Properties

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

$q$-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 \(-0.908495\pi\)
0.958964 0.283528i \(-0.0915048\pi\)
\(3\) 0 0
\(4\) 1.35690 0.678448
\(5\) − 0.246980i − 0.110453i −0.998474 0.0552263i \(-0.982412\pi\)
0.998474 0.0552263i \(-0.0175880\pi\)
\(6\) 0 0
\(7\) 2.35690i 0.890823i 0.895326 + 0.445411i \(0.146943\pi\)
−0.895326 + 0.445411i \(0.853057\pi\)
\(8\) − 2.69202i − 0.951773i
\(9\) 0 0
\(10\) −0.198062 −0.0626328
\(11\) − 4.24698i − 1.28051i −0.768161 0.640256i \(-0.778828\pi\)
0.768161 0.640256i \(-0.221172\pi\)
\(12\) 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 \(-0.996782\pi\)
0.999949 0.0101110i \(-0.00321850\pi\)
\(20\) − 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 \(-0.796961\pi\)
0.803368 0.595483i \(-0.203039\pi\)
\(32\) − 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 \(-0.845017\pi\)
0.883791 0.467881i \(-0.154983\pi\)
\(38\) −0.0706876 −0.0114670
\(39\) 0 0
\(40\) −0.664874 −0.105126
\(41\) 11.5918i 1.81033i 0.425056 + 0.905167i \(0.360254\pi\)
−0.425056 + 0.905167i \(0.639746\pi\)
\(42\) 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 \(-0.819725\pi\)
0.843864 0.536557i \(-0.180275\pi\)
\(48\) 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 \(-0.854499\pi\)
0.897334 0.441351i \(-0.145501\pi\)
\(60\) 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 \(0.155431\pi\)
−0.883131 + 0.469127i \(0.844569\pi\)
\(68\) 2.92931 0.355231
\(69\) 0 0
\(70\) − 0.466812i − 0.0557947i
\(71\) 8.66487i 1.02833i 0.857691 + 0.514166i \(0.171898\pi\)
−0.857691 + 0.514166i \(0.828102\pi\)
\(72\) 0 0
\(73\) − 6.73556i − 0.788338i −0.919038 0.394169i \(-0.871032\pi\)
0.919038 0.394169i \(-0.128968\pi\)
\(74\) −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 \(-0.971850\pi\)
0.996092 0.0883192i \(-0.0281495\pi\)
\(84\) 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 \(-0.951142\pi\)
0.988243 0.152889i \(-0.0488577\pi\)
\(90\) 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 \(-0.865838\pi\)
0.912483 0.409114i \(-0.134162\pi\)
\(98\) − 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 \(0.270899\pi\)
−0.659191 + 0.751976i \(0.729101\pi\)
\(110\) 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 \(0.0856024\pi\)
−0.964056 + 0.265698i \(0.914398\pi\)
\(138\) 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 \(-0.943175\pi\)
0.984108 0.177574i \(-0.0568248\pi\)
\(150\) 0 0
\(151\) − 3.94438i − 0.320989i −0.987037 0.160494i \(-0.948691\pi\)
0.987037 0.160494i \(-0.0513089\pi\)
\(152\) −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 \(-0.781894\pi\)
0.774292 0.632829i \(-0.218106\pi\)
\(164\) 15.7289i 1.22822i
\(165\) 0 0
\(166\) −1.29052 −0.100164
\(167\) 16.1172i 1.24719i 0.781749 + 0.623594i \(0.214328\pi\)
−0.781749 + 0.623594i \(0.785672\pi\)
\(168\) 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 \(0.162545\pi\)
−0.872427 + 0.488745i \(0.837455\pi\)
\(194\) −6.46250 −0.463980
\(195\) 0 0
\(196\) 1.96077 0.140055
\(197\) 0.560335i 0.0399222i 0.999801 + 0.0199611i \(0.00635424\pi\)
−0.999801 + 0.0199611i \(0.993646\pi\)
