 0
\(28\) − 3.19806i − 0.604377i
\(29\) −4.63102 −0.859959 −0.429980 0.902839i \(-0.641479\pi\)
−0.429980 + 0.902839i \(0.641479\pi\)
\(30\) 0 0
\(31\) 6.63102i 1.19097i 0.803368 + 0.595483i \(0.203039\pi\)
−0.803368 + 0.595483i \(0.796961\pi\)
\(32\) 5.82908i 1.03045i
\(33\) 0 0
\(34\) 1.73125i 0.296907i
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.69202i 0.935763i 0.883791 + 0.467881i \(0.154983\pi\)
−0.883791 + 0.467881i \(0.845017\pi\)
\(38\) −0.0706876 −0.0114670
\(39\) 0 0
\(40\) −0.664874 −0.105126
\(41\) − 11.5918i − 1.81033i −0.425056 0.905167i \(-0.639746\pi\)
0.425056 0.905167i \(-0.360254\pi\)
\(42\) 0 0
\(43\) 0.295897 0.0451239 0.0225619 0.999745i \(-0.492818\pi\)
0.0225619 + 0.999745i \(0.492818\pi\)
\(44\) 5.76271i 0.868761i
\(45\) 0 0
\(46\) 1.19806i 0.176645i
\(47\) 7.35690i 1.07311i 0.843864 + 0.536557i \(0.180275\pi\)
−0.843864 + 0.536557i \(0.819725\pi\)
\(48\) 0 0
\(49\) 1.44504 0.206435
\(50\) 3.96077i 0.560138i
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3937 1.42769 0.713844 0.700304i \(-0.246952\pi\)
0.713844 + 0.700304i \(0.246952\pi\)
\(54\) 0 0
\(55\) −1.04892 −0.141436
\(56\) 6.34481 0.847861
\(57\) 0 0
\(58\) − 3.71379i − 0.487645i
\(59\) 6.78017i 0.882703i 0.897334 + 0.441351i \(0.145501\pi\)
−0.897334 + 0.441351i \(0.854499\pi\)
\(60\) 0 0
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) −5.31767 −0.675344
\(63\) 0 0
\(64\) −3.56465 −0.445581
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.67994i − 0.938254i −0.883131 0.469127i \(-0.844569\pi\)
0.883131 0.469127i \(-0.155431\pi\)
\(68\) 2.92931 0.355231
\(69\) 0 0
\(70\) 0.466812i 0.0557947i
\(71\) − 8.66487i − 1.02833i −0.857691 0.514166i \(-0.828102\pi\)
0.857691 0.514166i \(-0.171898\pi\)
\(72\) 0 0
\(73\) 6.73556i 0.788338i 0.919038 + 0.394169i \(0.128968\pi\)
−0.919038 + 0.394169i \(0.871032\pi\)
\(74\) −4.56465 −0.530629
\(75\) 0 0
\(76\) 0.119605i 0.0137196i
\(77\) 10.0097 1.14071
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0.137063i 0.0153241i
\(81\) 0 0
\(82\) 9.29590 1.02656
\(83\) 1.60925i 0.176638i 0.996092 + 0.0883192i \(0.0281495\pi\)
−0.996092 + 0.0883192i \(0.971850\pi\)
\(84\) 0 0
\(85\) 0.533188i 0.0578323i
\(86\) 0.237291i 0.0255877i
\(87\) 0 0
\(88\) −11.4330 −1.21876
\(89\) 2.88471i 0.305778i 0.988243 + 0.152889i \(0.0488577\pi\)
−0.988243 + 0.152889i \(0.951142\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.02715 0.211345
\(93\) 0 0
\(94\) −5.89977 −0.608515
\(95\) −0.0217703 −0.00223358
\(96\) 0 0
\(97\) 8.05861i 0.818227i 0.912483 + 0.409114i \(0.134162\pi\)
−0.912483 + 0.409114i \(0.865838\pi\)
\(98\) 1.15883i 0.117060i
\(99\) 0 0
\(100\) 6.70171 0.670171
\(101\) −13.3545 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(102\) 0 0
\(103\) −1.36227 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.33513i 0.809579i
\(107\) −3.26875 −0.316002 −0.158001 0.987439i \(-0.550505\pi\)
−0.158001 + 0.987439i \(0.550505\pi\)
\(108\) 0 0
\(109\) − 15.7017i − 1.50395i −0.659191 0.751976i \(-0.729101\pi\)
0.659191 0.751976i \(-0.270899\pi\)
\(110\) − 0.841166i − 0.0802021i
\(111\) 0 0
\(112\) − 1.30798i − 0.123592i
\(113\) −12.0489 −1.13347 −0.566733 0.823901i \(-0.691793\pi\)
−0.566733 + 0.823901i \(0.691793\pi\)
\(114\) 0 0
\(115\) 0.368977i 0.0344073i
\(116\) −6.28382 −0.583438
\(117\) 0 0
\(118\) −5.43727 −0.500541
\(119\) − 5.08815i − 0.466430i
\(120\) 0 0
\(121\) −7.03684 −0.639712
\(122\) 2.78448i 0.252095i
\(123\) 0 0
\(124\) 8.99761i 0.808009i
\(125\) 2.45473i 0.219558i
\(126\) 0 0
\(127\) 9.80731 0.870258 0.435129 0.900368i \(-0.356703\pi\)
0.435129 + 0.900368i \(0.356703\pi\)
\(128\) 8.79954i 0.777777i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.57673 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(132\) 0 0
\(133\) 0.207751 0.0180143
\(134\) 6.15883 0.532042
\(135\) 0 0
\(136\) 5.81163i 0.498343i
\(137\) − 6.21983i − 0.531396i −0.964056 0.265698i \(-0.914398\pi\)
0.964056 0.265698i \(-0.0856024\pi\)
\(138\) 0 0
\(139\) −14.7071 −1.24744 −0.623719 0.781648i \(-0.714379\pi\)
−0.623719 + 0.781648i \(0.714379\pi\)
\(140\) 0.789856 0.0667550
\(141\) 0 0
\(142\) 6.94869 0.583121
\(143\) 0 0
\(144\) 0 0
\(145\) − 1.14377i − 0.0949848i
\(146\) −5.40150 −0.447031
\(147\) 0 0
