Properties

Label 4005.2.a.j.1.1
Level $4005$
Weight $2$
Character 4005.1
Self dual yes
Analytic conductor $31.980$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
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 1335)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 4005.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.801938 q^{2} -1.35690 q^{4} -1.00000 q^{5} -3.24698 q^{7} +2.69202 q^{8} +O(q^{10})\) \(q-0.801938 q^{2} -1.35690 q^{4} -1.00000 q^{5} -3.24698 q^{7} +2.69202 q^{8} +0.801938 q^{10} -2.49396 q^{11} -1.55496 q^{13} +2.60388 q^{14} +0.554958 q^{16} +6.29590 q^{17} -4.89008 q^{19} +1.35690 q^{20} +2.00000 q^{22} +1.10992 q^{23} +1.00000 q^{25} +1.24698 q^{26} +4.40581 q^{28} +9.40581 q^{29} +0.713792 q^{31} -5.82908 q^{32} -5.04892 q^{34} +3.24698 q^{35} +6.29590 q^{37} +3.92154 q^{38} -2.69202 q^{40} +4.53319 q^{41} -2.19806 q^{43} +3.38404 q^{44} -0.890084 q^{46} -2.53319 q^{47} +3.54288 q^{49} -0.801938 q^{50} +2.10992 q^{52} -3.56465 q^{53} +2.49396 q^{55} -8.74094 q^{56} -7.54288 q^{58} +14.2446 q^{59} -8.49396 q^{61} -0.572417 q^{62} +3.56465 q^{64} +1.55496 q^{65} -4.09783 q^{67} -8.54288 q^{68} -2.60388 q^{70} +9.08575 q^{71} -1.72587 q^{73} -5.04892 q^{74} +6.63533 q^{76} +8.09783 q^{77} -2.79656 q^{79} -0.554958 q^{80} -3.63533 q^{82} -17.7560 q^{83} -6.29590 q^{85} +1.76271 q^{86} -6.71379 q^{88} +1.00000 q^{89} +5.04892 q^{91} -1.50604 q^{92} +2.03146 q^{94} +4.89008 q^{95} +13.9758 q^{97} -2.84117 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 3 q^{5} - 5 q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{11} - 5 q^{13} - q^{14} + 2 q^{16} + 5 q^{17} - 14 q^{19} + 6 q^{22} + 4 q^{23} + 3 q^{25} - q^{26} + 15 q^{29} - 6 q^{31} - 7 q^{32} - 6 q^{34} + 5 q^{35} + 5 q^{37} - 14 q^{38} - 3 q^{40} + 17 q^{41} - 11 q^{43} - 2 q^{46} - 11 q^{47} - 8 q^{49} + 2 q^{50} + 7 q^{52} + 11 q^{53} - 2 q^{55} - 12 q^{56} - 4 q^{58} - 3 q^{59} - 16 q^{61} - 18 q^{62} - 11 q^{64} + 5 q^{65} + 6 q^{67} - 7 q^{68} + q^{70} - 10 q^{71} - 16 q^{73} - 6 q^{74} - 14 q^{76} + 6 q^{77} - 7 q^{79} - 2 q^{80} + 23 q^{82} - 14 q^{83} - 5 q^{85} - 12 q^{86} - 12 q^{88} + 3 q^{89} + 6 q^{91} - 14 q^{92} - 19 q^{94} + 14 q^{95} + 4 q^{97} - 17 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.801938 −0.567056 −0.283528 0.958964i \(-0.591505\pi\)
−0.283528 + 0.958964i \(0.591505\pi\)
\(3\) 0 0
\(4\) −1.35690 −0.678448
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.24698 −1.22724 −0.613621 0.789600i \(-0.710288\pi\)
−0.613621 + 0.789600i \(0.710288\pi\)
\(8\) 2.69202 0.951773
\(9\) 0 0
\(10\) 0.801938 0.253595
\(11\) −2.49396 −0.751957 −0.375978 0.926628i \(-0.622693\pi\)
−0.375978 + 0.926628i \(0.622693\pi\)
\(12\) 0 0
\(13\) −1.55496 −0.431268 −0.215634 0.976474i \(-0.569182\pi\)
−0.215634 + 0.976474i \(0.569182\pi\)
\(14\) 2.60388 0.695915
\(15\) 0 0
\(16\) 0.554958 0.138740
\(17\) 6.29590 1.52698 0.763490 0.645820i \(-0.223484\pi\)
0.763490 + 0.645820i \(0.223484\pi\)
\(18\) 0 0
\(19\) −4.89008 −1.12186 −0.560931 0.827863i \(-0.689557\pi\)
−0.560931 + 0.827863i \(0.689557\pi\)
\(20\) 1.35690 0.303411
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.10992 0.231434 0.115717 0.993282i \(-0.463084\pi\)
0.115717 + 0.993282i \(0.463084\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.24698 0.244553
\(27\) 0 0
\(28\) 4.40581 0.832620
\(29\) 9.40581 1.74662 0.873308 0.487169i \(-0.161970\pi\)
0.873308 + 0.487169i \(0.161970\pi\)
\(30\) 0 0
\(31\) 0.713792 0.128201 0.0641004 0.997943i \(-0.479582\pi\)
0.0641004 + 0.997943i \(0.479582\pi\)
\(32\) −5.82908 −1.03045
\(33\) 0 0
\(34\) −5.04892 −0.865882
\(35\) 3.24698 0.548840
\(36\) 0 0
\(37\) 6.29590 1.03504 0.517520 0.855671i \(-0.326855\pi\)
0.517520 + 0.855671i \(0.326855\pi\)
\(38\) 3.92154 0.636158
\(39\) 0 0
\(40\) −2.69202 −0.425646
\(41\) 4.53319 0.707965 0.353983 0.935252i \(-0.384827\pi\)
0.353983 + 0.935252i \(0.384827\pi\)
\(42\) 0 0
\(43\) −2.19806 −0.335201 −0.167601 0.985855i \(-0.553602\pi\)
−0.167601 + 0.985855i \(0.553602\pi\)
\(44\) 3.38404 0.510164
\(45\) 0 0
\(46\) −0.890084 −0.131236
\(47\) −2.53319 −0.369503 −0.184752 0.982785i \(-0.559148\pi\)
−0.184752 + 0.982785i \(0.559148\pi\)
\(48\) 0 0
\(49\) 3.54288 0.506125
\(50\) −0.801938 −0.113411
\(51\) 0 0
\(52\) 2.10992 0.292593
\(53\) −3.56465 −0.489642 −0.244821 0.969568i \(-0.578729\pi\)
−0.244821 + 0.969568i \(0.578729\pi\)
\(54\) 0 0
\(55\) 2.49396 0.336285
\(56\) −8.74094 −1.16806
\(57\) 0 0
\(58\) −7.54288 −0.990428