\(198\) 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 \(-0.975900\pi\)
0.997135 0.0756390i \(-0.0240996\pi\)
\(224\) 13.7385 0.917945
\(225\) 0 0
\(226\) 9.66248i 0.642739i
\(227\) 6.96615i 0.462359i 0.972911 + 0.231180i \(0.0742585\pi\)
−0.972911 + 0.231180i \(0.925741\pi\)
\(228\) 0 0
\(229\) 24.1739i 1.59746i 0.601692 + 0.798728i \(0.294493\pi\)
−0.601692 + 0.798728i \(0.705507\pi\)
\(230\) −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 \(0.302333\pi\)
−0.581839 + 0.813304i \(0.697667\pi\)
\(240\) 0 0
\(241\) 20.2664i 1.30547i 0.757586 + 0.652735i \(0.226379\pi\)
−0.757586 + 0.652735i \(0.773621\pi\)
\(242\) 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 \(-0.937255\pi\)
0.980635 0.195845i \(-0.0627449\pi\)
\(272\) 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 \(0.0480034\pi\)
−0.988650 + 0.150236i \(0.951997\pi\)
\(282\) 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 \(0.143813\pi\)
−0.899663 + 0.436586i \(0.856187\pi\)
\(294\) 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 \(-0.816194\pi\)
0.837861 0.545883i \(-0.183806\pi\)
\(308\) 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 \(0.128571\pi\)
−0.919528 + 0.393025i \(0.871429\pi\)
\(318\) 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 \(0.162921\pi\)
−0.871849 + 0.489774i \(0.837079\pi\)
\(332\) − 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 \(0.123966\pi\)
−0.925118 + 0.379679i \(0.876034\pi\)
\(350\) 9.33513 0.498983
\(351\) 0 0
\(352\) −24.7560 −1.31950
\(353\) 7.16852i 0.381542i 0.981635 + 0.190771i \(0.0610988\pi\)
−0.981635 + 0.190771i \(0.938901\pi\)
\(354\) 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 \(0.175842\pi\)
−0.851255 + 0.524753i \(0.824158\pi\)
\(360\) 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 \(-0.980295\pi\)
0.998084 0.0618658i \(-0.0197051\pi\)
\(380\) −0.0295400 −0.00151537
\(381\) 0 0
\(382\) − 11.5778i − 0.592371i
\(383\) 30.3913i 1.55292i 0.630164 + 0.776462i \(0.282988\pi\)
−0.630164 + 0.776462i \(0.717012\pi\)
\(384\) 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 \(-0.860462\pi\)
0.905444 0.424466i \(-0.139538\pi\)
\(398\) 9.21552i 0.461932i
\(399\) 0 0
\(400\) 2.74094 0.137047
\(401\) 26.6625i 1.33146i 0.746192 + 0.665730i \(0.231880\pi\)
−0.746192 + 0.665730i \(0.768120\pi\)
\(402\) 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 \(0.249062\pi\)
−0.709187 + 0.705021i \(0.750938\pi\)
\(410\) − 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 \(-0.908288\pi\)
0.958780 0.284151i \(-0.0917115\pi\)
\(422\) − 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 \(-0.966597\pi\)
0.994499 0.104745i \(-0.0334026\pi\)
\(432\) 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 \(0.0952386\pi\)
−0.955572 + 0.294757i \(0.904761\pi\)
\(450\) 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 \(0.0446764\pi\)
−0.990166 + 0.139895i \(0.955324\pi\)
\(458\) 19.3860 0.905847
\(459\) 0 0
\(460\) − 0.500664i − 0.0233436i
\(461\) 2.05669i 0.0957895i 0.998852 + 0.0478947i \(0.0152512\pi\)
−0.998852 + 0.0478947i \(0.984749\pi\)
\(462\) 0 0