\(148\) 7.72348i 0.634866i
\(149\) 4.33513i 0.355147i 0.984108 + 0.177574i \(0.0568248\pi\)
−0.984108 + 0.177574i \(0.943175\pi\)
\(150\) 0 0
\(151\) 3.94438i 0.320989i 0.987037 + 0.160494i \(0.0513089\pi\)
−0.987037 + 0.160494i \(0.948691\pi\)
\(152\) −0.237291 −0.0192468
\(153\) 0 0
\(154\) 8.02715i 0.646846i
\(155\) −1.63773 −0.131545
\(156\) 0 0
\(157\) 4.45473 0.355526 0.177763 0.984073i \(-0.443114\pi\)
0.177763 + 0.984073i \(0.443114\pi\)
\(158\) 7.99569i 0.636103i
\(159\) 0 0
\(160\) −1.43967 −0.113816
\(161\) − 3.52111i − 0.277502i
\(162\) 0 0
\(163\) 16.1588i 1.26566i 0.774292 + 0.632829i \(0.218106\pi\)
−0.774292 + 0.632829i \(0.781894\pi\)
\(164\) − 15.7289i − 1.22822i
\(165\) 0 0
\(166\) −1.29052 −0.100164
\(167\) − 16.1172i − 1.24719i −0.781749 0.623594i \(-0.785672\pi\)
0.781749 0.623594i \(-0.214328\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.427583 −0.0327942
\(171\) 0 0
\(172\) 0.401501 0.0306142
\(173\) −21.5362 −1.63736 −0.818682 0.574247i \(-0.805295\pi\)
−0.818682 + 0.574247i \(0.805295\pi\)
\(174\) 0 0
\(175\) − 11.6407i − 0.879955i
\(176\) 2.35690i 0.177658i
\(177\) 0 0
\(178\) −2.31336 −0.173393
\(179\) 11.4330 0.854540 0.427270 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(180\) 0 0
\(181\) −20.9705 −1.55872 −0.779361 0.626575i \(-0.784456\pi\)
−0.779361 + 0.626575i \(0.784456\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.02177i 0.296489i
\(185\) −1.40581 −0.103357
\(186\) 0 0
\(187\) 9.16852i 0.670469i
\(188\) 9.98254i 0.728052i
\(189\) 0 0
\(190\) − 0.0174584i − 0.00126657i
\(191\) 14.4373 1.04464 0.522322 0.852748i \(-0.325066\pi\)
0.522322 + 0.852748i \(0.325066\pi\)
\(192\) 0 0
\(193\) − 13.5797i − 0.977489i −0.872427 0.488745i \(-0.837455\pi\)
0.872427 0.488745i \(-0.162545\pi\)
\(194\) −6.46250 −0.463980
\(195\) 0 0
\(196\) 1.96077 0.140055
\(197\) − 0.560335i − 0.0399222i −0.999801 0.0199611i \(-0.993646\pi\)
0.999801 0.0199611i \(-0.00635424\pi\)
\(198\) 0 0
\(199\) −11.4916 −0.814616 −0.407308 0.913291i \(-0.633532\pi\)
−0.407308 + 0.913291i \(0.633532\pi\)
\(200\) 13.2959i 0.940162i
\(201\) 0 0
\(202\) − 10.7095i − 0.753516i
\(203\) 10.9148i 0.766071i
\(204\) 0 0
\(205\) 2.86294 0.199956
\(206\) − 1.09246i − 0.0761151i
\(207\) 0 0
\(208\) 0 0
\(209\) −0.374354 −0.0258946
\(210\) 0 0
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) 14.1032 0.968613
\(213\) 0 0
\(214\) − 2.62133i − 0.179191i
\(215\) 0.0730805i 0.00498405i
\(216\) 0 0
\(217\) 15.6286 1.06094
\(218\) 12.5918 0.852824
\(219\) 0 0
\(220\) −1.42327 −0.0959570
\(221\) 0 0
\(222\) 0 0
\(223\) 2.25906i 0.151278i 0.997135 + 0.0756390i \(0.0240996\pi\)
−0.997135 + 0.0756390i \(0.975900\pi\)
\(224\) 13.7385 0.917945
\(225\) 0 0
\(226\) − 9.66248i − 0.642739i
\(227\) − 6.96615i − 0.462359i −0.972911 0.231180i \(-0.925741\pi\)
0.972911 0.231180i \(-0.0742585\pi\)
\(228\) 0 0
\(229\) − 24.1739i − 1.59746i −0.601692 0.798728i \(-0.705507\pi\)
0.601692 0.798728i \(-0.294493\pi\)
\(230\) −0.295897 −0.0195109
\(231\) 0 0
\(232\) − 12.4668i − 0.818486i
\(233\) −3.06100 −0.200533 −0.100266 0.994961i \(-0.531969\pi\)
−0.100266 + 0.994961i \(0.531969\pi\)
\(234\) 0 0
\(235\) −1.81700 −0.118528
\(236\) 9.19998i 0.598868i
\(237\) 0 0
\(238\) 4.08038 0.264492
\(239\) − 25.1468i − 1.62661i −0.581839 0.813304i \(-0.697667\pi\)
0.581839 0.813304i \(-0.302333\pi\)
\(240\) 0 0
\(241\) − 20.2664i − 1.30547i −0.757586 0.652735i \(-0.773621\pi\)
0.757586 0.652735i \(-0.226379\pi\)
\(242\) − 5.64310i − 0.362752i
\(243\) 0 0
\(244\) 4.71140 0.301616
\(245\) 0.356896i 0.0228012i
\(246\) 0 0
\(247\) 0 0
\(248\) −17.8509 −1.13353
\(249\) 0 0
\(250\) −1.96854 −0.124501
\(251\) −23.7211 −1.49726 −0.748631 0.662987i \(-0.769288\pi\)
−0.748631 + 0.662987i \(0.769288\pi\)
\(252\) 0 0
\(253\) 6.34481i 0.398895i
\(254\) 7.86486i 0.493485i
\(255\) 0 0
\(256\) −14.1860 −0.886624
\(257\) 14.2241 0.887278 0.443639 0.896206i \(-0.353687\pi\)
0.443639 + 0.896206i \(0.353687\pi\)
\(258\) 0 0
\(259\) 13.4155 0.833599
\(260\) 0 0
\(261\) 0 0
\(262\) 5.27413i 0.325837i
\(263\) 17.0954 1.05415 0.527075 0.849819i \(-0.323289\pi\)
0.527075 + 0.849819i \(0.323289\pi\)
\(264\) 0 0
\(265\) 2.56704i 0.157692i
\(266\) 0.166603i 0.0102151i
\(267\) 0 0
\(268\) − 10.4209i − 0.636556i