\(59\) 14.2446 1.85449 0.927244 0.374459i \(-0.122171\pi\)
0.927244 + 0.374459i \(0.122171\pi\)
\(60\) 0 0
\(61\) −8.49396 −1.08754 −0.543770 0.839234i \(-0.683004\pi\)
−0.543770 + 0.839234i \(0.683004\pi\)
\(62\) −0.572417 −0.0726970
\(63\) 0 0
\(64\) 3.56465 0.445581
\(65\) 1.55496 0.192869
\(66\) 0 0
\(67\) −4.09783 −0.500630 −0.250315 0.968164i \(-0.580534\pi\)
−0.250315 + 0.968164i \(0.580534\pi\)
\(68\) −8.54288 −1.03598
\(69\) 0 0
\(70\) −2.60388 −0.311223
\(71\) 9.08575 1.07828 0.539140 0.842216i \(-0.318749\pi\)
0.539140 + 0.842216i \(0.318749\pi\)
\(72\) 0 0
\(73\) −1.72587 −0.201998 −0.100999 0.994887i \(-0.532204\pi\)
−0.100999 + 0.994887i \(0.532204\pi\)
\(74\) −5.04892 −0.586925
\(75\) 0 0
\(76\) 6.63533 0.761125
\(77\) 8.09783 0.922834
\(78\) 0 0
\(79\) −2.79656 −0.314638 −0.157319 0.987548i \(-0.550285\pi\)
−0.157319 + 0.987548i \(0.550285\pi\)
\(80\) −0.554958 −0.0620462
\(81\) 0 0
\(82\) −3.63533 −0.401456
\(83\) −17.7560 −1.94897 −0.974487 0.224442i \(-0.927944\pi\)
−0.974487 + 0.224442i \(0.927944\pi\)
\(84\) 0 0
\(85\) −6.29590 −0.682886
\(86\) 1.76271 0.190078
\(87\) 0 0
\(88\) −6.71379 −0.715693
\(89\) 1.00000 0.106000
\(90\) 0 0
\(91\) 5.04892 0.529270
\(92\) −1.50604 −0.157016
\(93\) 0 0
\(94\) 2.03146 0.209529
\(95\) 4.89008 0.501712
\(96\) 0 0
\(97\) 13.9758 1.41903 0.709516 0.704690i \(-0.248914\pi\)
0.709516 + 0.704690i \(0.248914\pi\)
\(98\) −2.84117 −0.287001
\(99\) 0 0
\(100\) −1.35690 −0.135690
\(101\) −9.97046 −0.992098 −0.496049 0.868295i \(-0.665216\pi\)
−0.496049 + 0.868295i \(0.665216\pi\)
\(102\) 0 0
\(103\) 8.58642 0.846045 0.423022 0.906119i \(-0.360969\pi\)
0.423022 + 0.906119i \(0.360969\pi\)
\(104\) −4.18598 −0.410469
\(105\) 0 0
\(106\) 2.85862 0.277654
\(107\) −3.58211 −0.346295 −0.173148 0.984896i \(-0.555394\pi\)
−0.173148 + 0.984896i \(0.555394\pi\)
\(108\) 0 0
\(109\) 10.4209 0.998139 0.499070 0.866562i \(-0.333675\pi\)
0.499070 + 0.866562i \(0.333675\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −1.80194 −0.170267
\(113\) 2.67025 0.251196 0.125598 0.992081i \(-0.459915\pi\)
0.125598 + 0.992081i \(0.459915\pi\)
\(114\) 0 0
\(115\) −1.10992 −0.103500
\(116\) −12.7627 −1.18499
\(117\) 0 0
\(118\) −11.4233 −1.05160
\(119\) −20.4426 −1.87397
\(120\) 0 0
\(121\) −4.78017 −0.434561
\(122\) 6.81163 0.616696
\(123\) 0 0
\(124\) −0.968541 −0.0869776
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.7463 −0.953581 −0.476791 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(128\) 8.79954 0.777777
\(129\) 0 0
\(130\) −1.24698 −0.109367
\(131\) −19.7995 −1.72989 −0.864947 0.501863i \(-0.832648\pi\)
−0.864947 + 0.501863i \(0.832648\pi\)
\(132\) 0 0
\(133\) 15.8780 1.37680
\(134\) 3.28621 0.283885
\(135\) 0 0
\(136\) 16.9487 1.45334
\(137\) 15.7453 1.34521 0.672604 0.740003i \(-0.265176\pi\)
0.672604 + 0.740003i \(0.265176\pi\)
\(138\) 0 0
\(139\) −15.4765 −1.31270 −0.656350 0.754457i \(-0.727900\pi\)
−0.656350 + 0.754457i \(0.727900\pi\)
\(140\) −4.40581 −0.372359
\(141\) 0 0
\(142\) −7.28621 −0.611445
\(143\) 3.87800 0.324295
\(144\) 0 0
\(145\) −9.40581 −0.781110
\(146\) 1.38404 0.114544
\(147\) 0 0
\(148\) −8.54288 −0.702220
\(149\) −4.41550 −0.361732 −0.180866 0.983508i \(-0.557890\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(150\) 0 0
\(151\) −12.5918 −1.02471 −0.512353 0.858775i \(-0.671226\pi\)
−0.512353 + 0.858775i \(0.671226\pi\)
\(152\) −13.1642 −1.06776
\(153\) 0 0
\(154\) −6.49396 −0.523298
\(155\) −0.713792 −0.0573331
\(156\) 0 0
\(157\) 22.9095 1.82837 0.914187 0.405293i \(-0.132830\pi\)
0.914187 + 0.405293i \(0.132830\pi\)
\(158\) 2.24267 0.178417
\(159\) 0 0
\(160\) 5.82908 0.460830
\(161\) −3.60388 −0.284025
\(162\) 0 0
\(163\) 12.0489 0.943744 0.471872 0.881667i \(-0.343578\pi\)
0.471872 + 0.881667i \(0.343578\pi\)
\(164\) −6.15106 −0.480317
\(165\) 0 0
\(166\) 14.2392 1.10518
\(167\) −1.37867 −0.106684 −0.0533422 0.998576i \(-0.516987\pi\)
−0.0533422 + 0.998576i \(0.516987\pi\)
\(168\) 0 0
\(169\) −10.5821 −0.814008
\(170\) 5.04892 0.387234
\(171\) 0 0
\(172\) 2.98254 0.227417
\(173\) −6.88471 −0.523435 −0.261717 0.965145i \(-0.584289\pi\)
−0.261717 + 0.965145i \(0.584289\pi\)
\(174\) 0 0
\(175\) −3.24698 −0.245449
\(176\) −1.38404 −0.104326
\(177\) 0 0
\(178\) −0.801938 −0.0601078
\(179\) 9.04221 0.675847 0.337923 0.941174i \(-0.390276\pi\)
0.337923 + 0.941174i \(0.390276\pi\)
\(180\) 0 0
\(181\) −26.4698 −1.96748 −0.983742 0.179586i \(-0.942524\pi\)
−0.983742 + 0.179586i \(0.942524\pi\)
\(182\) −4.04892 −0.300126
\(183\) 0 0
