Properties

Label 1521.2.a.o.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,2,Mod(1,1521)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

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: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24698 q^{2} +3.04892 q^{4} -1.44504 q^{5} -2.04892 q^{7} -2.35690 q^{8} +O(q^{10})\) \(q-2.24698 q^{2} +3.04892 q^{4} -1.44504 q^{5} -2.04892 q^{7} -2.35690 q^{8} +3.24698 q^{10} -2.55496 q^{11} +4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} +5.85086 q^{19} -4.40581 q^{20} +5.74094 q^{22} +1.89008 q^{23} -2.91185 q^{25} -6.24698 q^{28} -2.26875 q^{29} +4.26875 q^{31} +6.51573 q^{32} -11.8998 q^{34} +2.96077 q^{35} -5.35690 q^{37} -13.1468 q^{38} +3.40581 q^{40} +1.27413 q^{41} +6.13706 q^{43} -7.78986 q^{44} -4.24698 q^{46} -2.95108 q^{47} -2.80194 q^{49} +6.54288 q^{50} -5.52111 q^{53} +3.69202 q^{55} +4.82908 q^{56} +5.09783 q^{58} -12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -13.0368 q^{64} -0.576728 q^{67} +16.1468 q^{68} -6.65279 q^{70} -4.59419 q^{71} +10.5526 q^{73} +12.0368 q^{74} +17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +1.15883 q^{80} -2.86294 q^{82} +7.72348 q^{83} -7.65279 q^{85} -13.7899 q^{86} +6.02177 q^{88} +6.61356 q^{89} +5.76271 q^{92} +6.63102 q^{94} -8.45473 q^{95} -11.9269 q^{97} +6.29590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35} - 12 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 13 q^{43} - 8 q^{46} - 18 q^{47} - 4 q^{49} + q^{50} - q^{53} + 6 q^{55} + 4 q^{56} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} + q^{67} + 21 q^{68} - 2 q^{70} - 27 q^{71} - 9 q^{73} + 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - 5 q^{80} - 14 q^{82} - 7 q^{83} - 5 q^{85} - 18 q^{86} + 15 q^{88} - 11 q^{89} + 5 q^{94} - 3 q^{95} - 7 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.24698 −1.58885 −0.794427 0.607359i \(-0.792229\pi\)
−0.794427 + 0.607359i \(0.792229\pi\)
\(3\) 0 0
\(4\) 3.04892 1.52446
\(5\) −1.44504 −0.646242 −0.323121 0.946358i \(-0.604732\pi\)
−0.323121 + 0.946358i \(0.604732\pi\)
\(6\) 0 0
\(7\) −2.04892 −0.774418 −0.387209 0.921992i \(-0.626561\pi\)
−0.387209 + 0.921992i \(0.626561\pi\)
\(8\) −2.35690 −0.833289
\(9\) 0 0
\(10\) 3.24698 1.02679
\(11\) −2.55496 −0.770349 −0.385174 0.922844i \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.60388 1.23044
\(15\) 0 0
\(16\) −0.801938 −0.200484
\(17\) 5.29590 1.28444 0.642222 0.766519i \(-0.278013\pi\)
0.642222 + 0.766519i \(0.278013\pi\)
\(18\) 0 0
\(19\) 5.85086 1.34228 0.671139 0.741331i \(-0.265805\pi\)
0.671139 + 0.741331i \(0.265805\pi\)
\(20\) −4.40581 −0.985170
\(21\) 0 0
\(22\) 5.74094 1.22397
\(23\) 1.89008 0.394110 0.197055 0.980392i \(-0.436862\pi\)
0.197055 + 0.980392i \(0.436862\pi\)
\(24\) 0 0
\(25\) −2.91185 −0.582371
\(26\) 0 0
\(27\) 0 0
\(28\) −6.24698 −1.18057
\(29\) −2.26875 −0.421296 −0.210648 0.977562i \(-0.567557\pi\)
−0.210648 + 0.977562i \(0.567557\pi\)
\(30\) 0 0
\(31\) 4.26875 0.766690 0.383345 0.923605i \(-0.374772\pi\)
0.383345 + 0.923605i \(0.374772\pi\)
\(32\) 6.51573 1.15183
\(33\) 0 0
\(34\) −11.8998 −2.04079
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) −5.35690 −0.880668 −0.440334 0.897834i \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(38\) −13.1468 −2.13268
\(39\) 0 0
\(40\) 3.40581 0.538506
\(41\) 1.27413 0.198985 0.0994926 0.995038i \(-0.468278\pi\)
0.0994926 + 0.995038i \(0.468278\pi\)
\(42\) 0 0
\(43\) 6.13706 0.935893 0.467947 0.883757i \(-0.344994\pi\)
0.467947 + 0.883757i \(0.344994\pi\)
\(44\) −7.78986 −1.17437
\(45\) 0 0
\(46\) −4.24698 −0.626183
\(47\) −2.95108 −0.430460 −0.215230 0.976563i \(-0.569050\pi\)
−0.215230 + 0.976563i \(0.569050\pi\)
\(48\) 0 0
\(49\) −2.80194 −0.400277
\(50\) 6.54288 0.925302
\(51\) 0 0
\(52\) 0 0
\(53\) −5.52111 −0.758382 −0.379191 0.925318i \(-0.623798\pi\)
−0.379191 + 0.925318i \(0.623798\pi\)
\(54\) 0 0
\(55\) 3.69202 0.497832
\(56\) 4.82908 0.645314
\(57\) 0 0
\(58\) 5.09783 0.669378
\(59\) −12.2078 −1.58931 −0.794657 0.607059i \(-0.792349\pi\)
−0.794657 + 0.607059i \(0.792349\pi\)
\(60\) 0 0
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) −9.59179 −1.21816
\(63\) 0 0
\(64\) −13.0368 −1.62960
\(65\) 0 0
\(66\) 0 0
\(67\) −0.576728 −0.0704586 −0.0352293 0.999379i \(-0.511216\pi\)
−0.0352293 + 0.999379i \(0.511216\pi\)
\(68\) 16.1468 1.95808
\(69\) 0 0
\(70\) −6.65279 −0.795161
\(71\) −4.59419 −0.545230 −0.272615 0.962123i \(-0.587888\pi\)
−0.272615 + 0.962123i \(0.587888\pi\)
