Properties

Label 3040.2.a.k.1.1
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.08613 q^{3} -1.00000 q^{5} +0.648061 q^{7} +6.52420 q^{9} +O(q^{10})\) \(q-3.08613 q^{3} -1.00000 q^{5} +0.648061 q^{7} +6.52420 q^{9} -1.35194 q^{11} +4.43807 q^{13} +3.08613 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} +3.35194 q^{23} +1.00000 q^{25} -10.8761 q^{27} -4.17226 q^{29} -4.17226 q^{31} +4.17226 q^{33} -0.648061 q^{35} -2.43807 q^{37} -13.6965 q^{39} +10.1723 q^{41} -6.82032 q^{43} -6.52420 q^{45} +0.648061 q^{47} -6.58002 q^{49} -6.17226 q^{51} +6.43807 q^{53} +1.35194 q^{55} -3.08613 q^{57} -10.3445 q^{59} -2.11644 q^{61} +4.22808 q^{63} -4.43807 q^{65} +12.1345 q^{67} -10.3445 q^{69} +3.82774 q^{71} -4.17226 q^{73} -3.08613 q^{75} -0.876139 q^{77} -2.00000 q^{79} +13.9926 q^{81} -9.52420 q^{83} -2.00000 q^{85} +12.8761 q^{87} +3.82774 q^{89} +2.87614 q^{91} +12.8761 q^{93} -1.00000 q^{95} +7.73419 q^{97} -8.82032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 8 q^{23} + 3 q^{25} - 14 q^{27} + 2 q^{29} + 2 q^{31} - 2 q^{33} - 4 q^{35} + 2 q^{37} - 10 q^{39} + 16 q^{41} - 8 q^{43} - 3 q^{45} + 4 q^{47} + 3 q^{49} - 4 q^{51} + 10 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{59} + 2 q^{61} - 8 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} + 26 q^{71} + 2 q^{73} - 2 q^{75} + 16 q^{77} - 6 q^{79} + 15 q^{81} - 12 q^{83} - 6 q^{85} + 20 q^{87} + 26 q^{89} - 10 q^{91} + 20 q^{93} - 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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 0
\(3\) −3.08613 −1.78178 −0.890889 0.454221i \(-0.849918\pi\)
−0.890889 + 0.454221i \(0.849918\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.648061 0.244944 0.122472 0.992472i \(-0.460918\pi\)
0.122472 + 0.992472i \(0.460918\pi\)
\(8\) 0 0
\(9\) 6.52420 2.17473
\(10\) 0 0
\(11\) −1.35194 −0.407625 −0.203813 0.979010i \(-0.565333\pi\)
−0.203813 + 0.979010i \(0.565333\pi\)
\(12\) 0 0
\(13\) 4.43807 1.23090 0.615449 0.788176i \(-0.288975\pi\)
0.615449 + 0.788176i \(0.288975\pi\)
\(14\) 0 0
\(15\) 3.08613 0.796835
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 3.35194 0.698928 0.349464 0.936950i \(-0.386364\pi\)
0.349464 + 0.936950i \(0.386364\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −10.8761 −2.09311
\(28\) 0 0
\(29\) −4.17226 −0.774769 −0.387385 0.921918i \(-0.626621\pi\)
−0.387385 + 0.921918i \(0.626621\pi\)
\(30\) 0 0
\(31\) −4.17226 −0.749360 −0.374680 0.927154i \(-0.622247\pi\)
−0.374680 + 0.927154i \(0.622247\pi\)
\(32\) 0 0
\(33\) 4.17226 0.726297