\(184\) 2.98792 0.220272
\(185\) −6.29590 −0.462884
\(186\) 0 0
\(187\) −15.7017 −1.14822
\(188\) 3.43727 0.250689
\(189\) 0 0
\(190\) −3.92154 −0.284499
\(191\) −19.0368 −1.37746 −0.688729 0.725019i \(-0.741831\pi\)
−0.688729 + 0.725019i \(0.741831\pi\)
\(192\) 0 0
\(193\) −11.5646 −0.832441 −0.416221 0.909264i \(-0.636646\pi\)
−0.416221 + 0.909264i \(0.636646\pi\)
\(194\) −11.2078 −0.804670
\(195\) 0 0
\(196\) −4.80731 −0.343380
\(197\) 17.8538 1.27203 0.636017 0.771675i \(-0.280581\pi\)
0.636017 + 0.771675i \(0.280581\pi\)
\(198\) 0 0
\(199\) 21.6993 1.53822 0.769112 0.639114i \(-0.220699\pi\)
0.769112 + 0.639114i \(0.220699\pi\)
\(200\) 2.69202 0.190355
\(201\) 0 0
\(202\) 7.99569 0.562575
\(203\) −30.5405 −2.14352
\(204\) 0 0
\(205\) −4.53319 −0.316612
\(206\) −6.88577 −0.479754
\(207\) 0 0
\(208\) −0.862937 −0.0598339
\(209\) 12.1957 0.843592
\(210\) 0 0
\(211\) 9.70171 0.667893 0.333947 0.942592i \(-0.391619\pi\)
0.333947 + 0.942592i \(0.391619\pi\)
\(212\) 4.83685 0.332197
\(213\) 0 0
\(214\) 2.87263 0.196369
\(215\) 2.19806 0.149907
\(216\) 0 0
\(217\) −2.31767 −0.157334
\(218\) −8.35690 −0.566000
\(219\) 0 0
\(220\) −3.38404 −0.228152
\(221\) −9.78986 −0.658537
\(222\) 0 0
\(223\) −2.53750 −0.169924 −0.0849618 0.996384i \(-0.527077\pi\)
−0.0849618 + 0.996384i \(0.527077\pi\)
\(224\) 18.9269 1.26461
\(225\) 0 0
\(226\) −2.14138 −0.142442
\(227\) −13.4668 −0.893824 −0.446912 0.894578i \(-0.647476\pi\)
−0.446912 + 0.894578i \(0.647476\pi\)
\(228\) 0 0
\(229\) −13.2513 −0.875670 −0.437835 0.899055i \(-0.644255\pi\)
−0.437835 + 0.899055i \(0.644255\pi\)
\(230\) 0.890084 0.0586904
\(231\) 0 0
\(232\) 25.3207 1.66238
\(233\) −6.98254 −0.457442 −0.228721 0.973492i \(-0.573454\pi\)
−0.228721 + 0.973492i \(0.573454\pi\)
\(234\) 0 0
\(235\) 2.53319 0.165247
\(236\) −19.3284 −1.25817
\(237\) 0 0
\(238\) 16.3937 1.06265
\(239\) −18.1129 −1.17163 −0.585813 0.810446i \(-0.699225\pi\)
−0.585813 + 0.810446i \(0.699225\pi\)
\(240\) 0 0
\(241\) 3.70171 0.238448 0.119224 0.992867i \(-0.461959\pi\)
0.119224 + 0.992867i \(0.461959\pi\)
\(242\) 3.83340 0.246420
\(243\) 0 0
\(244\) 11.5254 0.737839
\(245\) −3.54288 −0.226346
\(246\) 0 0
\(247\) 7.60388 0.483823
\(248\) 1.92154 0.122018
\(249\) 0 0
\(250\) 0.801938 0.0507190
\(251\) 1.42758 0.0901083 0.0450541 0.998985i \(-0.485654\pi\)
0.0450541 + 0.998985i \(0.485654\pi\)
\(252\) 0 0
\(253\) −2.76809 −0.174028
\(254\) 8.61788 0.540734
\(255\) 0 0
\(256\) −14.1860 −0.886624
\(257\) −7.12498 −0.444444 −0.222222 0.974996i \(-0.571331\pi\)
−0.222222 + 0.974996i \(0.571331\pi\)
\(258\) 0 0
\(259\) −20.4426 −1.27024
\(260\) −2.10992 −0.130851
\(261\) 0 0
\(262\) 15.8780 0.980946
\(263\) −27.6209 −1.70318 −0.851588 0.524212i \(-0.824360\pi\)
−0.851588 + 0.524212i \(0.824360\pi\)
\(264\) 0 0
\(265\) 3.56465 0.218975
\(266\) −12.7332 −0.780721
\(267\) 0 0
\(268\) 5.56033 0.339652
\(269\) 21.9952 1.34107 0.670536 0.741877i \(-0.266064\pi\)
0.670536 + 0.741877i \(0.266064\pi\)
\(270\) 0 0
\(271\) −2.00431 −0.121753 −0.0608766 0.998145i \(-0.519390\pi\)
−0.0608766 + 0.998145i \(0.519390\pi\)
\(272\) 3.49396 0.211852
\(273\) 0 0
\(274\) −12.6267 −0.762807
\(275\) −2.49396 −0.150391
\(276\) 0 0
\(277\) −16.2935 −0.978982 −0.489491 0.872008i \(-0.662817\pi\)
−0.489491 + 0.872008i \(0.662817\pi\)
\(278\) 12.4112 0.744374
\(279\) 0 0
\(280\) 8.74094 0.522371
\(281\) 3.93469 0.234724 0.117362 0.993089i \(-0.462556\pi\)
0.117362 + 0.993089i \(0.462556\pi\)
\(282\) 0 0
\(283\) −9.04221 −0.537504 −0.268752 0.963209i \(-0.586611\pi\)
−0.268752 + 0.963209i \(0.586611\pi\)
\(284\) −12.3284 −0.731557
\(285\) 0 0
\(286\) −3.10992 −0.183893
\(287\) −14.7192 −0.868845
\(288\) 0 0
\(289\) 22.6383 1.33167
\(290\) 7.54288 0.442933
\(291\) 0 0
\(292\) 2.34183 0.137045
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −14.2446 −0.829352
\(296\) 16.9487 0.985123
\(297\) 0 0
\(298\) 3.54096 0.205122
\(299\) −1.72587 −0.0998098
\(300\) 0 0
\(301\) 7.13706 0.411373
\(302\) 10.0978 0.581065
\(303\) 0 0
\(304\) −2.71379 −0.155647
\(305\) 8.49396 0.486363
\(306\) 0 0
\(307\) −14.5676 −0.831419 −0.415709 0.909498i \(-0.636467\pi\)
−0.415709 + 0.909498i \(0.636467\pi\)
\(308\) −10.9879 −0.626095
\(309\) 0 0
\(310\) 0.572417 0.0325111
\(311\) −17.8237 −1.01069 −0.505345 0.862917i \(-0.668635\pi\)
−0.505345 + 0.862917i \(0.668635\pi\)
\(312\) 0 0
\(313\) −2.77910 −0.157084 −0.0785421 0.996911i \(-0.525026\pi\)
−0.0785421 + 0.996911i \(0.525026\pi\)
\(314\) −18.3720 −1.03679
\(315\) 0 0
\(316\) 3.79464 0.213465