\(269\) 6.46681 0.394288 0.197144 0.980374i \(-0.436833\pi\)
0.197144 + 0.980374i \(0.436833\pi\)
\(270\) 0 0
\(271\) 6.44803i 0.391690i 0.980635 + 0.195845i \(0.0627449\pi\)
−0.980635 + 0.195845i \(0.937255\pi\)
\(272\) 1.19806 0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) 20.9758i 1.26489i
\(276\) 0 0
\(277\) −13.4601 −0.808739 −0.404370 0.914596i \(-0.632509\pi\)
−0.404370 + 0.914596i \(0.632509\pi\)
\(278\) − 11.7942i − 0.707367i
\(279\) 0 0
\(280\) 1.56704i 0.0936485i
\(281\) − 5.03684i − 0.300472i −0.988650 0.150236i \(-0.951997\pi\)
0.988650 0.150236i \(-0.0480034\pi\)
\(282\) 0 0
\(283\) −22.1280 −1.31537 −0.657686 0.753293i \(-0.728464\pi\)
−0.657686 + 0.753293i \(0.728464\pi\)
\(284\) − 11.7573i − 0.697669i
\(285\) 0 0
\(286\) 0 0
\(287\) −27.3207 −1.61269
\(288\) 0 0
\(289\) −12.3394 −0.725849
\(290\) 0.917231 0.0538616
\(291\) 0 0
\(292\) 9.13946i 0.534846i
\(293\) − 14.9463i − 0.873172i −0.899663 0.436586i \(-0.856187\pi\)
0.899663 0.436586i \(-0.143813\pi\)
\(294\) 0 0
\(295\) −1.67456 −0.0974968
\(296\) −15.3230 −0.890634
\(297\) 0 0
\(298\) −3.47650 −0.201388
\(299\) 0 0
\(300\) 0 0
\(301\) − 0.697398i − 0.0401974i
\(302\) −3.16315 −0.182019
\(303\) 0 0
\(304\) 0.0489173i 0.00280560i
\(305\) 0.857560i 0.0491037i
\(306\) 0 0
\(307\) 19.1293i 1.09177i 0.837861 + 0.545883i \(0.183806\pi\)
−0.837861 + 0.545883i \(0.816194\pi\)
\(308\) 13.5821 0.773912
\(309\) 0 0
\(310\) − 1.31336i − 0.0745936i
\(311\) −0.269815 −0.0152998 −0.00764990 0.999971i \(-0.502435\pi\)
−0.00764990 + 0.999971i \(0.502435\pi\)
\(312\) 0 0
\(313\) −23.3937 −1.32229 −0.661146 0.750257i \(-0.729930\pi\)
−0.661146 + 0.750257i \(0.729930\pi\)
\(314\) 3.57242i 0.201603i
\(315\) 0 0
\(316\) 13.5289 0.761059
\(317\) − 13.9952i − 0.786050i −0.919528 0.393025i \(-0.871429\pi\)
0.919528 0.393025i \(-0.128571\pi\)
\(318\) 0 0
\(319\) − 19.6679i − 1.10119i
\(320\) − 0.880395i − 0.0492156i
\(321\) 0 0
\(322\) 2.82371 0.157359
\(323\) 0.190293i 0.0105882i
\(324\) 0 0
\(325\) 0 0
\(326\) −12.9584 −0.717698
\(327\) 0 0
\(328\) 31.2054 1.72303
\(329\) 17.3394 0.955954
\(330\) 0 0
\(331\) − 17.8213i − 0.979548i −0.871849 0.489774i \(-0.837079\pi\)
0.871849 0.489774i \(-0.162921\pi\)
\(332\) 2.18359i 0.119840i
\(333\) 0 0
\(334\) 12.9250 0.707225
\(335\) 1.89679 0.103633
\(336\) 0 0
\(337\) 27.8485 1.51700 0.758501 0.651672i \(-0.225932\pi\)
0.758501 + 0.651672i \(0.225932\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.723480i 0.0392362i
\(341\) −28.1618 −1.52505
\(342\) 0 0
\(343\) − 19.9041i − 1.07472i
\(344\) 0.796561i 0.0429477i
\(345\) 0 0
\(346\) − 17.2707i − 0.928477i
\(347\) −1.50365 −0.0807200 −0.0403600 0.999185i \(-0.512850\pi\)
−0.0403600 + 0.999185i \(0.512850\pi\)
\(348\) 0 0
\(349\) − 14.1860i − 0.759358i −0.925118 0.379679i \(-0.876034\pi\)
0.925118 0.379679i \(-0.123966\pi\)
\(350\) 9.33513 0.498983
\(351\) 0 0
\(352\) −24.7560 −1.31950
\(353\) − 7.16852i − 0.381542i −0.981635 0.190771i \(-0.938901\pi\)
0.981635 0.190771i \(-0.0610988\pi\)
\(354\) 0 0
\(355\) 2.14005 0.113582
\(356\) 3.91425i 0.207455i
\(357\) 0 0
\(358\) 9.16852i 0.484571i
\(359\) − 19.8853i − 1.04951i −0.851255 0.524753i \(-0.824158\pi\)
0.851255 0.524753i \(-0.175842\pi\)
\(360\) 0 0
\(361\) 18.9922 0.999591
\(362\) − 16.8170i − 0.883882i
\(363\) 0 0
\(364\) 0 0
\(365\) −1.66355 −0.0870740
\(366\) 0 0
\(367\) 1.08383 0.0565757 0.0282878 0.999600i \(-0.490994\pi\)
0.0282878 + 0.999600i \(0.490994\pi\)
\(368\) 0.829085 0.0432190
\(369\) 0 0
\(370\) − 1.12737i − 0.0586094i
\(371\) − 24.4969i − 1.27182i
\(372\) 0 0
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) −7.35258 −0.380193
\(375\) 0 0
\(376\) −19.8049 −1.02136
\(377\) 0 0
\(378\) 0 0
\(379\) 2.40880i 0.123732i 0.998084 + 0.0618658i \(0.0197051\pi\)
−0.998084 + 0.0618658i \(0.980295\pi\)
\(380\) −0.0295400 −0.00151537
\(381\) 0 0
\(382\) 11.5778i 0.592371i
\(383\) − 30.3913i − 1.55292i −0.630164 0.776462i \(-0.717012\pi\)
0.630164 0.776462i \(-0.282988\pi\)
\(384\) 0 0
\(385\) 2.47219i 0.125994i
\(386\) 10.8901 0.554291
\(387\) 0 0
\(388\) 10.9347i 0.555125i
\(389\) −15.9409 −0.808237 −0.404118 0.914707i \(-0.632422\pi\)
−0.404118 + 0.914707i \(0.632422\pi\)
\(390\) 0 0
\(391\) 3.22521 0.163106
\(392\) 3.89008i 0.196479i