\(34\) 0 0
\(35\) −0.648061 −0.109542
\(36\) 0 0
\(37\) −2.43807 −0.400816 −0.200408 0.979713i \(-0.564227\pi\)
−0.200408 + 0.979713i \(0.564227\pi\)
\(38\) 0 0
\(39\) −13.6965 −2.19319
\(40\) 0 0
\(41\) 10.1723 1.58864 0.794320 0.607499i \(-0.207827\pi\)
0.794320 + 0.607499i \(0.207827\pi\)
\(42\) 0 0
\(43\) −6.82032 −1.04009 −0.520045 0.854139i \(-0.674085\pi\)
−0.520045 + 0.854139i \(0.674085\pi\)
\(44\) 0 0
\(45\) −6.52420 −0.972570
\(46\) 0 0
\(47\) 0.648061 0.0945294 0.0472647 0.998882i \(-0.484950\pi\)
0.0472647 + 0.998882i \(0.484950\pi\)
\(48\) 0 0
\(49\) −6.58002 −0.940002
\(50\) 0 0
\(51\) −6.17226 −0.864289
\(52\) 0 0
\(53\) 6.43807 0.884337 0.442168 0.896932i \(-0.354209\pi\)
0.442168 + 0.896932i \(0.354209\pi\)
\(54\) 0 0
\(55\) 1.35194 0.182295
\(56\) 0 0
\(57\) −3.08613 −0.408768
\(58\) 0 0
\(59\) −10.3445 −1.34674 −0.673371 0.739305i \(-0.735154\pi\)
−0.673371 + 0.739305i \(0.735154\pi\)
\(60\) 0 0
\(61\) −2.11644 −0.270983 −0.135491 0.990779i \(-0.543261\pi\)
−0.135491 + 0.990779i \(0.543261\pi\)
\(62\) 0 0
\(63\) 4.22808 0.532688
\(64\) 0 0
\(65\) −4.43807 −0.550475
\(66\) 0 0
\(67\) 12.1345 1.48247 0.741234 0.671246i \(-0.234241\pi\)
0.741234 + 0.671246i \(0.234241\pi\)
\(68\) 0 0
\(69\) −10.3445 −1.24533
\(70\) 0 0
\(71\) 3.82774 0.454269 0.227135 0.973863i \(-0.427064\pi\)
0.227135 + 0.973863i \(0.427064\pi\)
\(72\) 0 0
\(73\) −4.17226 −0.488326 −0.244163 0.969734i \(-0.578513\pi\)
−0.244163 + 0.969734i \(0.578513\pi\)
\(74\) 0 0
\(75\) −3.08613 −0.356356
\(76\) 0 0
\(77\) −0.876139 −0.0998453
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 13.9926 1.55473
\(82\) 0 0
\(83\) −9.52420 −1.04542 −0.522708 0.852512i \(-0.675078\pi\)
−0.522708 + 0.852512i \(0.675078\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 12.8761 1.38047
\(88\) 0 0
\(89\) 3.82774 0.405740 0.202870 0.979206i \(-0.434973\pi\)
0.202870 + 0.979206i \(0.434973\pi\)
\(90\) 0 0
\(91\) 2.87614 0.301501
\(92\) 0 0
\(93\) 12.8761 1.33519
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 7.73419 0.785288 0.392644 0.919691i \(-0.371560\pi\)
0.392644 + 0.919691i \(0.371560\pi\)
\(98\) 0 0
\(99\) −8.82032 −0.886476
\(100\) 0 0
\(101\) −1.35194 −0.134523 −0.0672615 0.997735i \(-0.521426\pi\)
−0.0672615 + 0.997735i \(0.521426\pi\)
\(102\) 0 0
\(103\) 3.79001 0.373441 0.186720 0.982413i \(-0.440214\pi\)
0.186720 + 0.982413i \(0.440214\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 8.55451 0.826996 0.413498 0.910505i \(-0.364307\pi\)
0.413498 + 0.910505i \(0.364307\pi\)
\(108\) 0 0