\(463\) 8.44935i 0.392675i 0.980536 + 0.196337i \(0.0629048\pi\)
−0.980536 + 0.196337i \(0.937095\pi\)
\(464\) −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 \(-0.808876\pi\)
0.825091 0.565000i \(-0.191124\pi\)
\(480\) 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 \(-0.673291\pi\)
0.517913 0.855433i \(-0.326709\pi\)
\(488\) − 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 \(0.159708\pi\)
−0.876748 + 0.480951i \(0.840292\pi\)
\(500\) − 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 \(0.123780\pi\)
−0.925340 + 0.379139i \(0.876220\pi\)
\(510\) 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 \(-0.886862\pi\)
0.937496 0.347996i \(-0.113138\pi\)
\(542\) −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 \(0.0500463\pi\)
−0.987666 + 0.156578i \(0.949954\pi\)
\(558\) 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 \(0.175411\pi\)
−0.851965 + 0.523598i \(0.824589\pi\)
\(578\) 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 \(-0.640141\pi\)
0.426180 0.904639i \(-0.359859\pi\)
\(588\) 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 \(-0.828444\pi\)
0.858244 0.513242i \(-0.171556\pi\)
\(594\) 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 \(-0.824388\pi\)
0.851634 0.524137i \(-0.175612\pi\)
\(614\) −15.3405 −0.619092
\(615\) 0 0
\(616\) − 26.9463i − 1.08570i
\(617\) − 45.9396i − 1.84946i −0.380626 0.924729i \(-0.624291\pi\)
0.380626 0.924729i \(-0.375709\pi\)
\(618\) 0 0
\(619\) − 6.73556i − 0.270725i −0.990796 0.135363i \(-0.956780\pi\)
0.990796 0.135363i \(-0.0432199\pi\)
\(620\) −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 \(-0.645239\pi\)
0.440614 0.897696i \(-0.354761\pi\)
\(632\) − 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 \(0.168249\pi\)
−0.863530 + 0.504298i \(0.831751\pi\)
\(644\) 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 \(0.211911\pi\)
−0.786460 + 0.617641i \(0.788089\pi\)
\(662\) 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 \(-0.847576\pi\)
0.887524 0.460762i \(-0.152424\pi\)
\(684\) 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 \(0.0121997\pi\)
−0.999266 + 0.0383171i \(0.987800\pi\)
\(692\) −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 \(0.127764\pi\)
−0.920521 + 0.390693i \(0.872236\pi\)
\(710\) − 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 \(-0.702627\pi\)
0.594443 0.804138i \(-0.297373\pi\)
\(734\) − 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 \(-0.879750\pi\)
0.929487 0.368855i \(-0.120250\pi\)
\(740\) −1.90754 −0.0701226
\(741\) 0 0
\(742\) 19.6450 0.721191
\(743\) 33.1685i 1.21684i 0.793617 + 0.608418i \(0.208195\pi\)
−0.793617 + 0.608418i \(0.791805\pi\)
\(744\) 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 \(-0.870083\pi\)
0.917858 0.396909i \(-0.129917\pi\)
\(762\) 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 \(0.318737\pi\)
−0.539172 + 0.842196i \(0.681263\pi\)
\(770\) −1.98254 −0.0714458
\(771\) 0 0
\(772\) 18.4263i 0.663175i
\(773\) 30.2416i 1.08771i 0.839178 + 0.543857i \(0.183037\pi\)
−0.839178 + 0.543857i \(0.816963\pi\)
\(774\) 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 \(-0.829067\pi\)