\(72\) 0 0
\(73\) 10.5526 1.23508 0.617542 0.786538i \(-0.288128\pi\)
0.617542 + 0.786538i \(0.288128\pi\)
\(74\) 12.0368 1.39925
\(75\) 0 0
\(76\) 17.8388 2.04625
\(77\) 5.23490 0.596572
\(78\) 0 0
\(79\) −15.7778 −1.77514 −0.887569 0.460674i \(-0.847608\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(80\) 1.15883 0.129562
\(81\) 0 0
\(82\) −2.86294 −0.316158
\(83\) 7.72348 0.847762 0.423881 0.905718i \(-0.360667\pi\)
0.423881 + 0.905718i \(0.360667\pi\)
\(84\) 0 0
\(85\) −7.65279 −0.830062
\(86\) −13.7899 −1.48700
\(87\) 0 0
\(88\) 6.02177 0.641923
\(89\) 6.61356 0.701036 0.350518 0.936556i \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.76271 0.600804
\(93\) 0 0
\(94\) 6.63102 0.683938
\(95\) −8.45473 −0.867437
\(96\) 0 0
\(97\) −11.9269 −1.21100 −0.605498 0.795847i \(-0.707026\pi\)
−0.605498 + 0.795847i \(0.707026\pi\)
\(98\) 6.29590 0.635982
\(99\) 0 0
\(100\) −8.87800 −0.887800
\(101\) −13.0640 −1.29991 −0.649957 0.759971i \(-0.725213\pi\)
−0.649957 + 0.759971i \(0.725213\pi\)
\(102\) 0 0
\(103\) 9.16852 0.903401 0.451701 0.892170i \(-0.350818\pi\)
0.451701 + 0.892170i \(0.350818\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.4058 1.20496
\(107\) 6.89977 0.667026 0.333513 0.942745i \(-0.391766\pi\)
0.333513 + 0.942745i \(0.391766\pi\)
\(108\) 0 0
\(109\) −0.121998 −0.0116853 −0.00584264 0.999983i \(-0.501860\pi\)
−0.00584264 + 0.999983i \(0.501860\pi\)
\(110\) −8.29590 −0.790983
\(111\) 0 0
\(112\) 1.64310 0.155259
\(113\) −7.30798 −0.687477 −0.343738 0.939065i \(-0.611693\pi\)
−0.343738 + 0.939065i \(0.611693\pi\)
\(114\) 0 0
\(115\) −2.73125 −0.254690
\(116\) −6.91723 −0.642249
\(117\) 0 0
\(118\) 27.4306 2.52519
\(119\) −10.8509 −0.994696
\(120\) 0 0
\(121\) −4.47219 −0.406563
\(122\) −19.2446 −1.74232
\(123\) 0 0
\(124\) 13.0151 1.16879
\(125\) 11.4330 1.02260
\(126\) 0 0
\(127\) −18.9705 −1.68336 −0.841678 0.539980i \(-0.818432\pi\)
−0.841678 + 0.539980i \(0.818432\pi\)
\(128\) 16.2620 1.43738
\(129\) 0 0
\(130\) 0 0
\(131\) −3.25667 −0.284536 −0.142268 0.989828i \(-0.545440\pi\)
−0.142268 + 0.989828i \(0.545440\pi\)
\(132\) 0 0
\(133\) −11.9879 −1.03948
\(134\) 1.29590 0.111948
\(135\) 0 0
\(136\) −12.4819 −1.07031
\(137\) 0.792249 0.0676864 0.0338432 0.999427i \(-0.489225\pi\)
0.0338432 + 0.999427i \(0.489225\pi\)
\(138\) 0 0
\(139\) −11.3394 −0.961799 −0.480899 0.876776i \(-0.659690\pi\)
−0.480899 + 0.876776i \(0.659690\pi\)
\(140\) 9.02715 0.762933
\(141\) 0 0
\(142\) 10.3230 0.866291
\(143\) 0 0
\(144\) 0 0
\(145\) 3.27844 0.272260
\(146\) −23.7114 −1.96237
\(147\) 0 0
\(148\) −16.3327 −1.34254
\(149\) 8.40581 0.688631 0.344316 0.938854i \(-0.388111\pi\)
0.344316 + 0.938854i \(0.388111\pi\)
\(150\) 0 0
\(151\) −14.1293 −1.14983 −0.574913 0.818215i \(-0.694964\pi\)
−0.574913 + 0.818215i \(0.694964\pi\)
\(152\) −13.7899 −1.11851
\(153\) 0 0
\(154\) −11.7627 −0.947866
\(155\) −6.16852 −0.495468
\(156\) 0 0
\(157\) −9.43296 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(158\) 35.4523 2.82044
\(159\) 0 0
\(160\) −9.41550 −0.744361
\(161\) −3.87263 −0.305206
\(162\) 0 0
\(163\) −8.70410 −0.681758 −0.340879 0.940107i \(-0.610725\pi\)
−0.340879 + 0.940107i \(0.610725\pi\)
\(164\) 3.88471 0.303345
\(165\) 0 0
\(166\) −17.3545 −1.34697
\(167\) −23.8538 −1.84587 −0.922933 0.384961i \(-0.874215\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 17.1957 1.31885
\(171\) 0 0
\(172\) 18.7114 1.42673
\(173\) 18.8552 1.43353 0.716766 0.697314i \(-0.245622\pi\)
0.716766 + 0.697314i \(0.245622\pi\)
\(174\) 0 0
\(175\) 5.96615 0.450998
\(176\) 2.04892 0.154443
\(177\) 0 0
\(178\) −14.8605 −1.11384
\(179\) −6.02177 −0.450088 −0.225044 0.974349i \(-0.572253\pi\)
−0.225044 + 0.974349i \(0.572253\pi\)
\(180\) 0 0
\(181\) −4.77777 −0.355129 −0.177565 0.984109i \(-0.556822\pi\)
−0.177565 + 0.984109i \(0.556822\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.45473 −0.328407
\(185\) 7.74094 0.569125
\(186\) 0 0
\(187\) −13.5308 −0.989470
\(188\) −8.99761 −0.656218
\(189\) 0 0
\(190\) 18.9976 1.37823
\(191\) −18.4306 −1.33359 −0.666795 0.745242i \(-0.732334\pi\)
−0.666795 + 0.745242i \(0.732334\pi\)
\(192\) 0 0
\(193\) −6.05429 −0.435798 −0.217899 0.975971i \(-0.569920\pi\)
−0.217899 + 0.975971i \(0.569920\pi\)
\(194\) 26.7995 1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) −11.4155 −0.813321 −0.406660 0.913579i \(-0.633307\pi\)
−0.406660 + 0.913579i \(0.633307\pi\)
\(198\) 0 0
\(199\) −13.9051 −0.985710 −0.492855 0.870111i \(-0.664047\pi\)