\(109\) −7.04840 −0.675114 −0.337557 0.941305i \(-0.609601\pi\)
−0.337557 + 0.941305i \(0.609601\pi\)
\(110\) 0 0
\(111\) 7.52420 0.714165
\(112\) 0 0
\(113\) 18.6103 1.75071 0.875356 0.483478i \(-0.160627\pi\)
0.875356 + 0.483478i \(0.160627\pi\)
\(114\) 0 0
\(115\) −3.35194 −0.312570
\(116\) 0 0
\(117\) 28.9549 2.67688
\(118\) 0 0
\(119\) 1.29612 0.118815
\(120\) 0 0
\(121\) −9.17226 −0.833842
\(122\) 0 0
\(123\) −31.3929 −2.83060
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.79001 −0.158838 −0.0794188 0.996841i \(-0.525306\pi\)
−0.0794188 + 0.996841i \(0.525306\pi\)
\(128\) 0 0
\(129\) 21.0484 1.85321
\(130\) 0 0
\(131\) 17.2207 1.50458 0.752288 0.658834i \(-0.228950\pi\)
0.752288 + 0.658834i \(0.228950\pi\)
\(132\) 0 0
\(133\) 0.648061 0.0561940
\(134\) 0 0
\(135\) 10.8761 0.936069
\(136\) 0 0
\(137\) 10.1116 0.863895 0.431948 0.901899i \(-0.357827\pi\)
0.431948 + 0.901899i \(0.357827\pi\)
\(138\) 0 0
\(139\) 9.69646 0.822443 0.411222 0.911535i \(-0.365102\pi\)
0.411222 + 0.911535i \(0.365102\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 4.17226 0.346487
\(146\) 0 0
\(147\) 20.3068 1.67488
\(148\) 0 0
\(149\) 5.88356 0.482000 0.241000 0.970525i \(-0.422525\pi\)
0.241000 + 0.970525i \(0.422525\pi\)
\(150\) 0 0
\(151\) −18.6284 −1.51596 −0.757980 0.652278i \(-0.773813\pi\)
−0.757980 + 0.652278i \(0.773813\pi\)
\(152\) 0 0
\(153\) 13.0484 1.05490
\(154\) 0 0
\(155\) 4.17226 0.335124
\(156\) 0 0
\(157\) −17.3929 −1.38811 −0.694053 0.719924i \(-0.744177\pi\)
−0.694053 + 0.719924i \(0.744177\pi\)
\(158\) 0 0
\(159\) −19.8687 −1.57569
\(160\) 0 0
\(161\) 2.17226 0.171198
\(162\) 0 0
\(163\) −1.17968 −0.0923996 −0.0461998 0.998932i \(-0.514711\pi\)
−0.0461998 + 0.998932i \(0.514711\pi\)
\(164\) 0 0
\(165\) −4.17226 −0.324810
\(166\) 0 0
\(167\) 24.7268 1.91342 0.956708 0.291051i \(-0.0940049\pi\)
0.956708 + 0.291051i \(0.0940049\pi\)
\(168\) 0 0
\(169\) 6.69646 0.515112
\(170\) 0 0
\(171\) 6.52420 0.498918
\(172\) 0 0
\(173\) 17.2026 1.30789 0.653944 0.756543i \(-0.273113\pi\)
0.653944 + 0.756543i \(0.273113\pi\)
\(174\) 0 0
\(175\) 0.648061 0.0489888
\(176\) 0 0
\(177\) 31.9245 2.39960
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 16.0968 1.19647 0.598233 0.801322i \(-0.295870\pi\)
0.598233 + 0.801322i \(0.295870\pi\)
\(182\) 0 0
\(183\) 6.53162 0.482831
\(184\) 0 0
\(185\) 2.43807 0.179250
\(186\) 0 0
\(187\) −2.70388 −0.197727
\(188\) 0 0
\(189\) −7.04840 −0.512696
\(190\) 0 0
\(191\) −0.111635 −0.00807764 −0.00403882 0.999992i \(-0.501286\pi\)
−0.00403882 + 0.999992i \(0.501286\pi\)