\(393\) 0 0
\(394\) 0.449354 0.0226381
\(395\) 2.46250i 0.123902i
\(396\) 0 0
\(397\) 16.9148i 0.848931i 0.905444 + 0.424466i \(0.139538\pi\)
−0.905444 + 0.424466i \(0.860462\pi\)
\(398\) − 9.21552i − 0.461932i
\(399\) 0 0
\(400\) 2.74094 0.137047
\(401\) − 26.6625i − 1.33146i −0.746192 0.665730i \(-0.768120\pi\)
0.746192 0.665730i \(-0.231880\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −18.1207 −0.901537
\(405\) 0 0
\(406\) −8.75302 −0.434405
\(407\) −24.1739 −1.19826
\(408\) 0 0
\(409\) − 28.5163i − 1.41004i −0.709187 0.705021i \(-0.750938\pi\)
0.709187 0.705021i \(-0.249062\pi\)
\(410\) 2.29590i 0.113386i
\(411\) 0 0
\(412\) −1.84846 −0.0910672
\(413\) 15.9801 0.786332
\(414\) 0 0
\(415\) −0.397452 −0.0195102
\(416\) 0 0
\(417\) 0 0
\(418\) − 0.300209i − 0.0146837i
\(419\) 29.6093 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(420\) 0 0
\(421\) 11.6606i 0.568301i 0.958780 + 0.284151i \(0.0917115\pi\)
−0.958780 + 0.284151i \(0.908288\pi\)
\(422\) 7.04461i 0.342926i
\(423\) 0 0
\(424\) 27.9801i 1.35884i
\(425\) 10.6625 0.517206
\(426\) 0 0
\(427\) − 8.18359i − 0.396032i
\(428\) −4.43535 −0.214391
\(429\) 0 0
\(430\) −0.0586060 −0.00282623
\(431\) 4.34913i 0.209490i 0.994499 + 0.104745i \(0.0334026\pi\)
−0.994499 + 0.104745i \(0.966597\pi\)
\(432\) 0 0
\(433\) 14.3884 0.691460 0.345730 0.938334i \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(434\) 12.5332i 0.601612i
\(435\) 0 0
\(436\) − 21.3056i − 1.02035i
\(437\) 0.131687i 0.00629942i
\(438\) 0 0
\(439\) 20.2325 0.965645 0.482822 0.875718i \(-0.339612\pi\)
0.482822 + 0.875718i \(0.339612\pi\)
\(440\) − 2.82371i − 0.134615i
\(441\) 0 0
\(442\) 0 0
\(443\) −8.12200 −0.385888 −0.192944 0.981210i \(-0.561804\pi\)
−0.192944 + 0.981210i \(0.561804\pi\)
\(444\) 0 0
\(445\) −0.712464 −0.0337740
\(446\) −1.81163 −0.0857830
\(447\) 0 0
\(448\) 8.40150i 0.396934i
\(449\) − 12.4916i − 0.589513i −0.955572 0.294757i \(-0.904761\pi\)
0.955572 0.294757i \(-0.0952386\pi\)
\(450\) 0 0
\(451\) 49.2301 2.31816
\(452\) −16.3491 −0.768998
\(453\) 0 0
\(454\) 5.58642 0.262184
\(455\) 0 0
\(456\) 0 0
\(457\) − 5.98121i − 0.279789i −0.990166 0.139895i \(-0.955324\pi\)
0.990166 0.139895i \(-0.0446764\pi\)
\(458\) 19.3860 0.905847
\(459\) 0 0
\(460\) 0.500664i 0.0233436i
\(461\) − 2.05669i − 0.0957895i −0.998852 0.0478947i \(-0.984749\pi\)
0.998852 0.0478947i \(-0.0152512\pi\)
\(462\) 0 0
\(463\) − 8.44935i − 0.392675i −0.980536 0.196337i \(-0.937095\pi\)
0.980536 0.196337i \(-0.0629048\pi\)
\(464\) −2.57002 −0.119310
\(465\) 0 0
\(466\) − 2.45473i − 0.113713i
\(467\) 33.5139 1.55084 0.775420 0.631446i \(-0.217538\pi\)
0.775420 + 0.631446i \(0.217538\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) − 1.45712i − 0.0672121i
\(471\) 0 0
\(472\) −18.2524 −0.840133
\(473\) 1.25667i 0.0577817i
\(474\) 0 0
\(475\) 0.435353i 0.0199754i
\(476\) − 6.90408i − 0.316448i
\(477\) 0 0
\(478\) 20.1661 0.922377
\(479\) 24.7313i 1.13000i 0.825091 + 0.565000i \(0.191124\pi\)
−0.825091 + 0.565000i \(0.808876\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 16.2524 0.740275
\(483\) 0 0
\(484\) −9.54825 −0.434012
\(485\) −1.99031 −0.0903754
\(486\) 0 0
\(487\) 37.7555i 1.71087i 0.517913 + 0.855433i \(0.326709\pi\)
−0.517913 + 0.855433i \(0.673291\pi\)
\(488\) 9.34721i 0.423128i
\(489\) 0 0
\(490\) −0.286208 −0.0129296
\(491\) 31.3110 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(492\) 0 0
\(493\) −9.99761 −0.450270
\(494\) 0 0
\(495\) 0 0
\(496\) 3.67994i 0.165234i
\(497\) −20.4222 −0.916061
\(498\) 0 0
\(499\) − 21.4873i − 0.961902i −0.876748 0.480951i \(-0.840292\pi\)
0.876748 0.480951i \(-0.159708\pi\)
\(500\) 3.33081i 0.148959i
\(501\) 0 0
\(502\) − 19.0228i − 0.849031i
\(503\) −37.5924 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(504\) 0 0
\(505\) − 3.29829i − 0.146772i
\(506\) −5.08815 −0.226196
\(507\) 0 0
\(508\) 13.3075 0.590425
\(509\) − 17.1075i − 0.758278i −0.925340 0.379139i \(-0.876220\pi\)
0.925340 0.379139i \(-0.123780\pi\)
\(510\) 0 0
\(511\) 15.8750 0.702269
\(512\) 6.22282i 0.275012i
\(513\) 0 0
\(514\) 11.4069i 0.503136i
\(515\) − 0.336454i − 0.0148259i
\(516\) 0 0
\(517\) −31.2446 −1.37414
\(518\) 10.7584i 0.472697i
\(519\) 0 0
\(520\) 0 0