−0.492855 + 0.870111i \(0.664047\pi\)
\(200\) 6.86294 0.485283
\(201\) 0 0
\(202\) 29.3545 2.06538
\(203\) 4.64848 0.326259
\(204\) 0 0
\(205\) −1.84117 −0.128593
\(206\) −20.6015 −1.43537
\(207\) 0 0
\(208\) 0 0
\(209\) −14.9487 −1.03402
\(210\) 0 0
\(211\) −13.2446 −0.911795 −0.455897 0.890032i \(-0.650682\pi\)
−0.455897 + 0.890032i \(0.650682\pi\)
\(212\) −16.8334 −1.15612
\(213\) 0 0
\(214\) −15.5036 −1.05981
\(215\) −8.86831 −0.604814
\(216\) 0 0
\(217\) −8.74632 −0.593739
\(218\) 0.274127 0.0185662
\(219\) 0 0
\(220\) 11.2567 0.758924
\(221\) 0 0
\(222\) 0 0
\(223\) 7.33513 0.491196 0.245598 0.969372i \(-0.421016\pi\)
0.245598 + 0.969372i \(0.421016\pi\)
\(224\) −13.3502 −0.891997
\(225\) 0 0
\(226\) 16.4209 1.09230
\(227\) −8.67456 −0.575751 −0.287875 0.957668i \(-0.592949\pi\)
−0.287875 + 0.957668i \(0.592949\pi\)
\(228\) 0 0
\(229\) 13.6866 0.904439 0.452219 0.891907i \(-0.350632\pi\)
0.452219 + 0.891907i \(0.350632\pi\)
\(230\) 6.13706 0.404666
\(231\) 0 0
\(232\) 5.34721 0.351061
\(233\) 5.08815 0.333336 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(234\) 0 0
\(235\) 4.26444 0.278181
\(236\) −37.2204 −2.42284
\(237\) 0 0
\(238\) 24.3817 1.58043
\(239\) −10.9239 −0.706611 −0.353305 0.935508i \(-0.614942\pi\)
−0.353305 + 0.935508i \(0.614942\pi\)
\(240\) 0 0
\(241\) −11.9148 −0.767502 −0.383751 0.923437i \(-0.625368\pi\)
−0.383751 + 0.923437i \(0.625368\pi\)
\(242\) 10.0489 0.645969
\(243\) 0 0
\(244\) 26.1129 1.67171
\(245\) 4.04892 0.258676
\(246\) 0 0
\(247\) 0 0
\(248\) −10.0610 −0.638874
\(249\) 0 0
\(250\) −25.6896 −1.62475
\(251\) −22.3478 −1.41058 −0.705290 0.708919i \(-0.749183\pi\)
−0.705290 + 0.708919i \(0.749183\pi\)
\(252\) 0 0
\(253\) −4.82908 −0.303602
\(254\) 42.6262 2.67461
\(255\) 0 0
\(256\) −10.4668 −0.654176
\(257\) 18.6601 1.16398 0.581992 0.813194i \(-0.302273\pi\)
0.581992 + 0.813194i \(0.302273\pi\)
\(258\) 0 0
\(259\) 10.9758 0.682005
\(260\) 0 0
\(261\) 0 0
\(262\) 7.31767 0.452087
\(263\) −14.3991 −0.887887 −0.443944 0.896055i \(-0.646421\pi\)
−0.443944 + 0.896055i \(0.646421\pi\)
\(264\) 0 0
\(265\) 7.97823 0.490099
\(266\) 26.9366 1.65159
\(267\) 0 0
\(268\) −1.75840 −0.107411
\(269\) −0.652793 −0.0398015 −0.0199007 0.999802i \(-0.506335\pi\)
−0.0199007 + 0.999802i \(0.506335\pi\)
\(270\) 0 0
\(271\) 1.99569 0.121229 0.0606147 0.998161i \(-0.480694\pi\)
0.0606147 + 0.998161i \(0.480694\pi\)
\(272\) −4.24698 −0.257511
\(273\) 0 0
\(274\) −1.78017 −0.107544
\(275\) 7.43967 0.448629
\(276\) 0 0
\(277\) 11.7845 0.708061 0.354030 0.935234i \(-0.384811\pi\)
0.354030 + 0.935234i \(0.384811\pi\)
\(278\) 25.4795 1.52816
\(279\) 0 0
\(280\) −6.97823 −0.417029
\(281\) −6.47219 −0.386098 −0.193049 0.981189i \(-0.561838\pi\)
−0.193049 + 0.981189i \(0.561838\pi\)
\(282\) 0 0
\(283\) 6.58104 0.391202 0.195601 0.980684i \(-0.437334\pi\)
0.195601 + 0.980684i \(0.437334\pi\)
\(284\) −14.0073 −0.831180
\(285\) 0 0
\(286\) 0 0
\(287\) −2.61058 −0.154098
\(288\) 0 0
\(289\) 11.0465 0.649796
\(290\) −7.36658 −0.432581
\(291\) 0 0
\(292\) 32.1739 1.88284
\(293\) −24.3381 −1.42185 −0.710924 0.703269i \(-0.751723\pi\)
−0.710924 + 0.703269i \(0.751723\pi\)
\(294\) 0 0
\(295\) 17.6407 1.02708
\(296\) 12.6256 0.733851
\(297\) 0 0
\(298\) −18.8877 −1.09413
\(299\) 0 0
\(300\) 0 0
\(301\) −12.5743 −0.724773
\(302\) 31.7482 1.82691
\(303\) 0 0
\(304\) −4.69202 −0.269106
\(305\) −12.3763 −0.708663
\(306\) 0 0
\(307\) 14.0737 0.803227 0.401613 0.915809i \(-0.368450\pi\)
0.401613 + 0.915809i \(0.368450\pi\)
\(308\) 15.9608 0.909449
\(309\) 0 0
\(310\) 13.8605 0.787226
\(311\) 29.7700 1.68810 0.844051 0.536263i \(-0.180164\pi\)
0.844051 + 0.536263i \(0.180164\pi\)
\(312\) 0 0
\(313\) −7.47889 −0.422732 −0.211366 0.977407i \(-0.567791\pi\)
−0.211366 + 0.977407i \(0.567791\pi\)
\(314\) 21.1957 1.19614
\(315\) 0 0
\(316\) −48.1051 −2.70613
\(317\) 30.0301 1.68666 0.843330 0.537396i \(-0.180592\pi\)
0.843330 + 0.537396i \(0.180592\pi\)
\(318\) 0 0
\(319\) 5.79656 0.324545
\(320\) 18.8388 1.05312
\(321\) 0 0
\(322\) 8.70171 0.484927
\(323\) 30.9855 1.72408
\(324\) 0 0
\(325\) 0 0
\(326\) 19.5579 1.08321
\(327\) 0 0
\(328\) −3.00298 −0.165812
\(329\) 6.04652 0.333356
\(330\) 0 0
\(331\) 15.7168 0.863872 0.431936 0.901904i \(-0.357831\pi\)
0.431936 + 0.901904i \(0.357831\pi\)
\(332\) 23.5483 1.29238
\(333\) 0 0
\(334\) 53.5991 2.93281
\(335\) 0.833397 0.0455333
\(336\) 0 0
\(337\) 1.95407 0.106445 0.0532224 0.998583i \(-0.483051\pi\)