\(192\) 0 0
\(193\) −6.01809 −0.433191 −0.216596 0.976261i \(-0.569495\pi\)
−0.216596 + 0.976261i \(0.569495\pi\)
\(194\) 0 0
\(195\) 13.6965 0.980824
\(196\) 0 0
\(197\) 14.7645 1.05193 0.525964 0.850507i \(-0.323705\pi\)
0.525964 + 0.850507i \(0.323705\pi\)
\(198\) 0 0
\(199\) −9.75228 −0.691321 −0.345660 0.938360i \(-0.612345\pi\)
−0.345660 + 0.938360i \(0.612345\pi\)
\(200\) 0 0
\(201\) −37.4487 −2.64143
\(202\) 0 0
\(203\) −2.70388 −0.189775
\(204\) 0 0
\(205\) −10.1723 −0.710461
\(206\) 0 0
\(207\) 21.8687 1.51998
\(208\) 0 0
\(209\) −1.35194 −0.0935156
\(210\) 0 0
\(211\) −4.87614 −0.335687 −0.167844 0.985814i \(-0.553680\pi\)
−0.167844 + 0.985814i \(0.553680\pi\)
\(212\) 0 0
\(213\) −11.8129 −0.809407
\(214\) 0 0
\(215\) 6.82032 0.465142
\(216\) 0 0
\(217\) −2.70388 −0.183551
\(218\) 0 0
\(219\) 12.8761 0.870089
\(220\) 0 0
\(221\) 8.87614 0.597074
\(222\) 0 0
\(223\) 20.8384 1.39544 0.697721 0.716369i \(-0.254197\pi\)
0.697721 + 0.716369i \(0.254197\pi\)
\(224\) 0 0
\(225\) 6.52420 0.434947
\(226\) 0 0
\(227\) 25.3552 1.68288 0.841441 0.540348i \(-0.181707\pi\)
0.841441 + 0.540348i \(0.181707\pi\)
\(228\) 0 0
\(229\) 26.0410 1.72084 0.860418 0.509589i \(-0.170202\pi\)
0.860418 + 0.509589i \(0.170202\pi\)
\(230\) 0 0
\(231\) 2.70388 0.177902
\(232\) 0 0
\(233\) −15.1090 −0.989825 −0.494913 0.868943i \(-0.664800\pi\)
−0.494913 + 0.868943i \(0.664800\pi\)
\(234\) 0 0
\(235\) −0.648061 −0.0422748
\(236\) 0 0
\(237\) 6.17226 0.400931
\(238\) 0 0
\(239\) 20.9878 1.35759 0.678793 0.734330i \(-0.262503\pi\)
0.678793 + 0.734330i \(0.262503\pi\)
\(240\) 0 0
\(241\) −8.87614 −0.571762 −0.285881 0.958265i \(-0.592286\pi\)
−0.285881 + 0.958265i \(0.592286\pi\)
\(242\) 0 0
\(243\) −10.5545 −0.677072
\(244\) 0 0
\(245\) 6.58002 0.420382
\(246\) 0 0
\(247\) 4.43807 0.282388
\(248\) 0 0
\(249\) 29.3929 1.86270
\(250\) 0 0
\(251\) −4.11164 −0.259524 −0.129762 0.991545i \(-0.541421\pi\)
−0.129762 + 0.991545i \(0.541421\pi\)
\(252\) 0 0
\(253\) −4.53162 −0.284900
\(254\) 0 0
\(255\) 6.17226 0.386522
\(256\) 0 0
\(257\) −5.37483 −0.335273 −0.167636 0.985849i \(-0.553613\pi\)
−0.167636 + 0.985849i \(0.553613\pi\)
\(258\) 0 0
\(259\) −1.58002 −0.0981775
\(260\) 0 0
\(261\) −27.2207 −1.68492
\(262\) 0 0
\(263\) 7.58482 0.467700 0.233850 0.972273i \(-0.424867\pi\)
0.233850 + 0.972273i \(0.424867\pi\)
\(264\) 0 0
\(265\) −6.43807 −0.395487
\(266\) 0 0
\(267\) −11.8129 −0.722938
\(268\) 0 0
\(269\) −2.81551 −0.171665 −0.0858324 0.996310i \(-0.527355\pi\)
−0.0858324 + 0.996310i \(0.527355\pi\)