\(317\) 30.2476 1.69887 0.849436 0.527691i \(-0.176942\pi\)
0.849436 + 0.527691i \(0.176942\pi\)
\(318\) 0 0
\(319\) −23.4577 −1.31338
\(320\) −3.56465 −0.199270
\(321\) 0 0
\(322\) 2.89008 0.161058
\(323\) −30.7875 −1.71306
\(324\) 0 0
\(325\) −1.55496 −0.0862536
\(326\) −9.66248 −0.535155
\(327\) 0 0
\(328\) 12.2034 0.673822
\(329\) 8.22521 0.453470
\(330\) 0 0
\(331\) −19.2325 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(332\) 24.0930 1.32228
\(333\) 0 0
\(334\) 1.10560 0.0604960
\(335\) 4.09783 0.223889
\(336\) 0 0
\(337\) 13.7060 0.746615 0.373307 0.927708i \(-0.378224\pi\)
0.373307 + 0.927708i \(0.378224\pi\)
\(338\) 8.48619 0.461588
\(339\) 0 0
\(340\) 8.54288 0.463303
\(341\) −1.78017 −0.0964015
\(342\) 0 0
\(343\) 11.2252 0.606104
\(344\) −5.91723 −0.319036
\(345\) 0 0
\(346\) 5.52111 0.296817
\(347\) 19.7560 1.06056 0.530279 0.847823i \(-0.322087\pi\)
0.530279 + 0.847823i \(0.322087\pi\)
\(348\) 0 0
\(349\) −17.7560 −0.950457 −0.475229 0.879862i \(-0.657635\pi\)
−0.475229 + 0.879862i \(0.657635\pi\)
\(350\) 2.60388 0.139183
\(351\) 0 0
\(352\) 14.5375 0.774851
\(353\) 3.43834 0.183004 0.0915021 0.995805i \(-0.470833\pi\)
0.0915021 + 0.995805i \(0.470833\pi\)
\(354\) 0 0
\(355\) −9.08575 −0.482222
\(356\) −1.35690 −0.0719153
\(357\) 0 0
\(358\) −7.25129 −0.383243
\(359\) 1.56465 0.0825789 0.0412895 0.999147i \(-0.486853\pi\)
0.0412895 + 0.999147i \(0.486853\pi\)
\(360\) 0 0
\(361\) 4.91292 0.258575
\(362\) 21.2271 1.11567
\(363\) 0 0
\(364\) −6.85086 −0.359082
\(365\) 1.72587 0.0903363
\(366\) 0 0
\(367\) 10.3526 0.540400 0.270200 0.962804i \(-0.412910\pi\)
0.270200 + 0.962804i \(0.412910\pi\)
\(368\) 0.615957 0.0321090
\(369\) 0 0
\(370\) 5.04892 0.262481
\(371\) 11.5743 0.600910
\(372\) 0 0
\(373\) −17.1051 −0.885670 −0.442835 0.896603i \(-0.646027\pi\)
−0.442835 + 0.896603i \(0.646027\pi\)
\(374\) 12.5918 0.651106
\(375\) 0 0
\(376\) −6.81940 −0.351684
\(377\) −14.6256 −0.753259
\(378\) 0 0
\(379\) −11.3163 −0.581281 −0.290641 0.956832i \(-0.593868\pi\)
−0.290641 + 0.956832i \(0.593868\pi\)
\(380\) −6.63533 −0.340385
\(381\) 0 0
\(382\) 15.2664 0.781095
\(383\) 16.4940 0.842802 0.421401 0.906874i \(-0.361539\pi\)
0.421401 + 0.906874i \(0.361539\pi\)
\(384\) 0 0
\(385\) −8.09783 −0.412704
\(386\) 9.27413 0.472041
\(387\) 0 0
\(388\) −18.9638 −0.962739
\(389\) −3.57540 −0.181280 −0.0906400 0.995884i \(-0.528891\pi\)
−0.0906400 + 0.995884i \(0.528891\pi\)
\(390\) 0 0
\(391\) 6.98792 0.353394
\(392\) 9.53750 0.481716
\(393\) 0 0
\(394\) −14.3177 −0.721314
\(395\) 2.79656 0.140710
\(396\) 0 0
\(397\) 9.54048 0.478823 0.239412 0.970918i \(-0.423045\pi\)
0.239412 + 0.970918i \(0.423045\pi\)
\(398\) −17.4015 −0.872258
\(399\) 0 0
\(400\) 0.554958 0.0277479
\(401\) 10.9772 0.548173 0.274087 0.961705i \(-0.411624\pi\)
0.274087 + 0.961705i \(0.411624\pi\)
\(402\) 0 0
\(403\) −1.10992 −0.0552889
\(404\) 13.5289 0.673087
\(405\) 0 0
\(406\) 24.4916 1.21550
\(407\) −15.7017 −0.778305
\(408\) 0 0
\(409\) −13.4426 −0.664696 −0.332348 0.943157i \(-0.607841\pi\)
−0.332348 + 0.943157i \(0.607841\pi\)
\(410\) 3.63533 0.179536
\(411\) 0 0
\(412\) −11.6509 −0.573997
\(413\) −46.2519 −2.27591
\(414\) 0 0
\(415\) 17.7560 0.871608
\(416\) 9.06398 0.444398
\(417\) 0 0
\(418\) −9.78017 −0.478364
\(419\) −28.7114 −1.40264 −0.701322 0.712845i \(-0.747406\pi\)
−0.701322 + 0.712845i \(0.747406\pi\)
\(420\) 0 0
\(421\) −35.7754 −1.74359 −0.871793 0.489875i \(-0.837042\pi\)
−0.871793 + 0.489875i \(0.837042\pi\)
\(422\) −7.78017 −0.378733
\(423\) 0 0
\(424\) −9.59611 −0.466028
\(425\) 6.29590 0.305396
\(426\) 0 0
\(427\) 27.5797 1.33468
\(428\) 4.86054 0.234943
\(429\) 0 0
\(430\) −1.76271 −0.0850054
\(431\) 8.67264 0.417747 0.208873 0.977943i \(-0.433020\pi\)
0.208873 + 0.977943i \(0.433020\pi\)
\(432\) 0 0
\(433\) −23.1890 −1.11439 −0.557195 0.830382i \(-0.688123\pi\)
−0.557195 + 0.830382i \(0.688123\pi\)
\(434\) 1.85862 0.0892168
\(435\) 0 0
\(436\) −14.1400 −0.677185
\(437\) −5.42758 −0.259637
\(438\) 0 0
\(439\) 5.12067 0.244396 0.122198 0.992506i \(-0.461006\pi\)
0.122198 + 0.992506i \(0.461006\pi\)
\(440\) 6.71379 0.320067
\(441\) 0 0
\(442\) 7.85086 0.373427
\(443\) 17.5743 0.834982 0.417491 0.908681i \(-0.362910\pi\)
0.417491 + 0.908681i \(0.362910\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) 2.03492 0.0963561
\(447\) 0 0
\(448\) −11.5743 −0.546836
\(449\) −30.5676 −1.44258 −0.721288 0.692635i \(-0.756449\pi\)
−0.721288 + 0.692635i \(0.756449\pi\)
\(450\) 0 0
\(451\) −11.3056 −0.532359