0.0532224 + 0.998583i \(0.483051\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −23.3327 −1.26540
\(341\) −10.9065 −0.590619
\(342\) 0 0
\(343\) 20.0834 1.08440
\(344\) −14.4644 −0.779869
\(345\) 0 0
\(346\) −42.3672 −2.27767
\(347\) 17.1250 0.919317 0.459659 0.888096i \(-0.347972\pi\)
0.459659 + 0.888096i \(0.347972\pi\)
\(348\) 0 0
\(349\) 10.4668 0.560276 0.280138 0.959960i \(-0.409620\pi\)
0.280138 + 0.959960i \(0.409620\pi\)
\(350\) −13.4058 −0.716571
\(351\) 0 0
\(352\) −16.6474 −0.887310
\(353\) 15.5308 0.826621 0.413310 0.910590i \(-0.364372\pi\)
0.413310 + 0.910590i \(0.364372\pi\)
\(354\) 0 0
\(355\) 6.63879 0.352351
\(356\) 20.1642 1.06870
\(357\) 0 0
\(358\) 13.5308 0.715125
\(359\) −21.4263 −1.13083 −0.565417 0.824805i \(-0.691285\pi\)
−0.565417 + 0.824805i \(0.691285\pi\)
\(360\) 0 0
\(361\) 15.2325 0.801711
\(362\) 10.7356 0.564249
\(363\) 0 0
\(364\) 0 0
\(365\) −15.2489 −0.798164
\(366\) 0 0
\(367\) 34.3032 1.79061 0.895306 0.445452i \(-0.146957\pi\)
0.895306 + 0.445452i \(0.146957\pi\)
\(368\) −1.51573 −0.0790129
\(369\) 0 0
\(370\) −17.3937 −0.904257
\(371\) 11.3123 0.587305
\(372\) 0 0
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) 30.4034 1.57212
\(375\) 0 0
\(376\) 6.95539 0.358697
\(377\) 0 0
\(378\) 0 0
\(379\) −16.5386 −0.849529 −0.424765 0.905304i \(-0.639643\pi\)
−0.424765 + 0.905304i \(0.639643\pi\)
\(380\) −25.7778 −1.32237
\(381\) 0 0
\(382\) 41.4131 2.11888
\(383\) 7.53617 0.385080 0.192540 0.981289i \(-0.438327\pi\)
0.192540 + 0.981289i \(0.438327\pi\)
\(384\) 0 0
\(385\) −7.56465 −0.385530
\(386\) 13.6039 0.692419
\(387\) 0 0
\(388\) −36.3642 −1.84611
\(389\) −35.5555 −1.80274 −0.901369 0.433052i \(-0.857437\pi\)
−0.901369 + 0.433052i \(0.857437\pi\)
\(390\) 0 0
\(391\) 10.0097 0.506212
\(392\) 6.60388 0.333546
\(393\) 0 0
\(394\) 25.6504 1.29225
\(395\) 22.7995 1.14717
\(396\) 0 0
\(397\) −1.35152 −0.0678308 −0.0339154 0.999425i \(-0.510798\pi\)
−0.0339154 + 0.999425i \(0.510798\pi\)
\(398\) 31.2446 1.56615
\(399\) 0 0
\(400\) 2.33513 0.116756
\(401\) 0.579121 0.0289199 0.0144600 0.999895i \(-0.495397\pi\)
0.0144600 + 0.999895i \(0.495397\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −39.8310 −1.98167
\(405\) 0 0
\(406\) −10.4450 −0.518379
\(407\) 13.6866 0.678422
\(408\) 0 0
\(409\) 15.1575 0.749490 0.374745 0.927128i \(-0.377730\pi\)
0.374745 + 0.927128i \(0.377730\pi\)
\(410\) 4.13706 0.204315
\(411\) 0 0
\(412\) 27.9541 1.37720
\(413\) 25.0127 1.23079
\(414\) 0 0
\(415\) −11.1608 −0.547860
\(416\) 0 0
\(417\) 0 0
\(418\) 33.5894 1.64291
\(419\) 35.7235 1.74521 0.872603 0.488430i \(-0.162430\pi\)
0.872603 + 0.488430i \(0.162430\pi\)
\(420\) 0 0
\(421\) 35.0465 1.70806 0.854032 0.520221i \(-0.174151\pi\)
0.854032 + 0.520221i \(0.174151\pi\)
\(422\) 29.7603 1.44871
\(423\) 0 0
\(424\) 13.0127 0.631951
\(425\) −15.4209 −0.748022
\(426\) 0 0
\(427\) −17.5483 −0.849220
\(428\) 21.0368 1.01685
\(429\) 0 0
\(430\) 19.9269 0.960961
\(431\) −34.2814 −1.65128 −0.825639 0.564199i \(-0.809185\pi\)
−0.825639 + 0.564199i \(0.809185\pi\)
\(432\) 0 0
\(433\) 13.7385 0.660232 0.330116 0.943940i \(-0.392912\pi\)
0.330116 + 0.943940i \(0.392912\pi\)
\(434\) 19.6528 0.943364
\(435\) 0 0
\(436\) −0.371961 −0.0178137
\(437\) 11.0586 0.529005
\(438\) 0 0
\(439\) 10.2403 0.488742 0.244371 0.969682i \(-0.421419\pi\)
0.244371 + 0.969682i \(0.421419\pi\)
\(440\) −8.70171 −0.414838
\(441\) 0 0
\(442\) 0 0
\(443\) −12.1763 −0.578513 −0.289257 0.957252i \(-0.593408\pi\)
−0.289257 + 0.957252i \(0.593408\pi\)
\(444\) 0 0
\(445\) −9.55688 −0.453039
\(446\) −16.4819 −0.780440
\(447\) 0 0
\(448\) 26.7114 1.26199
\(449\) −12.9051 −0.609032 −0.304516 0.952507i \(-0.598495\pi\)
−0.304516 + 0.952507i \(0.598495\pi\)
\(450\) 0 0
\(451\) −3.25534 −0.153288
\(452\) −22.2814 −1.04803
\(453\) 0 0
\(454\) 19.4916 0.914785
\(455\) 0 0
\(456\) 0 0
\(457\) −4.65710 −0.217850 −0.108925 0.994050i \(-0.534741\pi\)
−0.108925 + 0.994050i \(0.534741\pi\)
\(458\) −30.7536 −1.43702
\(459\) 0 0
\(460\) −8.32736 −0.388265
\(461\) −31.5405 −1.46899 −0.734493 0.678616i \(-0.762580\pi\)
−0.734493 + 0.678616i \(0.762580\pi\)
\(462\) 0 0
\(463\) −17.6504 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(464\) 1.81940 0.0844633
\(465\) 0 0
\(466\) −11.4330 −0.529622
\(467\) 32.1726 1.48877 0.744385 0.667751i \(-0.232743\pi\)
0.744385 + 0.667751i \(0.232743\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) −9.58211 −0.441990
\(471\) 0 0
\(472\) 28.7724 1.32436
\(473\) −15.6799 −0.720964
\(474\) 0 0