\(270\) 0 0
\(271\) 16.7449 1.01718 0.508589 0.861009i \(-0.330167\pi\)
0.508589 + 0.861009i \(0.330167\pi\)
\(272\) 0 0
\(273\) −8.87614 −0.537208
\(274\) 0 0
\(275\) −1.35194 −0.0815250
\(276\) 0 0
\(277\) 25.0484 1.50501 0.752506 0.658585i \(-0.228845\pi\)
0.752506 + 0.658585i \(0.228845\pi\)
\(278\) 0 0
\(279\) −27.2207 −1.62966
\(280\) 0 0
\(281\) −26.8613 −1.60241 −0.801205 0.598389i \(-0.795808\pi\)
−0.801205 + 0.598389i \(0.795808\pi\)
\(282\) 0 0
\(283\) 21.5242 1.27948 0.639740 0.768591i \(-0.279042\pi\)
0.639740 + 0.768591i \(0.279042\pi\)
\(284\) 0 0
\(285\) 3.08613 0.182807
\(286\) 0 0
\(287\) 6.59224 0.389128
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −23.8687 −1.39921
\(292\) 0 0
\(293\) 18.6710 1.09077 0.545384 0.838186i \(-0.316384\pi\)
0.545384 + 0.838186i \(0.316384\pi\)
\(294\) 0 0
\(295\) 10.3445 0.602281
\(296\) 0 0
\(297\) 14.7039 0.853206
\(298\) 0 0
\(299\) 14.8761 0.860309
\(300\) 0 0
\(301\) −4.41998 −0.254764
\(302\) 0 0
\(303\) 4.17226 0.239690
\(304\) 0 0
\(305\) 2.11644 0.121187
\(306\) 0 0
\(307\) 10.3068 0.588240 0.294120 0.955769i \(-0.404974\pi\)
0.294120 + 0.955769i \(0.404974\pi\)
\(308\) 0 0
\(309\) −11.6965 −0.665388
\(310\) 0 0
\(311\) 12.4003 0.703159 0.351579 0.936158i \(-0.385645\pi\)
0.351579 + 0.936158i \(0.385645\pi\)
\(312\) 0 0
\(313\) −27.4535 −1.55177 −0.775883 0.630877i \(-0.782695\pi\)
−0.775883 + 0.630877i \(0.782695\pi\)
\(314\) 0 0
\(315\) −4.22808 −0.238225
\(316\) 0 0
\(317\) −34.1149 −1.91608 −0.958041 0.286630i \(-0.907465\pi\)
−0.958041 + 0.286630i \(0.907465\pi\)
\(318\) 0 0
\(319\) 5.64064 0.315815
\(320\) 0 0
\(321\) −26.4003 −1.47352
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 4.43807 0.246180
\(326\) 0 0
\(327\) 21.7523 1.20290
\(328\) 0 0
\(329\) 0.419983 0.0231544
\(330\) 0 0
\(331\) 16.2691 0.894228 0.447114 0.894477i \(-0.352452\pi\)
0.447114 + 0.894477i \(0.352452\pi\)
\(332\) 0 0
\(333\) −15.9065 −0.871668
\(334\) 0 0
\(335\) −12.1345 −0.662980
\(336\) 0 0
\(337\) 23.0665 1.25651 0.628256 0.778007i \(-0.283769\pi\)
0.628256 + 0.778007i \(0.283769\pi\)
\(338\) 0 0
\(339\) −57.4339 −3.11938
\(340\) 0 0
\(341\) 5.64064 0.305458
\(342\) 0 0
\(343\) −8.80068 −0.475192
\(344\) 0 0
\(345\) 10.3445 0.556930
\(346\) 0 0
\(347\) −26.4003 −1.41724 −0.708622 0.705588i \(-0.750683\pi\)
−0.708622 + 0.705588i \(0.750683\pi\)
\(348\) 0 0
\(349\) −35.4535 −1.89778 −0.948892 0.315600i \(-0.897794\pi\)
−0.948892 + 0.315600i \(0.897794\pi\)
\(350\) 0 0
\(351\) −48.2691 −2.57641
\(352\) 0 0
\(353\) −19.0336 −1.01305 −0.506527 0.862224i \(-0.669071\pi\)