\(521\) 19.8465 0.869493 0.434746 0.900553i \(-0.356838\pi\)
0.434746 + 0.900553i \(0.356838\pi\)
\(522\) 0 0
\(523\) −11.4300 −0.499798 −0.249899 0.968272i \(-0.580397\pi\)
−0.249899 + 0.968272i \(0.580397\pi\)
\(524\) 8.92394 0.389844
\(525\) 0 0
\(526\) 13.7095i 0.597762i
\(527\) 14.3153i 0.623583i
\(528\) 0 0
\(529\) −20.7681 −0.902960
\(530\) −2.05861 −0.0894201
\(531\) 0 0
\(532\) 0.281896 0.0122218
\(533\) 0 0
\(534\) 0 0
\(535\) − 0.807315i − 0.0349033i
\(536\) 20.6746 0.893005
\(537\) 0 0
\(538\) 5.18598i 0.223584i
\(539\) 6.13706i 0.264342i
\(540\) 0 0
\(541\) 16.1884i 0.695993i 0.937496 + 0.347996i \(0.113138\pi\)
−0.937496 + 0.347996i \(0.886862\pi\)
\(542\) −5.17092 −0.222110
\(543\) 0 0
\(544\) 12.5840i 0.539536i
\(545\) 3.87800 0.166115
\(546\) 0 0
\(547\) 5.33081 0.227929 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(548\) − 8.43967i − 0.360525i
\(549\) 0 0
\(550\) −16.8213 −0.717263
\(551\) − 0.408206i − 0.0173902i
\(552\) 0 0
\(553\) − 23.4993i − 0.999293i
\(554\) − 10.7942i − 0.458600i
\(555\) 0 0
\(556\) −19.9560 −0.846322
\(557\) − 7.39075i − 0.313156i −0.987666 0.156578i \(-0.949954\pi\)
0.987666 0.156578i \(-0.0500463\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.323044 0.0136511
\(561\) 0 0
\(562\) 4.03923 0.170385
\(563\) −9.47889 −0.399488 −0.199744 0.979848i \(-0.564011\pi\)
−0.199744 + 0.979848i \(0.564011\pi\)
\(564\) 0 0
\(565\) − 2.97584i − 0.125194i
\(566\) − 17.7453i − 0.745889i
\(567\) 0 0
\(568\) 23.3260 0.978738
\(569\) −10.1438 −0.425249 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(570\) 0 0
\(571\) 14.0925 0.589751 0.294876 0.955536i \(-0.404722\pi\)
0.294876 + 0.955536i \(0.404722\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 21.9095i − 0.914483i
\(575\) 7.37867 0.307712
\(576\) 0 0
\(577\) − 25.1545i − 1.04720i −0.851965 0.523598i \(-0.824589\pi\)
0.851965 0.523598i \(-0.175411\pi\)
\(578\) − 9.89546i − 0.411597i
\(579\) 0 0
\(580\) − 1.55197i − 0.0644422i
\(581\) 3.79284 0.157354
\(582\) 0 0
\(583\) 44.1420i 1.82817i
\(584\) −18.1323 −0.750319
\(585\) 0 0
\(586\) 11.9860 0.495137
\(587\) 43.8353i 1.80928i 0.426180 + 0.904639i \(0.359859\pi\)
−0.426180 + 0.904639i \(0.640141\pi\)
\(588\) 0 0
\(589\) −0.584498 −0.0240838
\(590\) − 1.34290i − 0.0552861i
\(591\) 0 0
\(592\) 3.15883i 0.129827i
\(593\) 24.9965i 1.02648i 0.858244 + 0.513242i \(0.171556\pi\)
−0.858244 + 0.513242i \(0.828444\pi\)
\(594\) 0 0
\(595\) 1.25667 0.0515184
\(596\) 5.88231i 0.240949i
\(597\) 0 0
\(598\) 0 0
\(599\) 6.24027 0.254971 0.127485 0.991840i \(-0.459309\pi\)
0.127485 + 0.991840i \(0.459309\pi\)
\(600\) 0 0
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) 0.559270 0.0227941
\(603\) 0 0
\(604\) 5.35211i 0.217774i
\(605\) − 1.73795i − 0.0706579i
\(606\) 0 0
\(607\) −43.6480 −1.77162 −0.885809 0.464050i \(-0.846396\pi\)
−0.885809 + 0.464050i \(0.846396\pi\)
\(608\) −0.513811 −0.0208378
\(609\) 0 0
\(610\) −0.687710 −0.0278445
\(611\) 0 0
\(612\) 0 0
\(613\) 25.9541i 1.04827i 0.851634 + 0.524137i \(0.175612\pi\)
−0.851634 + 0.524137i \(0.824388\pi\)
\(614\) −15.3405 −0.619092
\(615\) 0 0
\(616\) 26.9463i 1.08570i
\(617\) 45.9396i 1.84946i 0.380626 + 0.924729i \(0.375709\pi\)
−0.380626 + 0.924729i \(0.624291\pi\)
\(618\) 0 0
\(619\) 6.73556i 0.270725i 0.990796 + 0.135363i \(0.0432199\pi\)
−0.990796 + 0.135363i \(0.956780\pi\)
\(620\) −2.22223 −0.0892467
\(621\) 0 0
\(622\) − 0.216375i − 0.00867583i
\(623\) 6.79895 0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) − 18.7603i − 0.749813i
\(627\) 0 0
\(628\) 6.04461 0.241206
\(629\) 12.2881i 0.489960i
\(630\) 0 0
\(631\) 45.0998i 1.79539i 0.440614 + 0.897696i \(0.354761\pi\)
−0.440614 + 0.897696i \(0.645239\pi\)
\(632\) 26.8407i 1.06767i
\(633\) 0 0
\(634\) 11.2233 0.445734
\(635\) 2.42221i 0.0961223i
\(636\) 0 0
\(637\) 0 0
\(638\) 15.7724 0.624435
\(639\) 0 0
\(640\) −2.17331 −0.0859075
\(641\) 32.5821 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(642\) 0 0
\(643\) − 25.5754i − 1.00860i −0.863530 0.504298i \(-0.831751\pi\)
0.863530 0.504298i \(-0.168249\pi\)
\(644\) − 4.77777i − 0.188271i
\(645\) 0 0
\(646\) −0.152603 −0.00600408
\(647\) −30.1715 −1.18616 −0.593082 0.805142i \(-0.702089\pi\)
−0.593082 + 0.805142i \(0.702089\pi\)
\(648\) 0 0