\(475\) −17.0368 −0.781704
\(476\) −33.0834 −1.51637
\(477\) 0 0
\(478\) 24.5459 1.12270
\(479\) −34.8998 −1.59461 −0.797306 0.603576i \(-0.793742\pi\)
−0.797306 + 0.603576i \(0.793742\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 26.7724 1.21945
\(483\) 0 0
\(484\) −13.6353 −0.619788
\(485\) 17.2349 0.782596
\(486\) 0 0
\(487\) −41.8351 −1.89573 −0.947864 0.318676i \(-0.896762\pi\)
−0.947864 + 0.318676i \(0.896762\pi\)
\(488\) −20.1860 −0.913776
\(489\) 0 0
\(490\) −9.09783 −0.410998
\(491\) −21.8455 −0.985873 −0.492936 0.870065i \(-0.664076\pi\)
−0.492936 + 0.870065i \(0.664076\pi\)
\(492\) 0 0
\(493\) −12.0151 −0.541131
\(494\) 0 0
\(495\) 0 0
\(496\) −3.42327 −0.153709
\(497\) 9.41311 0.422236
\(498\) 0 0
\(499\) −23.5472 −1.05412 −0.527058 0.849829i \(-0.676705\pi\)
−0.527058 + 0.849829i \(0.676705\pi\)
\(500\) 34.8582 1.55890
\(501\) 0 0
\(502\) 50.2150 2.24121
\(503\) 7.08682 0.315986 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(504\) 0 0
\(505\) 18.8780 0.840060
\(506\) 10.8509 0.482379
\(507\) 0 0
\(508\) −57.8394 −2.56621
\(509\) 7.61894 0.337704 0.168852 0.985641i \(-0.445994\pi\)
0.168852 + 0.985641i \(0.445994\pi\)
\(510\) 0 0
\(511\) −21.6213 −0.956471
\(512\) −9.00538 −0.397985
\(513\) 0 0
\(514\) −41.9288 −1.84940
\(515\) −13.2489 −0.583816
\(516\) 0 0
\(517\) 7.53989 0.331604
\(518\) −24.6625 −1.08361
\(519\) 0 0
\(520\) 0 0
\(521\) 39.5133 1.73111 0.865555 0.500813i \(-0.166966\pi\)
0.865555 + 0.500813i \(0.166966\pi\)
\(522\) 0 0
\(523\) −15.8194 −0.691734 −0.345867 0.938284i \(-0.612415\pi\)
−0.345867 + 0.938284i \(0.612415\pi\)
\(524\) −9.92931 −0.433764
\(525\) 0 0
\(526\) 32.3545 1.41072
\(527\) 22.6069 0.984770
\(528\) 0 0
\(529\) −19.4276 −0.844678
\(530\) −17.9269 −0.778696
\(531\) 0 0
\(532\) −36.5502 −1.58465
\(533\) 0 0
\(534\) 0 0
\(535\) −9.97046 −0.431061
\(536\) 1.35929 0.0587123
\(537\) 0 0
\(538\) 1.46681 0.0632388
\(539\) 7.15883 0.308353
\(540\) 0 0
\(541\) −34.4819 −1.48249 −0.741246 0.671234i \(-0.765765\pi\)
−0.741246 + 0.671234i \(0.765765\pi\)
\(542\) −4.48427 −0.192616
\(543\) 0 0
\(544\) 34.5066 1.47946
\(545\) 0.176292 0.00755152
\(546\) 0 0
\(547\) 36.8582 1.57594 0.787970 0.615713i \(-0.211132\pi\)
0.787970 + 0.615713i \(0.211132\pi\)
\(548\) 2.41550 0.103185
\(549\) 0 0
\(550\) −16.7168 −0.712806
\(551\) −13.2741 −0.565497
\(552\) 0 0
\(553\) 32.3274 1.37470
\(554\) −26.4795 −1.12501
\(555\) 0 0
\(556\) −34.5730 −1.46622
\(557\) 1.27652 0.0540879 0.0270439 0.999634i \(-0.491391\pi\)
0.0270439 + 0.999634i \(0.491391\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.37435 −0.100335
\(561\) 0 0
\(562\) 14.5429 0.613454
\(563\) 9.12737 0.384673 0.192336 0.981329i \(-0.438393\pi\)
0.192336 + 0.981329i \(0.438393\pi\)
\(564\) 0 0
\(565\) 10.5603 0.444277
\(566\) −14.7875 −0.621563
\(567\) 0 0
\(568\) 10.8280 0.454334
\(569\) 5.72156 0.239860 0.119930 0.992782i \(-0.461733\pi\)
0.119930 + 0.992782i \(0.461733\pi\)
\(570\) 0 0
\(571\) 7.60148 0.318112 0.159056 0.987270i \(-0.449155\pi\)
0.159056 + 0.987270i \(0.449155\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 5.86592 0.244839
\(575\) −5.50365 −0.229518
\(576\) 0 0
\(577\) −45.1564 −1.87989 −0.939944 0.341330i \(-0.889123\pi\)
−0.939944 + 0.341330i \(0.889123\pi\)
\(578\) −24.8213 −1.03243
\(579\) 0 0
\(580\) 9.99569 0.415048
\(581\) −15.8248 −0.656522
\(582\) 0 0
\(583\) 14.1062 0.584219
\(584\) −24.8713 −1.02918
\(585\) 0 0
\(586\) 54.6872 2.25911
\(587\) −32.4040 −1.33746 −0.668728 0.743507i \(-0.733161\pi\)
−0.668728 + 0.743507i \(0.733161\pi\)
\(588\) 0 0
\(589\) 24.9758 1.02911
\(590\) −39.6383 −1.63188
\(591\) 0 0
\(592\) 4.29590 0.176560
\(593\) 36.6848 1.50647 0.753233 0.657754i \(-0.228493\pi\)
0.753233 + 0.657754i \(0.228493\pi\)
\(594\) 0 0
\(595\) 15.6799 0.642815
\(596\) 25.6286 1.04979
\(597\) 0 0
\(598\) 0 0
\(599\) 9.99223 0.408271 0.204136 0.978943i \(-0.434562\pi\)
0.204136 + 0.978943i \(0.434562\pi\)
\(600\) 0 0
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) 28.2543 1.15156
\(603\) 0 0
\(604\) −43.0790 −1.75286
\(605\) 6.46250 0.262738
\(606\) 0 0
\(607\) 11.2161 0.455248 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(608\) 38.1226 1.54608
\(609\) 0 0
\(610\) 27.8092 1.12596
\(611\) 0 0
\(612\) 0 0
\(613\) 20.8944 0.843917 0.421958 0.906615i \(-0.361343\pi\)
0.421958 + 0.906615i \(0.361343\pi\)
\(614\) −31.6233 −1.27621
\(615\) 0 0
\(616\) −12.3381 −0.497117
\(617\) 12.0992 0.487094 0.243547 0.969889i \(-0.421689\pi\)