−0.506527 + 0.862224i \(0.669071\pi\)
\(354\) 0 0
\(355\) −3.82774 −0.203155
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 4.16745 0.219950 0.109975 0.993934i \(-0.464923\pi\)
0.109975 + 0.993934i \(0.464923\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 28.3068 1.48572
\(364\) 0 0
\(365\) 4.17226 0.218386
\(366\) 0 0
\(367\) 34.3249 1.79174 0.895872 0.444312i \(-0.146552\pi\)
0.895872 + 0.444312i \(0.146552\pi\)
\(368\) 0 0
\(369\) 66.3659 3.45487
\(370\) 0 0
\(371\) 4.17226 0.216613
\(372\) 0 0
\(373\) −1.32905 −0.0688153 −0.0344077 0.999408i \(-0.510954\pi\)
−0.0344077 + 0.999408i \(0.510954\pi\)
\(374\) 0 0
\(375\) 3.08613 0.159367
\(376\) 0 0
\(377\) −18.5168 −0.953663
\(378\) 0 0
\(379\) −10.5922 −0.544087 −0.272043 0.962285i \(-0.587699\pi\)
−0.272043 + 0.962285i \(0.587699\pi\)
\(380\) 0 0
\(381\) 5.52420 0.283013
\(382\) 0 0
\(383\) −2.38225 −0.121727 −0.0608637 0.998146i \(-0.519385\pi\)
−0.0608637 + 0.998146i \(0.519385\pi\)
\(384\) 0 0
\(385\) 0.876139 0.0446522
\(386\) 0 0
\(387\) −44.4971 −2.26192
\(388\) 0 0
\(389\) 23.3323 1.18299 0.591497 0.806307i \(-0.298537\pi\)
0.591497 + 0.806307i \(0.298537\pi\)
\(390\) 0 0
\(391\) 6.70388 0.339030
\(392\) 0 0
\(393\) −53.1452 −2.68082
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 34.3807 1.72552 0.862759 0.505616i \(-0.168735\pi\)
0.862759 + 0.505616i \(0.168735\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) 24.3297 1.21497 0.607483 0.794333i \(-0.292179\pi\)
0.607483 + 0.794333i \(0.292179\pi\)
\(402\) 0 0
\(403\) −18.5168 −0.922387
\(404\) 0 0
\(405\) −13.9926 −0.695297
\(406\) 0 0
\(407\) 3.29612 0.163383
\(408\) 0 0
\(409\) −9.75228 −0.482219 −0.241110 0.970498i \(-0.577511\pi\)
−0.241110 + 0.970498i \(0.577511\pi\)
\(410\) 0 0
\(411\) −31.2058 −1.53927
\(412\) 0 0
\(413\) −6.70388 −0.329876
\(414\) 0 0
\(415\) 9.52420 0.467525
\(416\) 0 0
\(417\) −29.9245 −1.46541
\(418\) 0 0
\(419\) −6.51678 −0.318366 −0.159183 0.987249i \(-0.550886\pi\)
−0.159183 + 0.987249i \(0.550886\pi\)
\(420\) 0 0
\(421\) 2.40515 0.117220 0.0586098 0.998281i \(-0.481333\pi\)
0.0586098 + 0.998281i \(0.481333\pi\)
\(422\) 0 0
\(423\) 4.22808 0.205576
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −1.37158 −0.0663756
\(428\) 0 0
\(429\) 18.5168 0.893999
\(430\) 0 0
\(431\) −13.1090 −0.631439 −0.315720 0.948852i \(-0.602246\pi\)
−0.315720 + 0.948852i \(0.602246\pi\)
\(432\) 0 0
\(433\) −20.8942 −1.00411 −0.502056 0.864835i \(-0.667423\pi\)
−0.502056 + 0.864835i \(0.667423\pi\)
\(434\) 0 0