\(452\) −3.62325 −0.170423
\(453\) 0 0
\(454\) 10.7995 0.506848
\(455\) −5.04892 −0.236697
\(456\) 0 0
\(457\) −35.8189 −1.67554 −0.837769 0.546024i \(-0.816141\pi\)
−0.837769 + 0.546024i \(0.816141\pi\)
\(458\) 10.6267 0.496554
\(459\) 0 0
\(460\) 1.50604 0.0702195
\(461\) 4.23921 0.197440 0.0987198 0.995115i \(-0.468525\pi\)
0.0987198 + 0.995115i \(0.468525\pi\)
\(462\) 0 0
\(463\) 41.3793 1.92306 0.961529 0.274705i \(-0.0885802\pi\)
0.961529 + 0.274705i \(0.0885802\pi\)
\(464\) 5.21983 0.242325
\(465\) 0 0
\(466\) 5.59956 0.259395
\(467\) −8.93900 −0.413648 −0.206824 0.978378i \(-0.566313\pi\)
−0.206824 + 0.978378i \(0.566313\pi\)
\(468\) 0 0
\(469\) 13.3056 0.614395
\(470\) −2.03146 −0.0937042
\(471\) 0 0
\(472\) 38.3467 1.76505
\(473\) 5.48188 0.252057
\(474\) 0 0
\(475\) −4.89008 −0.224372
\(476\) 27.7385 1.27139
\(477\) 0 0
\(478\) 14.5254 0.664377
\(479\) 38.1608 1.74361 0.871805 0.489854i \(-0.162950\pi\)
0.871805 + 0.489854i \(0.162950\pi\)
\(480\) 0 0
\(481\) −9.78986 −0.446379
\(482\) −2.96854 −0.135213
\(483\) 0 0
\(484\) 6.48619 0.294827
\(485\) −13.9758 −0.634610
\(486\) 0 0
\(487\) 28.2741 1.28122 0.640611 0.767865i \(-0.278681\pi\)
0.640611 + 0.767865i \(0.278681\pi\)
\(488\) −22.8659 −1.03509
\(489\) 0 0
\(490\) 2.84117 0.128351
\(491\) 27.4034 1.23670 0.618350 0.785903i \(-0.287802\pi\)
0.618350 + 0.785903i \(0.287802\pi\)
\(492\) 0 0
\(493\) 59.2180 2.66705
\(494\) −6.09783 −0.274355
\(495\) 0 0
\(496\) 0.396125 0.0177865
\(497\) −29.5013 −1.32331
\(498\) 0 0
\(499\) −20.3284 −0.910025 −0.455013 0.890485i \(-0.650365\pi\)
−0.455013 + 0.890485i \(0.650365\pi\)
\(500\) 1.35690 0.0606822
\(501\) 0 0
\(502\) −1.14483 −0.0510964
\(503\) 21.7995 0.971994 0.485997 0.873961i \(-0.338457\pi\)
0.485997 + 0.873961i \(0.338457\pi\)
\(504\) 0 0
\(505\) 9.97046 0.443680
\(506\) 2.21983 0.0986836
\(507\) 0 0
\(508\) 14.5816 0.646955
\(509\) −42.7875 −1.89652 −0.948260 0.317493i \(-0.897159\pi\)
−0.948260 + 0.317493i \(0.897159\pi\)
\(510\) 0 0
\(511\) 5.60388 0.247901
\(512\) −6.22282 −0.275012
\(513\) 0 0
\(514\) 5.71379 0.252025
\(515\) −8.58642 −0.378363
\(516\) 0 0
\(517\) 6.31767 0.277851
\(518\) 16.3937 0.720299
\(519\) 0 0
\(520\) 4.18598 0.183567
\(521\) −27.6142 −1.20980 −0.604899 0.796302i \(-0.706787\pi\)
−0.604899 + 0.796302i \(0.706787\pi\)
\(522\) 0 0
\(523\) −18.2198 −0.796698 −0.398349 0.917234i \(-0.630417\pi\)
−0.398349 + 0.917234i \(0.630417\pi\)
\(524\) 26.8659 1.17364
\(525\) 0 0
\(526\) 22.1502 0.965795
\(527\) 4.49396 0.195760
\(528\) 0 0
\(529\) −21.7681 −0.946439
\(530\) −2.85862 −0.124171
\(531\) 0 0
\(532\) −21.5448 −0.934085
\(533\) −7.04892 −0.305323
\(534\) 0 0
\(535\) 3.58211 0.154868
\(536\) −11.0315 −0.476486
\(537\) 0 0
\(538\) −17.6388 −0.760462
\(539\) −8.83579 −0.380584
\(540\) 0 0
\(541\) 19.7560 0.849377 0.424688 0.905340i \(-0.360384\pi\)
0.424688 + 0.905340i \(0.360384\pi\)
\(542\) 1.60733 0.0690409
\(543\) 0 0
\(544\) −36.6993 −1.57347
\(545\) −10.4209 −0.446381
\(546\) 0 0
\(547\) 16.4403 0.702935 0.351467 0.936200i \(-0.385683\pi\)
0.351467 + 0.936200i \(0.385683\pi\)
\(548\) −21.3647 −0.912653
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) −45.9952 −1.95946
\(552\) 0 0
\(553\) 9.08038 0.386137
\(554\) 13.0664 0.555137
\(555\) 0 0
\(556\) 21.0000 0.890598
\(557\) 1.10992 0.0470287 0.0235143 0.999723i \(-0.492514\pi\)
0.0235143 + 0.999723i \(0.492514\pi\)
\(558\) 0 0
\(559\) 3.41789 0.144562
\(560\) 1.80194 0.0761458
\(561\) 0 0
\(562\) −3.15538 −0.133101
\(563\) 15.2078 0.640930 0.320465 0.947260i \(-0.396161\pi\)
0.320465 + 0.947260i \(0.396161\pi\)
\(564\) 0 0
\(565\) −2.67025 −0.112338
\(566\) 7.25129 0.304795
\(567\) 0 0
\(568\) 24.4590 1.02628
\(569\) 40.1312 1.68239 0.841194 0.540733i \(-0.181853\pi\)
0.841194 + 0.540733i \(0.181853\pi\)
\(570\) 0 0
\(571\) −3.84787 −0.161028 −0.0805142 0.996753i \(-0.525656\pi\)
−0.0805142 + 0.996753i \(0.525656\pi\)
\(572\) −5.26205 −0.220017
\(573\) 0 0
\(574\) 11.8039 0.492683
\(575\) 1.10992 0.0462867
\(576\) 0 0
\(577\) 4.60089 0.191538 0.0957688 0.995404i \(-0.469469\pi\)
0.0957688 + 0.995404i \(0.469469\pi\)
\(578\) −18.1545 −0.755129
\(579\) 0 0
\(580\) 12.7627 0.529943
\(581\) 57.6534 2.39187
\(582\) 0 0
\(583\) 8.89008 0.368190
\(584\) −4.64609 −0.192256
\(585\) 0 0
\(586\) 11.2271 0.463788
\(587\) 1.58450 0.0653992 0.0326996 0.999465i \(-0.489590\pi\)
0.0326996 + 0.999465i \(0.489590\pi\)
\(588\) 0 0
\(589\) −3.49050 −0.143824
\(590\) 11.4233 0.470289
\(591\) 0 0
\(592\) 3.49396 0.143601