0.243547 + 0.969889i \(0.421689\pi\)
\(618\) 0 0
\(619\) 10.5526 0.424143 0.212072 0.977254i \(-0.431979\pi\)
0.212072 + 0.977254i \(0.431979\pi\)
\(620\) −18.8073 −0.755320
\(621\) 0 0
\(622\) −66.8926 −2.68215
\(623\) −13.5506 −0.542895
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 16.8049 0.671660
\(627\) 0 0
\(628\) −28.7603 −1.14766
\(629\) −28.3696 −1.13117
\(630\) 0 0
\(631\) 13.8514 0.551417 0.275709 0.961241i \(-0.411087\pi\)
0.275709 + 0.961241i \(0.411087\pi\)
\(632\) 37.1866 1.47920
\(633\) 0 0
\(634\) −67.4771 −2.67986
\(635\) 27.4131 1.08786
\(636\) 0 0
\(637\) 0 0
\(638\) −13.0248 −0.515655
\(639\) 0 0
\(640\) −23.4993 −0.928893
\(641\) −34.9608 −1.38087 −0.690434 0.723396i \(-0.742580\pi\)
−0.690434 + 0.723396i \(0.742580\pi\)
\(642\) 0 0
\(643\) −33.3980 −1.31709 −0.658545 0.752541i \(-0.728828\pi\)
−0.658545 + 0.752541i \(0.728828\pi\)
\(644\) −11.8073 −0.465273
\(645\) 0 0
\(646\) −69.6238 −2.73931
\(647\) −2.32842 −0.0915397 −0.0457698 0.998952i \(-0.514574\pi\)
−0.0457698 + 0.998952i \(0.514574\pi\)
\(648\) 0 0
\(649\) 31.1903 1.22433
\(650\) 0 0
\(651\) 0 0
\(652\) −26.5381 −1.03931
\(653\) −14.5714 −0.570221 −0.285111 0.958495i \(-0.592030\pi\)
−0.285111 + 0.958495i \(0.592030\pi\)
\(654\) 0 0
\(655\) 4.70602 0.183879
\(656\) −1.02177 −0.0398934
\(657\) 0 0
\(658\) −13.5864 −0.529654
\(659\) −11.1395 −0.433932 −0.216966 0.976179i \(-0.569616\pi\)
−0.216966 + 0.976179i \(0.569616\pi\)
\(660\) 0 0
\(661\) 13.8498 0.538694 0.269347 0.963043i \(-0.413192\pi\)
0.269347 + 0.963043i \(0.413192\pi\)
\(662\) −35.3153 −1.37257
\(663\) 0 0
\(664\) −18.2034 −0.706430
\(665\) 17.3230 0.671759
\(666\) 0 0
\(667\) −4.28813 −0.166037
\(668\) −72.7284 −2.81395
\(669\) 0 0
\(670\) −1.87263 −0.0723458
\(671\) −21.8823 −0.844757
\(672\) 0 0
\(673\) −6.52973 −0.251703 −0.125851 0.992049i \(-0.540166\pi\)
−0.125851 + 0.992049i \(0.540166\pi\)
\(674\) −4.39075 −0.169125
\(675\) 0 0
\(676\) 0 0
\(677\) 11.3104 0.434693 0.217346 0.976095i \(-0.430260\pi\)
0.217346 + 0.976095i \(0.430260\pi\)
\(678\) 0 0
\(679\) 24.4373 0.937816
\(680\) 18.0368 0.691681
\(681\) 0 0
\(682\) 24.5066 0.938407
\(683\) 14.1793 0.542555 0.271277 0.962501i \(-0.412554\pi\)
0.271277 + 0.962501i \(0.412554\pi\)
\(684\) 0 0
\(685\) −1.14483 −0.0437418
\(686\) −45.1269 −1.72295
\(687\) 0 0
\(688\) −4.92154 −0.187632
\(689\) 0 0
\(690\) 0 0
\(691\) −30.7952 −1.17151 −0.585753 0.810490i \(-0.699201\pi\)
−0.585753 + 0.810490i \(0.699201\pi\)
\(692\) 57.4878 2.18536
\(693\) 0 0
\(694\) −38.4795 −1.46066
\(695\) 16.3860 0.621555
\(696\) 0 0
\(697\) 6.74764 0.255585
\(698\) −23.5187 −0.890196
\(699\) 0 0
\(700\) 18.1903 0.687528
\(701\) −6.73184 −0.254258 −0.127129 0.991886i \(-0.540576\pi\)
−0.127129 + 0.991886i \(0.540576\pi\)
\(702\) 0 0
\(703\) −31.3424 −1.18210
\(704\) 33.3086 1.25536
\(705\) 0 0
\(706\) −34.8974 −1.31338
\(707\) 26.7670 1.00668
\(708\) 0 0
\(709\) 47.6252 1.78860 0.894300 0.447467i \(-0.147674\pi\)
0.894300 + 0.447467i \(0.147674\pi\)
\(710\) −14.9172 −0.559834
\(711\) 0 0
\(712\) −15.5875 −0.584166
\(713\) 8.06829 0.302160
\(714\) 0 0
\(715\) 0 0
\(716\) −18.3599 −0.686141
\(717\) 0 0
\(718\) 48.1444 1.79673
\(719\) 5.99330 0.223512 0.111756 0.993736i \(-0.464352\pi\)
0.111756 + 0.993736i \(0.464352\pi\)
\(720\) 0 0
\(721\) −18.7855 −0.699610
\(722\) −34.2271 −1.27380
\(723\) 0 0
\(724\) −14.5670 −0.541380
\(725\) 6.60627 0.245351
\(726\) 0 0
\(727\) −24.1226 −0.894657 −0.447329 0.894370i \(-0.647625\pi\)
−0.447329 + 0.894370i \(0.647625\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 34.2640 1.26817
\(731\) 32.5013 1.20210
\(732\) 0 0
\(733\) −36.0646 −1.33208 −0.666038 0.745918i \(-0.732011\pi\)
−0.666038 + 0.745918i \(0.732011\pi\)
\(734\) −77.0786 −2.84502
\(735\) 0 0
\(736\) 12.3153 0.453947
\(737\) 1.47352 0.0542777
\(738\) 0 0
\(739\) −27.5254 −1.01254 −0.506269 0.862375i \(-0.668976\pi\)
−0.506269 + 0.862375i \(0.668976\pi\)
\(740\) 23.6015 0.867608
\(741\) 0 0
\(742\) −25.4185 −0.933142
\(743\) −10.4692 −0.384078 −0.192039 0.981387i \(-0.561510\pi\)
−0.192039 + 0.981387i \(0.561510\pi\)
\(744\) 0 0
\(745\) −12.1468 −0.445023
\(746\) 28.3032 1.03625
\(747\) 0 0
\(748\) −41.2543 −1.50841
\(749\) −14.1371 −0.516557
\(750\) 0 0
\(751\) 4.06770 0.148433 0.0742163 0.997242i \(-0.476354\pi\)
0.0742163 + 0.997242i \(0.476354\pi\)
\(752\) 2.36658 0.0863005
\(753\) 0 0
\(754\) 0 0
\(755\) 20.4174 0.743066
\(756\) 0 0
\(757\) 20.4336 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(758\) 37.1618 1.34978