\(435\) −12.8761 −0.617364
\(436\) 0 0
\(437\) 3.35194 0.160345
\(438\) 0 0
\(439\) 41.3929 1.97558 0.987788 0.155803i \(-0.0497965\pi\)
0.987788 + 0.155803i \(0.0497965\pi\)
\(440\) 0 0
\(441\) −42.9293 −2.04425
\(442\) 0 0
\(443\) −10.0558 −0.477766 −0.238883 0.971048i \(-0.576781\pi\)
−0.238883 + 0.971048i \(0.576781\pi\)
\(444\) 0 0
\(445\) −3.82774 −0.181452
\(446\) 0 0
\(447\) −18.1574 −0.858817
\(448\) 0 0
\(449\) 12.1116 0.571583 0.285792 0.958292i \(-0.407743\pi\)
0.285792 + 0.958292i \(0.407743\pi\)
\(450\) 0 0
\(451\) −13.7523 −0.647569
\(452\) 0 0
\(453\) 57.4897 2.70110
\(454\) 0 0
\(455\) −2.87614 −0.134835
\(456\) 0 0
\(457\) −30.9123 −1.44602 −0.723008 0.690839i \(-0.757241\pi\)
−0.723008 + 0.690839i \(0.757241\pi\)
\(458\) 0 0
\(459\) −21.7523 −1.01531
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −31.0894 −1.44485 −0.722423 0.691451i \(-0.756972\pi\)
−0.722423 + 0.691451i \(0.756972\pi\)
\(464\) 0 0
\(465\) −12.8761 −0.597117
\(466\) 0 0
\(467\) 12.0410 0.557190 0.278595 0.960409i \(-0.410131\pi\)
0.278595 + 0.960409i \(0.410131\pi\)
\(468\) 0 0
\(469\) 7.86391 0.363122
\(470\) 0 0
\(471\) 53.6768 2.47330
\(472\) 0 0
\(473\) 9.22066 0.423966
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 42.0032 1.92320
\(478\) 0 0
\(479\) −10.3035 −0.470781 −0.235390 0.971901i \(-0.575637\pi\)
−0.235390 + 0.971901i \(0.575637\pi\)
\(480\) 0 0
\(481\) −10.8203 −0.493364
\(482\) 0 0
\(483\) −6.70388 −0.305037
\(484\) 0 0
\(485\) −7.73419 −0.351192
\(486\) 0 0
\(487\) 22.4939 1.01930 0.509648 0.860383i \(-0.329776\pi\)
0.509648 + 0.860383i \(0.329776\pi\)
\(488\) 0 0
\(489\) 3.64064 0.164636
\(490\) 0 0
\(491\) 35.6162 1.60734 0.803668 0.595078i \(-0.202879\pi\)
0.803668 + 0.595078i \(0.202879\pi\)
\(492\) 0 0
\(493\) −8.34452 −0.375818
\(494\) 0 0
\(495\) 8.82032 0.396444
\(496\) 0 0
\(497\) 2.48061 0.111270
\(498\) 0 0
\(499\) 0.0558176 0.00249874 0.00124937 0.999999i \(-0.499602\pi\)
0.00124937 + 0.999999i \(0.499602\pi\)
\(500\) 0 0
\(501\) −76.3100 −3.40928
\(502\) 0 0
\(503\) −23.8081 −1.06155 −0.530775 0.847513i \(-0.678099\pi\)
−0.530775 + 0.847513i \(0.678099\pi\)
\(504\) 0 0
\(505\) 1.35194 0.0601605
\(506\) 0 0
\(507\) −20.6661 −0.917816
\(508\) 0 0
\(509\) 10.5316 0.466806 0.233403 0.972380i \(-0.425014\pi\)
0.233403 + 0.972380i \(0.425014\pi\)
\(510\) 0 0
\(511\) −2.70388 −0.119613
\(512\) 0 0
\(513\) −10.8761 −0.480193
\(514\) 0 0
\(515\) −3.79001 −0.167008
\(516\) 0 0
\(517\) −0.876139 −0.0385325
\(518\) 0 0
\(519\) −53.0894 −2.33037
\(520\) 0 0