\(593\) 23.7211 0.974108 0.487054 0.873372i \(-0.338071\pi\)
0.487054 + 0.873372i \(0.338071\pi\)
\(594\) 0 0
\(595\) 20.4426 0.838067
\(596\) 5.99138 0.245416
\(597\) 0 0
\(598\) 1.38404 0.0565977
\(599\) −25.5646 −1.04454 −0.522272 0.852779i \(-0.674915\pi\)
−0.522272 + 0.852779i \(0.674915\pi\)
\(600\) 0 0
\(601\) −28.1739 −1.14924 −0.574619 0.818421i \(-0.694850\pi\)
−0.574619 + 0.818421i \(0.694850\pi\)
\(602\) −5.72348 −0.233272
\(603\) 0 0
\(604\) 17.0858 0.695209
\(605\) 4.78017 0.194341
\(606\) 0 0
\(607\) −33.1836 −1.34688 −0.673440 0.739242i \(-0.735184\pi\)
−0.673440 + 0.739242i \(0.735184\pi\)
\(608\) 28.5047 1.15602
\(609\) 0 0
\(610\) −6.81163 −0.275795
\(611\) 3.93900 0.159355
\(612\) 0 0
\(613\) −3.42758 −0.138439 −0.0692194 0.997601i \(-0.522051\pi\)
−0.0692194 + 0.997601i \(0.522051\pi\)
\(614\) 11.6823 0.471461
\(615\) 0 0
\(616\) 21.7995 0.878329
\(617\) 19.5496 0.787037 0.393518 0.919317i \(-0.371258\pi\)
0.393518 + 0.919317i \(0.371258\pi\)
\(618\) 0 0
\(619\) −20.3811 −0.819184 −0.409592 0.912269i \(-0.634329\pi\)
−0.409592 + 0.912269i \(0.634329\pi\)
\(620\) 0.968541 0.0388975
\(621\) 0 0
\(622\) 14.2935 0.573117
\(623\) −3.24698 −0.130087
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.22867 0.0890755
\(627\) 0 0
\(628\) −31.0858 −1.24046
\(629\) 39.6383 1.58048
\(630\) 0 0
\(631\) 26.7713 1.06575 0.532875 0.846194i \(-0.321112\pi\)
0.532875 + 0.846194i \(0.321112\pi\)
\(632\) −7.52840 −0.299464
\(633\) 0 0
\(634\) −24.2567 −0.963355
\(635\) 10.7463 0.426455
\(636\) 0 0
\(637\) −5.50902 −0.218276
\(638\) 18.8116 0.744759
\(639\) 0 0
\(640\) −8.79954 −0.347833
\(641\) 30.0737 1.18784 0.593919 0.804525i \(-0.297580\pi\)
0.593919 + 0.804525i \(0.297580\pi\)
\(642\) 0 0
\(643\) −27.4470 −1.08240 −0.541201 0.840893i \(-0.682030\pi\)
−0.541201 + 0.840893i \(0.682030\pi\)
\(644\) 4.89008 0.192696
\(645\) 0 0
\(646\) 24.6896 0.971400
\(647\) −18.5483 −0.729207 −0.364604 0.931163i \(-0.618795\pi\)
−0.364604 + 0.931163i \(0.618795\pi\)
\(648\) 0 0
\(649\) −35.5254 −1.39449
\(650\) 1.24698 0.0489106
\(651\) 0 0
\(652\) −16.3491 −0.640281
\(653\) −21.6582 −0.847550 −0.423775 0.905768i \(-0.639295\pi\)
−0.423775 + 0.905768i \(0.639295\pi\)
\(654\) 0 0
\(655\) 19.7995 0.773632
\(656\) 2.51573 0.0982227
\(657\) 0 0
\(658\) −6.59611 −0.257143
\(659\) −6.90084 −0.268818 −0.134409 0.990926i \(-0.542914\pi\)
−0.134409 + 0.990926i \(0.542914\pi\)
\(660\) 0 0
\(661\) −3.03624 −0.118096 −0.0590481 0.998255i \(-0.518807\pi\)
−0.0590481 + 0.998255i \(0.518807\pi\)
\(662\) 15.4233 0.599442
\(663\) 0 0
\(664\) −47.7995 −1.85498
\(665\) −15.8780 −0.615723
\(666\) 0 0
\(667\) 10.4397 0.404225
\(668\) 1.87071 0.0723798
\(669\) 0 0
\(670\) −3.28621 −0.126957
\(671\) 21.1836 0.817783
\(672\) 0 0
\(673\) −40.0844 −1.54514 −0.772571 0.634929i \(-0.781029\pi\)
−0.772571 + 0.634929i \(0.781029\pi\)
\(674\) −10.9914 −0.423372
\(675\) 0 0
\(676\) 14.3588 0.552262
\(677\) 32.9181 1.26514 0.632572 0.774501i \(-0.281999\pi\)
0.632572 + 0.774501i \(0.281999\pi\)
\(678\) 0 0
\(679\) −45.3793 −1.74150
\(680\) −16.9487 −0.649953
\(681\) 0 0
\(682\) 1.42758 0.0546650
\(683\) 47.6883 1.82474 0.912371 0.409364i \(-0.134249\pi\)
0.912371 + 0.409364i \(0.134249\pi\)
\(684\) 0 0
\(685\) −15.7453 −0.601595
\(686\) −9.00192 −0.343695
\(687\) 0 0
\(688\) −1.21983 −0.0465057
\(689\) 5.54288 0.211167
\(690\) 0 0
\(691\) 16.7192 0.636027 0.318014 0.948086i \(-0.396984\pi\)
0.318014 + 0.948086i \(0.396984\pi\)
\(692\) 9.34183 0.355123
\(693\) 0 0
\(694\) −15.8431 −0.601395
\(695\) 15.4765 0.587057
\(696\) 0 0
\(697\) 28.5405 1.08105
\(698\) 14.2392 0.538962
\(699\) 0 0
\(700\) 4.40581 0.166524
\(701\) −26.1957 −0.989397 −0.494698 0.869065i \(-0.664721\pi\)
−0.494698 + 0.869065i \(0.664721\pi\)
\(702\) 0 0
\(703\) −30.7875 −1.16117
\(704\) −8.89008 −0.335058
\(705\) 0 0
\(706\) −2.75733 −0.103774
\(707\) 32.3739 1.21755
\(708\) 0 0
\(709\) −47.2900 −1.77601 −0.888007 0.459829i \(-0.847911\pi\)
−0.888007 + 0.459829i \(0.847911\pi\)
\(710\) 7.28621 0.273447
\(711\) 0 0
\(712\) 2.69202 0.100888
\(713\) 0.792249 0.0296700
\(714\) 0 0
\(715\) −3.87800 −0.145029
\(716\) −12.2693 −0.458527
\(717\) 0 0
\(718\) −1.25475 −0.0468268
\(719\) −47.2403 −1.76176 −0.880882 0.473335i \(-0.843050\pi\)
−0.880882 + 0.473335i \(0.843050\pi\)
\(720\) 0 0
\(721\) −27.8799 −1.03830
\(722\) −3.93986 −0.146626
\(723\) 0 0
\(724\) 35.9168 1.33484
\(725\) 9.40581 0.349323
\(726\) 0 0
\(727\) 14.5894 0.541091 0.270545 0.962707i \(-0.412796\pi\)