\(759\) 0 0
\(760\) 19.9269 0.722825
\(761\) −27.0237 −0.979608 −0.489804 0.871833i \(-0.662932\pi\)
−0.489804 + 0.871833i \(0.662932\pi\)
\(762\) 0 0
\(763\) 0.249964 0.00904929
\(764\) −56.1933 −2.03300
\(765\) 0 0
\(766\) −16.9336 −0.611837
\(767\) 0 0
\(768\) 0 0
\(769\) 37.9407 1.36818 0.684088 0.729400i \(-0.260201\pi\)
0.684088 + 0.729400i \(0.260201\pi\)
\(770\) 16.9976 0.612551
\(771\) 0 0
\(772\) −18.4590 −0.664355
\(773\) −16.3375 −0.587620 −0.293810 0.955864i \(-0.594923\pi\)
−0.293810 + 0.955864i \(0.594923\pi\)
\(774\) 0 0
\(775\) −12.4300 −0.446498
\(776\) 28.1105 1.00911
\(777\) 0 0
\(778\) 79.8926 2.86429
\(779\) 7.45473 0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) −22.4916 −0.804297
\(783\) 0 0
\(784\) 2.24698 0.0802493
\(785\) 13.6310 0.486512
\(786\) 0 0
\(787\) 18.6907 0.666251 0.333126 0.942882i \(-0.391897\pi\)
0.333126 + 0.942882i \(0.391897\pi\)
\(788\) −34.8049 −1.23987
\(789\) 0 0
\(790\) −51.2301 −1.82269
\(791\) 14.9734 0.532394
\(792\) 0 0
\(793\) 0 0
\(794\) 3.03684 0.107773
\(795\) 0 0
\(796\) −42.3957 −1.50267
\(797\) −29.2519 −1.03615 −0.518077 0.855334i \(-0.673352\pi\)
−0.518077 + 0.855334i \(0.673352\pi\)
\(798\) 0 0
\(799\) −15.6286 −0.552901
\(800\) −18.9729 −0.670792
\(801\) 0 0
\(802\) −1.30127 −0.0459496
\(803\) −26.9614 −0.951446
\(804\) 0 0
\(805\) 5.59611 0.197237
\(806\) 0 0
\(807\) 0 0
\(808\) 30.7904 1.08320
\(809\) 6.65087 0.233832 0.116916 0.993142i \(-0.462699\pi\)
0.116916 + 0.993142i \(0.462699\pi\)
\(810\) 0 0
\(811\) −3.89200 −0.136667 −0.0683333 0.997663i \(-0.521768\pi\)
−0.0683333 + 0.997663i \(0.521768\pi\)
\(812\) 14.1728 0.497369
\(813\) 0 0
\(814\) −30.7536 −1.07791
\(815\) 12.5778 0.440581
\(816\) 0 0
\(817\) 35.9071 1.25623
\(818\) −34.0586 −1.19083
\(819\) 0 0
\(820\) −5.61356 −0.196034
\(821\) 45.9982 1.60535 0.802674 0.596418i \(-0.203410\pi\)
0.802674 + 0.596418i \(0.203410\pi\)
\(822\) 0 0
\(823\) 7.95300 0.277224 0.138612 0.990347i \(-0.455736\pi\)
0.138612 + 0.990347i \(0.455736\pi\)
\(824\) −21.6093 −0.752794
\(825\) 0 0
\(826\) −56.2030 −1.95555
\(827\) 27.9648 0.972432 0.486216 0.873839i \(-0.338377\pi\)
0.486216 + 0.873839i \(0.338377\pi\)
\(828\) 0 0
\(829\) 27.6310 0.959665 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(830\) 25.0780 0.870470
\(831\) 0 0
\(832\) 0 0
\(833\) −14.8388 −0.514133
\(834\) 0 0
\(835\) 34.4698 1.19288
\(836\) −45.5773 −1.57632
\(837\) 0 0
\(838\) −80.2699 −2.77288
\(839\) 28.6848 0.990311 0.495155 0.868804i \(-0.335111\pi\)
0.495155 + 0.868804i \(0.335111\pi\)
\(840\) 0 0
\(841\) −23.8528 −0.822509
\(842\) −78.7488 −2.71386
\(843\) 0 0
\(844\) −40.3817 −1.38999
\(845\) 0 0
\(846\) 0 0
\(847\) 9.16315 0.314849
\(848\) 4.42758 0.152044
\(849\) 0 0
\(850\) 34.6504 1.18850
\(851\) −10.1250 −0.347080
\(852\) 0 0
\(853\) 43.2078 1.47941 0.739703 0.672934i \(-0.234966\pi\)
0.739703 + 0.672934i \(0.234966\pi\)
\(854\) 39.4306 1.34929
\(855\) 0 0
\(856\) −16.2620 −0.555825
\(857\) −35.1685 −1.20133 −0.600667 0.799499i \(-0.705098\pi\)
−0.600667 + 0.799499i \(0.705098\pi\)
\(858\) 0 0
\(859\) 27.3793 0.934168 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(860\) −27.0388 −0.922014
\(861\) 0 0
\(862\) 77.0297 2.62364
\(863\) −41.3913 −1.40898 −0.704489 0.709715i \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(864\) 0 0
\(865\) −27.2465 −0.926409
\(866\) −30.8702 −1.04901
\(867\) 0 0
\(868\) −26.6668 −0.905130
\(869\) 40.3116 1.36748
\(870\) 0 0
\(871\) 0 0
\(872\) 0.287536 0.00973721
\(873\) 0 0
\(874\) −24.8485 −0.840512
\(875\) −23.4252 −0.791916
\(876\) 0 0
\(877\) 24.7472 0.835653 0.417826 0.908527i \(-0.362792\pi\)
0.417826 + 0.908527i \(0.362792\pi\)
\(878\) −23.0097 −0.776539
\(879\) 0 0
\(880\) −2.96077 −0.0998076
\(881\) 28.5875 0.963137 0.481568 0.876409i \(-0.340067\pi\)
0.481568 + 0.876409i \(0.340067\pi\)
\(882\) 0 0
\(883\) 9.61702 0.323639 0.161819 0.986820i \(-0.448264\pi\)
0.161819 + 0.986820i \(0.448264\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 27.3599 0.919173
\(887\) −15.9661 −0.536091 −0.268045 0.963406i \(-0.586378\pi\)
−0.268045 + 0.963406i \(0.586378\pi\)
\(888\) 0 0
\(889\) 38.8689 1.30362
\(890\) 21.4741 0.719814
\(891\) 0 0
\(892\) 22.3642 0.748809
\(893\) −17.2664 −0.577797
\(894\) 0 0
\(895\) 8.70171 0.290866
\(896\) −33.3196 −1.11313
\(897\) 0 0
\(898\) 28.9976 0.967663
\(899\) −9.68473 −0.323004
\(900\) 0 0
\(901\) −29.2392 −0.974099
\(902\) 7.31468 0.243552
\(903\) 0 0