0.859246 0.511563i \(-0.170933\pi\)
\(788\) 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 \(-0.943660\pi\)
0.984377 0.176074i \(-0.0563397\pi\)
\(812\) − 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 \(0.150961\pi\)
−0.889632 + 0.456677i \(0.849039\pi\)
\(822\) 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 \(-0.810229\pi\)
0.827485 0.561488i \(-0.189771\pi\)
\(828\) 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 \(-0.807104\pi\)
0.821933 0.569584i \(-0.192896\pi\)
\(840\) 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 \(0.223887\pi\)
−0.762673 + 0.646784i \(0.776113\pi\)
\(854\) 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 \(0.0333518\pi\)
−0.994516 + 0.104586i \(0.966648\pi\)
\(864\) 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 \(-0.926761\pi\)
0.973647 0.228061i \(-0.0732386\pi\)
\(878\) − 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 \(-0.869981\pi\)
0.917731 0.397203i \(-0.130019\pi\)
\(930\) 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 \(0.0837891\pi\)
−0.965554 + 0.260202i \(0.916211\pi\)
\(942\) 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 \(0.0337560\pi\)
−0.994382 + 0.105849i \(0.966244\pi\)
\(948\) 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 \(-0.751192\pi\)
0.709751 0.704453i \(-0.248808\pi\)
\(968\) 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 \(-0.858959\pi\)
0.903430 0.428735i \(-0.141041\pi\)
\(978\) 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 \(0.143097\pi\)
−0.900642 + 0.434562i \(0.856903\pi\)
\(984\) 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.1351.2 6
3.2 odd 2 169.2.b.b.168.5 6
12.11 even 2 2704.2.f.o.337.6 6
13.5 odd 4 1521.2.a.r.1.1 3
13.8 odd 4 1521.2.a.o.1.3 3
13.12 even 2 inner 1521.2.b.l.1351.5 6
39.2 even 12 169.2.c.c.22.1 6
39.5 even 4 169.2.a.b.1.3 3
39.8 even 4 169.2.a.c.1.1 yes 3
39.11 even 12 169.2.c.b.22.3 6
39.17 odd 6 169.2.e.b.23.2 12
39.20 even 12 169.2.c.b.146.3 6
39.23 odd 6 169.2.e.b.147.5 12
39.29 odd 6 169.2.e.b.147.2 12
39.32 even 12 169.2.c.c.146.1 6
39.35 odd 6 169.2.e.b.23.5 12
39.38 odd 2 169.2.b.b.168.2 6
156.47 odd 4 2704.2.a.ba.1.3 3
156.83 odd 4 2704.2.a.z.1.3 3
156.155 even 2 2704.2.f.o.337.5 6
195.44 even 4 4225.2.a.bg.1.1 3
195.164 even 4 4225.2.a.bb.1.3 3
273.83 odd 4 8281.2.a.bf.1.3 3
273.125 odd 4 8281.2.a.bj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 39.5 even 4
169.2.a.c.1.1 yes 3 39.8 even 4
169.2.b.b.168.2 6 39.38 odd 2
169.2.b.b.168.5 6 3.2 odd 2
169.2.c.b.22.3 6 39.11 even 12
169.2.c.b.146.3 6 39.20 even 12
169.2.c.c.22.1 6 39.2 even 12
169.2.c.c.146.1 6 39.32 even 12
169.2.e.b.23.2 12 39.17 odd 6
169.2.e.b.23.5 12 39.35 odd 6
169.2.e.b.147.2 12 39.29 odd 6
169.2.e.b.147.5 12 39.23 odd 6
1521.2.a.o.1.3 3 13.8 odd 4
1521.2.a.r.1.1 3 13.5 odd 4
1521.2.b.l.1351.2 6 1.1 even 1 trivial
1521.2.b.l.1351.5 6 13.12 even 2 inner
2704.2.a.z.1.3 3 156.83 odd 4
2704.2.a.ba.1.3 3 156.47 odd 4
2704.2.f.o.337.5 6 156.155 even 2
2704.2.f.o.337.6 6 12.11 even 2
4225.2.a.bb.1.3 3 195.164 even 4
4225.2.a.bg.1.1 3 195.44 even 4
8281.2.a.bf.1.3 3 273.83 odd 4
8281.2.a.bj.1.1 3 273.125 odd 4