0.270545 + 0.962707i \(0.412796\pi\)
\(728\) 13.5918 0.503745
\(729\) 0 0
\(730\) −1.38404 −0.0512257
\(731\) −13.8388 −0.511846
\(732\) 0 0
\(733\) 8.04354 0.297095 0.148547 0.988905i \(-0.452540\pi\)
0.148547 + 0.988905i \(0.452540\pi\)
\(734\) −8.30213 −0.306437
\(735\) 0 0
\(736\) −6.46980 −0.238480
\(737\) 10.2198 0.376452
\(738\) 0 0
\(739\) −42.7633 −1.57307 −0.786537 0.617544i \(-0.788128\pi\)
−0.786537 + 0.617544i \(0.788128\pi\)
\(740\) 8.54288 0.314042
\(741\) 0 0
\(742\) −9.28190 −0.340749
\(743\) −6.06292 −0.222427 −0.111213 0.993797i \(-0.535474\pi\)
−0.111213 + 0.993797i \(0.535474\pi\)
\(744\) 0 0
\(745\) 4.41550 0.161771
\(746\) 13.7172 0.502224
\(747\) 0 0
\(748\) 21.3056 0.779009
\(749\) 11.6310 0.424988
\(750\) 0 0
\(751\) −33.4553 −1.22080 −0.610401 0.792093i \(-0.708992\pi\)
−0.610401 + 0.792093i \(0.708992\pi\)
\(752\) −1.40581 −0.0512647
\(753\) 0 0
\(754\) 11.7289 0.427140
\(755\) 12.5918 0.458262
\(756\) 0 0
\(757\) −31.4926 −1.14462 −0.572310 0.820038i \(-0.693952\pi\)
−0.572310 + 0.820038i \(0.693952\pi\)
\(758\) 9.07500 0.329619
\(759\) 0 0
\(760\) 13.1642 0.477516
\(761\) 30.9336 1.12134 0.560672 0.828038i \(-0.310543\pi\)
0.560672 + 0.828038i \(0.310543\pi\)
\(762\) 0 0
\(763\) −33.8364 −1.22496
\(764\) 25.8310 0.934533
\(765\) 0 0
\(766\) −13.2271 −0.477916
\(767\) −22.1497 −0.799781
\(768\) 0 0
\(769\) 36.8092 1.32737 0.663687 0.748010i \(-0.268991\pi\)
0.663687 + 0.748010i \(0.268991\pi\)
\(770\) 6.49396 0.234026
\(771\) 0 0
\(772\) 15.6920 0.564768
\(773\) −40.1280 −1.44330 −0.721651 0.692257i \(-0.756617\pi\)
−0.721651 + 0.692257i \(0.756617\pi\)
\(774\) 0 0
\(775\) 0.713792 0.0256402
\(776\) 37.6233 1.35060
\(777\) 0 0
\(778\) 2.86725 0.102796
\(779\) −22.1677 −0.794239
\(780\) 0 0
\(781\) −22.6595 −0.810821
\(782\) −5.60388 −0.200394
\(783\) 0 0
\(784\) 1.96615 0.0702196
\(785\) −22.9095 −0.817674
\(786\) 0 0
\(787\) −26.4547 −0.943009 −0.471505 0.881864i \(-0.656289\pi\)
−0.471505 + 0.881864i \(0.656289\pi\)
\(788\) −24.2258 −0.863008
\(789\) 0 0
\(790\) −2.24267 −0.0797905
\(791\) −8.67025 −0.308279
\(792\) 0 0
\(793\) 13.2078 0.469021
\(794\) −7.65087 −0.271519
\(795\) 0 0
\(796\) −29.4437 −1.04360
\(797\) 18.8847 0.668931 0.334465 0.942408i \(-0.391444\pi\)
0.334465 + 0.942408i \(0.391444\pi\)
\(798\) 0 0
\(799\) −15.9487 −0.564224
\(800\) −5.82908 −0.206089
\(801\) 0 0
\(802\) −8.80300 −0.310845
\(803\) 4.30426 0.151894
\(804\) 0 0
\(805\) 3.60388 0.127020
\(806\) 0.890084 0.0313519
\(807\) 0 0
\(808\) −26.8407 −0.944252
\(809\) −21.8974 −0.769871 −0.384935 0.922944i \(-0.625776\pi\)
−0.384935 + 0.922944i \(0.625776\pi\)
\(810\) 0 0
\(811\) 25.4118 0.892328 0.446164 0.894951i \(-0.352790\pi\)
0.446164 + 0.894951i \(0.352790\pi\)
\(812\) 41.4403 1.45427
\(813\) 0 0
\(814\) 12.5918 0.441342
\(815\) −12.0489 −0.422055
\(816\) 0 0
\(817\) 10.7487 0.376050
\(818\) 10.7802 0.376920
\(819\) 0 0
\(820\) 6.15106 0.214804
\(821\) −15.4228 −0.538259 −0.269130 0.963104i \(-0.586736\pi\)
−0.269130 + 0.963104i \(0.586736\pi\)
\(822\) 0 0
\(823\) 6.63533 0.231293 0.115647 0.993290i \(-0.463106\pi\)
0.115647 + 0.993290i \(0.463106\pi\)
\(824\) 23.1148 0.805243
\(825\) 0 0
\(826\) 37.0911 1.29057
\(827\) −8.22462 −0.285998 −0.142999 0.989723i \(-0.545675\pi\)
−0.142999 + 0.989723i \(0.545675\pi\)
\(828\) 0 0
\(829\) −13.0315 −0.452601 −0.226301 0.974058i \(-0.572663\pi\)
−0.226301 + 0.974058i \(0.572663\pi\)
\(830\) −14.2392 −0.494250
\(831\) 0 0
\(832\) −5.54288 −0.192165
\(833\) 22.3056 0.772843
\(834\) 0 0
\(835\) 1.37867 0.0477107
\(836\) −16.5483 −0.572333
\(837\) 0 0
\(838\) 23.0248 0.795377
\(839\) 33.9861 1.17333 0.586666 0.809829i \(-0.300440\pi\)
0.586666 + 0.809829i \(0.300440\pi\)
\(840\) 0 0
\(841\) 59.4693 2.05067
\(842\) 28.6896 0.988710
\(843\) 0 0
\(844\) −13.1642 −0.453131
\(845\) 10.5821 0.364035
\(846\) 0 0
\(847\) 15.5211 0.533312
\(848\) −1.97823 −0.0679327
\(849\) 0 0
\(850\) −5.04892 −0.173176
\(851\) 6.98792 0.239543
\(852\) 0 0
\(853\) 11.0398 0.377996 0.188998 0.981977i \(-0.439476\pi\)
0.188998 + 0.981977i \(0.439476\pi\)
\(854\) −22.1172 −0.756835
\(855\) 0 0
\(856\) −9.64310 −0.329595
\(857\) −38.3672 −1.31060 −0.655299 0.755370i \(-0.727457\pi\)
−0.655299 + 0.755370i \(0.727457\pi\)
\(858\) 0 0
\(859\) 30.4215 1.03797 0.518984 0.854784i \(-0.326311\pi\)
0.518984 + 0.854784i \(0.326311\pi\)
\(860\) −2.98254 −0.101704
\(861\) 0 0
\(862\) −6.95492 −0.236886
\(863\) 23.9892 0.816603 0.408302 0.912847i \(-0.366121\pi\)
0.408302 + 0.912847i \(0.366121\pi\)