\(904\) 17.2241 0.572867
\(905\) 6.90408 0.229500
\(906\) 0 0
\(907\) −28.8364 −0.957496 −0.478748 0.877952i \(-0.658909\pi\)
−0.478748 + 0.877952i \(0.658909\pi\)
\(908\) −26.4480 −0.877709
\(909\) 0 0
\(910\) 0 0
\(911\) −38.5633 −1.27766 −0.638830 0.769348i \(-0.720581\pi\)
−0.638830 + 0.769348i \(0.720581\pi\)
\(912\) 0 0
\(913\) −19.7332 −0.653073
\(914\) 10.4644 0.346132
\(915\) 0 0
\(916\) 41.7294 1.37878
\(917\) 6.67264 0.220350
\(918\) 0 0
\(919\) 8.87502 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(920\) 6.43727 0.212231
\(921\) 0 0
\(922\) 70.8708 2.33401
\(923\) 0 0
\(924\) 0 0
\(925\) 15.5985 0.512875
\(926\) 39.6601 1.30331
\(927\) 0 0
\(928\) −14.7826 −0.485261
\(929\) 24.2295 0.794945 0.397472 0.917614i \(-0.369887\pi\)
0.397472 + 0.917614i \(0.369887\pi\)
\(930\) 0 0
\(931\) −16.3937 −0.537283
\(932\) 15.5133 0.508156
\(933\) 0 0
\(934\) −72.2911 −2.36544
\(935\) 19.5526 0.639437
\(936\) 0 0
\(937\) 17.2644 0.564005 0.282002 0.959414i \(-0.409001\pi\)
0.282002 + 0.959414i \(0.409001\pi\)
\(938\) −2.65519 −0.0866949
\(939\) 0 0
\(940\) 13.0019 0.424076
\(941\) 4.34050 0.141496 0.0707482 0.997494i \(-0.477461\pi\)
0.0707482 + 0.997494i \(0.477461\pi\)
\(942\) 0 0
\(943\) 2.40821 0.0784220
\(944\) 9.78986 0.318633
\(945\) 0 0
\(946\) 35.2325 1.14551
\(947\) −45.0146 −1.46278 −0.731389 0.681961i \(-0.761128\pi\)
−0.731389 + 0.681961i \(0.761128\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 38.2814 1.24201
\(951\) 0 0
\(952\) 25.5743 0.828869
\(953\) 46.8859 1.51878 0.759391 0.650634i \(-0.225497\pi\)
0.759391 + 0.650634i \(0.225497\pi\)
\(954\) 0 0
\(955\) 26.6329 0.861822
\(956\) −33.3062 −1.07720
\(957\) 0 0
\(958\) 78.4191 2.53361
\(959\) −1.62325 −0.0524176
\(960\) 0 0
\(961\) −12.7778 −0.412186
\(962\) 0 0
\(963\) 0 0
\(964\) −36.3274 −1.17003
\(965\) 8.74871 0.281631
\(966\) 0 0
\(967\) −6.29457 −0.202420 −0.101210 0.994865i \(-0.532271\pi\)
−0.101210 + 0.994865i \(0.532271\pi\)
\(968\) 10.5405 0.338784
\(969\) 0 0
\(970\) −38.7265 −1.24343
\(971\) 41.8068 1.34165 0.670823 0.741618i \(-0.265941\pi\)
0.670823 + 0.741618i \(0.265941\pi\)
\(972\) 0 0
\(973\) 23.2336 0.744834
\(974\) 94.0025 3.01203
\(975\) 0 0
\(976\) −6.86831 −0.219849
\(977\) 23.7530 0.759926 0.379963 0.925002i \(-0.375937\pi\)
0.379963 + 0.925002i \(0.375937\pi\)
\(978\) 0 0
\(979\) −16.8974 −0.540043
\(980\) 12.3448 0.394341
\(981\) 0 0
\(982\) 49.0863 1.56641
\(983\) −55.7251 −1.77736 −0.888678 0.458532i \(-0.848375\pi\)
−0.888678 + 0.458532i \(0.848375\pi\)
\(984\) 0 0
\(985\) 16.4959 0.525602
\(986\) 26.9976 0.859779
\(987\) 0 0
\(988\) 0 0
\(989\) 11.5996 0.368845
\(990\) 0 0
\(991\) −35.5512 −1.12932 −0.564661 0.825323i \(-0.690993\pi\)
−0.564661 + 0.825323i \(0.690993\pi\)
\(992\) 27.8140 0.883096
\(993\) 0 0
\(994\) −21.1511 −0.670871
\(995\) 20.0935 0.637007
\(996\) 0 0
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) 52.9101 1.67484
\(999\) 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.a.o.1.1 3
3.2 odd 2 169.2.a.c.1.3 yes 3
12.11 even 2 2704.2.a.ba.1.2 3
13.5 odd 4 1521.2.b.l.1351.6 6
13.8 odd 4 1521.2.b.l.1351.1 6
13.12 even 2 1521.2.a.r.1.3 3
15.14 odd 2 4225.2.a.bb.1.1 3
21.20 even 2 8281.2.a.bj.1.3 3
39.2 even 12 169.2.e.b.147.6 12
39.5 even 4 169.2.b.b.168.1 6
39.8 even 4 169.2.b.b.168.6 6
39.11 even 12 169.2.e.b.147.1 12
39.17 odd 6 169.2.c.c.146.3 6
39.20 even 12 169.2.e.b.23.6 12
39.23 odd 6 169.2.c.c.22.3 6
39.29 odd 6 169.2.c.b.22.1 6
39.32 even 12 169.2.e.b.23.1 12
39.35 odd 6 169.2.c.b.146.1 6
39.38 odd 2 169.2.a.b.1.1 3
156.47 odd 4 2704.2.f.o.337.4 6
156.83 odd 4 2704.2.f.o.337.3 6
156.155 even 2 2704.2.a.z.1.2 3
195.194 odd 2 4225.2.a.bg.1.3 3
273.272 even 2 8281.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 39.38 odd 2
169.2.a.c.1.3 yes 3 3.2 odd 2
169.2.b.b.168.1 6 39.5 even 4
169.2.b.b.168.6 6 39.8 even 4
169.2.c.b.22.1 6 39.29 odd 6
169.2.c.b.146.1 6 39.35 odd 6
169.2.c.c.22.3 6 39.23 odd 6
169.2.c.c.146.3 6 39.17 odd 6
169.2.e.b.23.1 12 39.32 even 12
169.2.e.b.23.6 12 39.20 even 12
169.2.e.b.147.1 12 39.11 even 12
169.2.e.b.147.6 12 39.2 even 12
1521.2.a.o.1.1 3 1.1 even 1 trivial
1521.2.a.r.1.3 3 13.12 even 2
1521.2.b.l.1351.1 6 13.8 odd 4
1521.2.b.l.1351.6 6 13.5 odd 4
2704.2.a.z.1.2 3 156.155 even 2
2704.2.a.ba.1.2 3 12.11 even 2
2704.2.f.o.337.3 6 156.83 odd 4
2704.2.f.o.337.4 6 156.47 odd 4
4225.2.a.bb.1.1 3 15.14 odd 2
4225.2.a.bg.1.3 3 195.194 odd 2
8281.2.a.bf.1.1 3 273.272 even 2
8281.2.a.bj.1.3 3 21.20 even 2