\(649\) −28.7952 −1.13031
\(650\) 0 0
\(651\) 0 0
\(652\) 21.9259i 0.858683i
\(653\) −36.9028 −1.44412 −0.722058 0.691832i \(-0.756804\pi\)
−0.722058 + 0.691832i \(0.756804\pi\)
\(654\) 0 0
\(655\) 1.62432i 0.0634673i
\(656\) − 6.43296i − 0.251165i
\(657\) 0 0
\(658\) 13.9051i 0.542079i
\(659\) −23.6866 −0.922701 −0.461350 0.887218i \(-0.652635\pi\)
−0.461350 + 0.887218i \(0.652635\pi\)
\(660\) 0 0
\(661\) − 31.7590i − 1.23528i −0.786460 0.617641i \(-0.788089\pi\)
0.786460 0.617641i \(-0.211911\pi\)
\(662\) 14.2916 0.555458
\(663\) 0 0
\(664\) −4.33214 −0.168120
\(665\) 0.0513102i 0.00198973i
\(666\) 0 0
\(667\) −6.91856 −0.267888
\(668\) − 21.8694i − 0.846152i
\(669\) 0 0
\(670\) 1.52111i 0.0587655i
\(671\) 14.7463i 0.569275i
\(672\) 0 0
\(673\) 7.50232 0.289193 0.144597 0.989491i \(-0.453812\pi\)
0.144597 + 0.989491i \(0.453812\pi\)
\(674\) 22.3327i 0.860225i
\(675\) 0 0
\(676\) 0 0
\(677\) 35.0315 1.34637 0.673184 0.739475i \(-0.264926\pi\)
0.673184 + 0.739475i \(0.264926\pi\)
\(678\) 0 0
\(679\) 18.9933 0.728896
\(680\) −1.43535 −0.0550433
\(681\) 0 0
\(682\) − 22.5840i − 0.864787i
\(683\) 24.0834i 0.921524i 0.887524 + 0.460762i \(0.152424\pi\)
−0.887524 + 0.460762i \(0.847576\pi\)
\(684\) 0 0
\(685\) 1.53617 0.0586941
\(686\) 15.9618 0.609426
\(687\) 0 0
\(688\) 0.164210 0.00626046
\(689\) 0 0
\(690\) 0 0
\(691\) − 2.01447i − 0.0766342i −0.999266 0.0383171i \(-0.987800\pi\)
0.999266 0.0383171i \(-0.0121997\pi\)
\(692\) −29.2223 −1.11087
\(693\) 0 0
\(694\) − 1.20583i − 0.0457728i
\(695\) − 3.63235i − 0.137783i
\(696\) 0 0
\(697\) − 25.0248i − 0.947880i
\(698\) 11.3763 0.430598
\(699\) 0 0
\(700\) − 15.7952i − 0.597004i
\(701\) −48.8189 −1.84387 −0.921933 0.387350i \(-0.873390\pi\)
−0.921933 + 0.387350i \(0.873390\pi\)
\(702\) 0 0
\(703\) −0.501729 −0.0189231
\(704\) − 15.1390i − 0.570572i
\(705\) 0 0
\(706\) 5.74871 0.216355
\(707\) 31.4752i 1.18375i
\(708\) 0 0
\(709\) − 20.8060i − 0.781385i −0.920521 0.390693i \(-0.872236\pi\)
0.920521 0.390693i \(-0.127764\pi\)
\(710\) 1.71618i 0.0644073i
\(711\) 0 0
\(712\) −7.76569 −0.291032
\(713\) 9.90648i 0.371000i
\(714\) 0 0
\(715\) 0 0
\(716\) 15.5133 0.579761
\(717\) 0 0
\(718\) 15.9468 0.595128
\(719\) 21.4306 0.799225 0.399613 0.916684i \(-0.369145\pi\)
0.399613 + 0.916684i \(0.369145\pi\)
\(720\) 0 0
\(721\) 3.21073i 0.119574i
\(722\) 15.2306i 0.566824i
\(723\) 0 0
\(724\) −28.4547 −1.05751
\(725\) −22.8726 −0.849468
\(726\) 0 0
\(727\) −13.4862 −0.500175 −0.250088 0.968223i \(-0.580459\pi\)
−0.250088 + 0.968223i \(0.580459\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 1.33406i − 0.0493758i
\(731\) 0.638792 0.0236266
\(732\) 0 0
\(733\) 43.5424i 1.60828i 0.594443 + 0.804138i \(0.297373\pi\)
−0.594443 + 0.804138i \(0.702627\pi\)
\(734\) 0.869167i 0.0320816i
\(735\) 0 0
\(736\) 8.70841i 0.320996i
\(737\) 32.6165 1.20145
\(738\) 0 0
\(739\) 20.0543i 0.737709i 0.929487 + 0.368855i \(0.120250\pi\)
−0.929487 + 0.368855i \(0.879750\pi\)
\(740\) −1.90754 −0.0701226
\(741\) 0 0
\(742\) 19.6450 0.721191
\(743\) − 33.1685i − 1.21684i −0.793617 0.608418i \(-0.791805\pi\)
0.793617 0.608418i \(-0.208195\pi\)
\(744\) 0 0
\(745\) −1.07069 −0.0392270
\(746\) − 4.91617i − 0.179994i
\(747\) 0 0
\(748\) 12.4407i 0.454878i
\(749\) 7.70410i 0.281502i
\(750\) 0 0
\(751\) −39.2814 −1.43340 −0.716700 0.697382i \(-0.754348\pi\)
−0.716700 + 0.697382i \(0.754348\pi\)
\(752\) 4.08277i 0.148883i
\(753\) 0 0
\(754\) 0 0
\(755\) −0.974181 −0.0354541
\(756\) 0 0
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) −1.93171 −0.0701627
\(759\) 0 0
\(760\) − 0.0586060i − 0.00212586i
\(761\) 21.8984i 0.793818i 0.917858 + 0.396909i \(0.129917\pi\)
−0.917858 + 0.396909i \(0.870083\pi\)
\(762\) 0 0
\(763\) −37.0073 −1.33975
\(764\) 19.5899 0.708737
\(765\) 0 0
\(766\) 24.3720 0.880595
\(767\) 0 0
\(768\) 0 0
\(769\) − 46.7096i − 1.68439i −0.539172 0.842196i \(-0.681263\pi\)
0.539172 0.842196i \(-0.318737\pi\)
\(770\) −1.98254 −0.0714458
\(771\) 0 0
\(772\) − 18.4263i − 0.663175i
\(773\) − 30.2416i − 1.08771i −0.839178 0.543857i \(-0.816963\pi\)
0.839178 0.543857i \(-0.183037\pi\)
\(774\) 0 0
\(775\) 32.7506i 1.17644i
\(776\) −21.6939 −0.778767
\(777\) 0 0
\(778\) − 12.7836i − 0.458315i