\(864\) 0 0
\(865\) 6.88471 0.234087
\(866\) 18.5961 0.631921
\(867\) 0 0
\(868\) 3.14483 0.106743
\(869\) 6.97451 0.236594
\(870\) 0 0
\(871\) 6.37196 0.215906
\(872\) 28.0532 0.950002
\(873\) 0 0
\(874\) 4.35258 0.147228
\(875\) 3.24698 0.109768
\(876\) 0 0
\(877\) 38.9396 1.31490 0.657448 0.753500i \(-0.271636\pi\)
0.657448 + 0.753500i \(0.271636\pi\)
\(878\) −4.10646 −0.138586
\(879\) 0 0
\(880\) 1.38404 0.0466561
\(881\) −19.3104 −0.650583 −0.325291 0.945614i \(-0.605462\pi\)
−0.325291 + 0.945614i \(0.605462\pi\)
\(882\) 0 0
\(883\) −4.28322 −0.144142 −0.0720710 0.997400i \(-0.522961\pi\)
−0.0720710 + 0.997400i \(0.522961\pi\)
\(884\) 13.2838 0.446783
\(885\) 0 0
\(886\) −14.0935 −0.473481
\(887\) −12.3720 −0.415410 −0.207705 0.978192i \(-0.566599\pi\)
−0.207705 + 0.978192i \(0.566599\pi\)
\(888\) 0 0
\(889\) 34.8931 1.17028
\(890\) 0.801938 0.0268810
\(891\) 0 0
\(892\) 3.44312 0.115284
\(893\) 12.3875 0.414532
\(894\) 0 0
\(895\) −9.04221 −0.302248
\(896\) −28.5719 −0.954522
\(897\) 0 0
\(898\) 24.5133 0.818021
\(899\) 6.71379 0.223917
\(900\) 0 0
\(901\) −22.4426 −0.747673
\(902\) 9.06638 0.301877
\(903\) 0 0
\(904\) 7.18837 0.239082
\(905\) 26.4698 0.879886
\(906\) 0 0
\(907\) 35.4276 1.17635 0.588177 0.808732i \(-0.299846\pi\)
0.588177 + 0.808732i \(0.299846\pi\)
\(908\) 18.2731 0.606413
\(909\) 0 0
\(910\) 4.04892 0.134220
\(911\) −0.792249 −0.0262484 −0.0131242 0.999914i \(-0.504178\pi\)
−0.0131242 + 0.999914i \(0.504178\pi\)
\(912\) 0 0
\(913\) 44.2828 1.46555
\(914\) 28.7245 0.950124
\(915\) 0 0
\(916\) 17.9806 0.594096
\(917\) 64.2887 2.12300
\(918\) 0 0
\(919\) 27.1099 0.894274 0.447137 0.894466i \(-0.352444\pi\)
0.447137 + 0.894466i \(0.352444\pi\)
\(920\) −2.98792 −0.0985088
\(921\) 0 0
\(922\) −3.39958 −0.111959
\(923\) −14.1280 −0.465028
\(924\) 0 0
\(925\) 6.29590 0.207008
\(926\) −33.1836 −1.09048
\(927\) 0 0
\(928\) −54.8273 −1.79979
\(929\) 13.5013 0.442962 0.221481 0.975165i \(-0.428911\pi\)
0.221481 + 0.975165i \(0.428911\pi\)
\(930\) 0 0
\(931\) −17.3250 −0.567803
\(932\) 9.47458 0.310350
\(933\) 0 0
\(934\) 7.16852 0.234561
\(935\) 15.7017 0.513501
\(936\) 0 0
\(937\) −23.5582 −0.769613 −0.384807 0.922997i \(-0.625732\pi\)
−0.384807 + 0.922997i \(0.625732\pi\)
\(938\) −10.6703 −0.348396
\(939\) 0 0
\(940\) −3.43727 −0.112111
\(941\) 17.5579 0.572373 0.286186 0.958174i \(-0.407612\pi\)
0.286186 + 0.958174i \(0.407612\pi\)
\(942\) 0 0
\(943\) 5.03146 0.163847
\(944\) 7.90515 0.257291
\(945\) 0 0
\(946\) −4.39612 −0.142930
\(947\) 32.4510 1.05452 0.527258 0.849705i \(-0.323220\pi\)
0.527258 + 0.849705i \(0.323220\pi\)
\(948\) 0 0
\(949\) 2.68366 0.0871153
\(950\) 3.92154 0.127232
\(951\) 0 0
\(952\) −55.0320 −1.78360
\(953\) 49.0025 1.58735 0.793673 0.608344i \(-0.208166\pi\)
0.793673 + 0.608344i \(0.208166\pi\)
\(954\) 0 0
\(955\) 19.0368 0.616018
\(956\) 24.5773 0.794887
\(957\) 0 0
\(958\) −30.6025 −0.988723
\(959\) −51.1245 −1.65090
\(960\) 0 0
\(961\) −30.4905 −0.983565
\(962\) 7.85086 0.253122
\(963\) 0 0
\(964\) −5.02284 −0.161775
\(965\) 11.5646 0.372279
\(966\) 0 0
\(967\) 48.4510 1.55808 0.779040 0.626975i \(-0.215707\pi\)
0.779040 + 0.626975i \(0.215707\pi\)
\(968\) −12.8683 −0.413603
\(969\) 0 0
\(970\) 11.2078 0.359859
\(971\) −30.6305 −0.982981 −0.491491 0.870883i \(-0.663548\pi\)
−0.491491 + 0.870883i \(0.663548\pi\)
\(972\) 0 0
\(973\) 50.2519 1.61100
\(974\) −22.6741 −0.726525
\(975\) 0 0
\(976\) −4.71379 −0.150885
\(977\) −42.2629 −1.35211 −0.676055 0.736851i \(-0.736312\pi\)
−0.676055 + 0.736851i \(0.736312\pi\)
\(978\) 0 0
\(979\) −2.49396 −0.0797073
\(980\) 4.80731 0.153564
\(981\) 0 0
\(982\) −21.9758 −0.701277
\(983\) 57.1355 1.82234 0.911170 0.412030i \(-0.135180\pi\)
0.911170 + 0.412030i \(0.135180\pi\)
\(984\) 0 0
\(985\) −17.8538 −0.568871
\(986\) −47.4892 −1.51236
\(987\) 0 0
\(988\) −10.3177 −0.328249
\(989\) −2.43967 −0.0775768
\(990\) 0 0
\(991\) 26.8418 0.852657 0.426328 0.904569i \(-0.359807\pi\)
0.426328 + 0.904569i \(0.359807\pi\)
\(992\) −4.16075 −0.132104
\(993\) 0 0
\(994\) 23.6582 0.750392
\(995\) −21.6993 −0.687915
\(996\) 0 0
\(997\) 12.6595 0.400930 0.200465 0.979701i \(-0.435755\pi\)
0.200465 + 0.979701i \(0.435755\pi\)
\(998\) 16.3021 0.516035
\(999\) 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 4005.2.a.j.1.1 3
3.2 odd 2 1335.2.a.d.1.3 3
15.14 odd 2 6675.2.a.o.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1335.2.a.d.1.3 3 3.2 odd 2
4005.2.a.j.1.1 3 1.1 even 1 trivial
6675.2.a.o.1.1 3 15.14 odd 2