\(779\) 1.02177 0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 2.58642i 0.0924901i
\(783\) 0 0
\(784\) 0.801938 0.0286406
\(785\) 1.10023i 0.0392688i
\(786\) 0 0
\(787\) 28.7023i 1.02313i 0.859246 + 0.511563i \(0.170933\pi\)
−0.859246 + 0.511563i \(0.829067\pi\)
\(788\) − 0.760316i − 0.0270851i
\(789\) 0 0
\(790\) −1.97477 −0.0702592
\(791\) 28.3980i 1.00972i
\(792\) 0 0
\(793\) 0 0
\(794\) −13.5646 −0.481391
\(795\) 0 0
\(796\) −15.5929 −0.552674
\(797\) −18.5418 −0.656785 −0.328392 0.944541i \(-0.606507\pi\)
−0.328392 + 0.944541i \(0.606507\pi\)
\(798\) 0 0
\(799\) 15.8823i 0.561876i
\(800\) 28.7899i 1.01788i
\(801\) 0 0
\(802\) 21.3817 0.755012
\(803\) −28.6058 −1.00948
\(804\) 0 0
\(805\) 0.869641 0.0306508
\(806\) 0 0
\(807\) 0 0
\(808\) − 35.9506i − 1.26474i
\(809\) 10.0677 0.353962 0.176981 0.984214i \(-0.443367\pi\)
0.176981 + 0.984214i \(0.443367\pi\)
\(810\) 0 0
\(811\) 10.0285i 0.352147i 0.984377 + 0.176074i \(0.0563397\pi\)
−0.984377 + 0.176074i \(0.943660\pi\)
\(812\) 14.8103i 0.519740i
\(813\) 0 0
\(814\) − 19.3860i − 0.679478i
\(815\) −3.99090 −0.139795
\(816\) 0 0
\(817\) 0.0260821i 0 0.000912498i
\(818\) 22.8683 0.799572
\(819\) 0 0
\(820\) 3.88471 0.135660
\(821\) − 26.1704i − 0.913355i −0.889632 0.456677i \(-0.849039\pi\)
0.889632 0.456677i \(-0.150961\pi\)
\(822\) 0 0
\(823\) −1.82238 −0.0635242 −0.0317621 0.999495i \(-0.510112\pi\)
−0.0317621 + 0.999495i \(0.510112\pi\)
\(824\) − 3.66727i − 0.127755i
\(825\) 0 0
\(826\) 12.8151i 0.445894i
\(827\) 32.2941i 1.12298i 0.827485 + 0.561488i \(0.189771\pi\)
−0.827485 + 0.561488i \(0.810229\pi\)
\(828\) 0 0
\(829\) −15.1002 −0.524453 −0.262226 0.965006i \(-0.584457\pi\)
−0.262226 + 0.965006i \(0.584457\pi\)
\(830\) − 0.318732i − 0.0110634i
\(831\) 0 0
\(832\) 0 0
\(833\) 3.11960 0.108088
\(834\) 0 0
\(835\) 3.98062 0.137755
\(836\) −0.507960 −0.0175682
\(837\) 0 0
\(838\) 23.7448i 0.820250i
\(839\) 32.9965i 1.13917i 0.821933 + 0.569584i \(0.192896\pi\)
−0.821933 + 0.569584i \(0.807104\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) −9.35105 −0.322258
\(843\) 0 0
\(844\) 11.9196 0.410290
\(845\) 0 0
\(846\) 0 0
\(847\) 16.5851i 0.569870i
\(848\) 5.76809 0.198077
\(849\) 0 0
\(850\) 8.55065i 0.293285i
\(851\) 8.50365i 0.291501i
\(852\) 0 0
\(853\) − 37.7802i − 1.29357i −0.762673 0.646784i \(-0.776113\pi\)
0.762673 0.646784i \(-0.223887\pi\)
\(854\) 6.56273 0.224572
\(855\) 0 0
\(856\) − 8.79954i − 0.300762i
\(857\) 27.3623 0.934677 0.467339 0.884078i \(-0.345213\pi\)
0.467339 + 0.884078i \(0.345213\pi\)
\(858\) 0 0
\(859\) −20.0629 −0.684538 −0.342269 0.939602i \(-0.611195\pi\)
−0.342269 + 0.939602i \(0.611195\pi\)
\(860\) 0.0991626i 0.00338142i
\(861\) 0 0
\(862\) −3.48773 −0.118792
\(863\) − 6.14483i − 0.209173i −0.994516 0.104586i \(-0.966648\pi\)
0.994516 0.104586i \(-0.0333518\pi\)
\(864\) 0 0
\(865\) − 5.31900i − 0.180851i
\(866\) 11.5386i 0.392096i
\(867\) 0 0
\(868\) 21.2064 0.719793
\(869\) 42.3443i 1.43643i
\(870\) 0 0
\(871\) 0 0
\(872\) 42.2693 1.43142
\(873\) 0 0
\(874\) −0.105604 −0.00357212
\(875\) 5.78554 0.195587
\(876\) 0 0
\(877\) 13.5077i 0.456123i 0.973647 + 0.228061i \(0.0732386\pi\)
−0.973647 + 0.228061i \(0.926761\pi\)
\(878\) 16.2252i 0.547574i
\(879\) 0 0
\(880\) −0.582105 −0.0196228
\(881\) −5.23431 −0.176348 −0.0881741 0.996105i \(-0.528103\pi\)
−0.0881741 + 0.996105i \(0.528103\pi\)
\(882\) 0 0
\(883\) 4.57301 0.153894 0.0769470 0.997035i \(-0.475483\pi\)
0.0769470 + 0.997035i \(0.475483\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 6.51334i − 0.218820i
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) 0 0
\(889\) − 23.1148i − 0.775246i
\(890\) − 0.571352i − 0.0191517i
\(891\) 0 0
\(892\) 3.06531i 0.102634i
\(893\) −0.648481 −0.0217006
\(894\) 0 0
\(895\) 2.82371i 0.0943861i
\(896\) 20.7396 0.692862
\(897\) 0 0
\(898\) 10.0175 0.334287
\(899\) − 30.7084i − 1.02418i
\(900\) 0 0
\(901\) 22.4383 0.747529
\(902\) 39.4795i 1.31452i
\(903\) 0 0
\(904\) − 32.4359i − 1.07880i
\(905\) − 5.17928i − 0.172165i
\(906\) 0 0
\(907\) −8.10215 −0.269027 −0.134514 0.990912i \(-0.542947\pi\)
−0.134514 + 0.990912i \(0.542947\pi\)
\(908\) − 9.45234i − 0.313687i
\(909\) 0 0
\(910\) 0 0
\(911\) 9.18119 0.304187 0.152093 0.988366i \(-0.451399\pi\)
0.152093 + 0.988366i \(0.451399\pi\)
\(912\) 0 0
\(913\) −6.83446 −0.226188
\(914\) 4.79656 0.158656
\(915\) 0 0
\(916\) − 32.8015i − 1.08379i
\(917\) − 15.5007i − 0.511877i
\(918\) 0 0
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) −0.993295 −0.0327480
\(921\) 0 0
\(922\) 1.64933 0.0543180
\(923\) 0 0
\(924\) 0 0
\(925\) 28.1129i 0.924346i
\(926\) 6.77586 0.222668
\(927\) 0 0
\(928\) − 26.9946i − 0.886142i
\(929\) 24.2131i 0.794407i 0.917731 + 0.397203i \(0.130019\pi\)
−0.917731 + 0.397203i \(0.869981\pi\)
\(930\) 0 0
\(931\) 0.127375i 0.00417454i
\(932\) −4.15346 −0.136051
\(933\) 0 0
\(934\) 26.8761i 0.879412i
\(935\) −2.26444 −0.0740550
\(936\) 0 0
\(937\) 11.1830 0.365333 0.182666 0.983175i \(-0.441527\pi\)
0.182666 + 0.983175i \(0.441527\pi\)
\(938\) − 14.5157i − 0.473955i
\(939\) 0 0
\(940\) −2.46548 −0.0804152
\(941\) − 15.9638i − 0.520404i −0.965554 0.260202i \(-0.916211\pi\)
0.965554 0.260202i \(-0.0837891\pi\)
\(942\) 0 0
\(943\) − 17.3177i − 0.563941i
\(944\) 3.76271i 0.122466i
\(945\) 0 0
\(946\) −1.00777 −0.0327654
\(947\) − 6.51466i − 0.211698i −0.994382 0.105849i \(-0.966244\pi\)
0.994382 0.105849i \(-0.0337560\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.349126 −0.0113271
\(951\) 0 0
\(952\) 13.6974 0.443935
\(953\) 47.6469 1.54344 0.771718 0.635965i \(-0.219398\pi\)
0.771718 + 0.635965i \(0.219398\pi\)
\(954\) 0 0
\(955\) 3.56571i 0.115384i
\(956\) − 34.1215i − 1.10357i
\(957\) 0 0
\(958\) −19.8329 −0.640773
\(959\) −14.6595 −0.473380
\(960\) 0 0
\(961\) −12.9705 −0.418402
\(962\) 0 0
\(963\) 0 0
\(964\) − 27.4993i − 0.885694i
\(965\) 3.35391 0.107966
\(966\) 0 0
\(967\) 43.8122i 1.40891i 0.709751 + 0.704453i \(0.248808\pi\)
−0.709751 + 0.704453i \(0.751192\pi\)
\(968\) − 18.9433i − 0.608861i
\(969\) 0 0
\(970\) − 1.59611i − 0.0512479i
\(971\) −4.29483 −0.137828 −0.0689139 0.997623i \(-0.521953\pi\)
−0.0689139 + 0.997623i \(0.521953\pi\)
\(972\) 0 0
\(973\) 34.6631i 1.11125i
\(974\) −30.2776 −0.970156
\(975\) 0 0
\(976\) 1.92692 0.0616792
\(977\) 26.8019i 0.857470i 0.903430 + 0.428735i \(0.141041\pi\)
−0.903430 + 0.428735i \(0.858959\pi\)
\(978\) 0 0
\(979\) −12.2513 −0.391553
\(980\) 0.484271i 0.0154695i
\(981\) 0 0
\(982\) 25.1094i 0.801275i
\(983\) − 27.2495i − 0.869124i −0.900642 0.434562i \(-0.856903\pi\)
0.900642 0.434562i \(-0.143097\pi\)
\(984\) 0 0
\(985\) 0.138391 0.00440951
\(986\) − 8.01746i − 0.255328i
\(987\) 0 0
\(988\) 0 0
\(989\) 0.442058 0.0140566
\(990\) 0 0
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) −38.6528 −1.22723
\(993\) 0 0
\(994\) − 16.3773i − 0.519458i
\(995\) − 2.83818i − 0.0899764i
\(996\) 0 0
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) 17.2314 0.545452
\(999\) 0 0
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 1521.2.b.l.1351.5 6
3.2 odd 2 169.2.b.b.168.2 6
12.11 even 2 2704.2.f.o.337.5 6
13.5 odd 4 1521.2.a.o.1.3 3
13.8 odd 4 1521.2.a.r.1.1 3
13.12 even 2 inner 1521.2.b.l.1351.2 6
39.2 even 12 169.2.c.b.22.3 6
39.5 even 4 169.2.a.c.1.1 yes 3
39.8 even 4 169.2.a.b.1.3 3
39.11 even 12 169.2.c.c.22.1 6
39.17 odd 6 169.2.e.b.23.5 12
39.20 even 12 169.2.c.c.146.1 6
39.23 odd 6 169.2.e.b.147.2 12
39.29 odd 6 169.2.e.b.147.5 12
39.32 even 12 169.2.c.b.146.3 6
39.35 odd 6 169.2.e.b.23.2 12
39.38 odd 2 169.2.b.b.168.5 6
156.47 odd 4 2704.2.a.z.1.3 3
156.83 odd 4 2704.2.a.ba.1.3 3
156.155 even 2 2704.2.f.o.337.6 6
195.44 even 4 4225.2.a.bb.1.3 3
195.164 even 4 4225.2.a.bg.1.1 3
273.83 odd 4 8281.2.a.bj.1.1 3
273.125 odd 4 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 39.8 even 4
169.2.a.c.1.1 yes 3 39.5 even 4
169.2.b.b.168.2 6 3.2 odd 2
169.2.b.b.168.5 6 39.38 odd 2
169.2.c.b.22.3 6 39.2 even 12
169.2.c.b.146.3 6 39.32 even 12
169.2.c.c.22.1 6 39.11 even 12
169.2.c.c.146.1 6 39.20 even 12
169.2.e.b.23.2 12 39.35 odd 6
169.2.e.b.23.5 12 39.17 odd 6
169.2.e.b.147.2 12 39.23 odd 6
169.2.e.b.147.5 12 39.29 odd 6
1521.2.a.o.1.3 3 13.5 odd 4
1521.2.a.r.1.1 3 13.8 odd 4
1521.2.b.l.1351.2 6 13.12 even 2 inner
1521.2.b.l.1351.5 6 1.1 even 1 trivial
2704.2.a.z.1.3 3 156.47 odd 4
2704.2.a.ba.1.3 3 156.83 odd 4
2704.2.f.o.337.5 6 12.11 even 2
2704.2.f.o.337.6 6 156.155 even 2
4225.2.a.bb.1.3 3 195.44 even 4
4225.2.a.bg.1.1 3 195.164 even 4
8281.2.a.bf.1.3 3 273.125 odd 4
8281.2.a.bj.1.1 3 273.83 odd 4