\(521\) −11.2961 −0.494892 −0.247446 0.968902i \(-0.579591\pi\)
−0.247446 + 0.968902i \(0.579591\pi\)
\(522\) 0 0
\(523\) 11.0861 0.484763 0.242381 0.970181i \(-0.422071\pi\)
0.242381 + 0.970181i \(0.422071\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) −8.34452 −0.363493
\(528\) 0 0
\(529\) −11.7645 −0.511500
\(530\) 0 0
\(531\) −67.4897 −2.92880
\(532\) 0 0
\(533\) 45.1452 1.95546
\(534\) 0 0
\(535\) −8.55451 −0.369844
\(536\) 0 0
\(537\) 18.5168 0.799058
\(538\) 0 0
\(539\) 8.89578 0.383169
\(540\) 0 0
\(541\) 16.8565 0.724717 0.362359 0.932039i \(-0.381972\pi\)
0.362359 + 0.932039i \(0.381972\pi\)
\(542\) 0 0
\(543\) −49.6768 −2.13184
\(544\) 0 0
\(545\) 7.04840 0.301920
\(546\) 0 0
\(547\) −16.6151 −0.710412 −0.355206 0.934788i \(-0.615589\pi\)
−0.355206 + 0.934788i \(0.615589\pi\)
\(548\) 0 0
\(549\) −13.8081 −0.589315
\(550\) 0 0
\(551\) −4.17226 −0.177744
\(552\) 0 0
\(553\) −1.29612 −0.0551167
\(554\) 0 0
\(555\) −7.52420 −0.319384
\(556\) 0 0
\(557\) 38.4562 1.62944 0.814720 0.579855i \(-0.196891\pi\)
0.814720 + 0.579855i \(0.196891\pi\)
\(558\) 0 0
\(559\) −30.2691 −1.28024
\(560\) 0 0
\(561\) 8.34452 0.352306
\(562\) 0 0
\(563\) −40.4184 −1.70343 −0.851717 0.524002i \(-0.824438\pi\)
−0.851717 + 0.524002i \(0.824438\pi\)
\(564\) 0 0
\(565\) −18.6103 −0.782942
\(566\) 0 0
\(567\) 9.06804 0.380822
\(568\) 0 0
\(569\) 26.9368 1.12925 0.564624 0.825348i \(-0.309021\pi\)
0.564624 + 0.825348i \(0.309021\pi\)
\(570\) 0 0
\(571\) 21.4636 0.898223 0.449111 0.893476i \(-0.351741\pi\)
0.449111 + 0.893476i \(0.351741\pi\)
\(572\) 0 0
\(573\) 0.344521 0.0143926
\(574\) 0 0
\(575\) 3.35194 0.139786
\(576\) 0 0
\(577\) −12.8613 −0.535423 −0.267712 0.963499i \(-0.586267\pi\)
−0.267712 + 0.963499i \(0.586267\pi\)
\(578\) 0 0
\(579\) 18.5726 0.771851
\(580\) 0 0
\(581\) −6.17226 −0.256069
\(582\) 0 0
\(583\) −8.70388 −0.360478
\(584\) 0 0
\(585\) −28.9549 −1.19714
\(586\) 0 0
\(587\) 13.8687 0.572423 0.286212 0.958166i \(-0.407604\pi\)
0.286212 + 0.958166i \(0.407604\pi\)
\(588\) 0 0
\(589\) −4.17226 −0.171915
\(590\) 0 0
\(591\) −45.5652 −1.87430
\(592\) 0 0
\(593\) −20.9368 −0.859770 −0.429885 0.902884i \(-0.641446\pi\)
−0.429885 + 0.902884i \(0.641446\pi\)
\(594\) 0 0
\(595\) −1.29612 −0.0531358
\(596\) 0 0
\(597\) 30.0968 1.23178
\(598\) 0 0
\(599\) 0.703878 0.0287597 0.0143798 0.999897i \(-0.495423\pi\)
0.0143798 + 0.999897i \(0.495423\pi\)
\(600\) 0 0
\(601\) 5.22066 0.212955 0.106478 0.994315i \(-0.466043\pi\)
0.106478 + 0.994315i \(0.466043\pi\)
\(602\) 0 0
\(603\) 79.1681 3.22397
\(604\) 0 0
\(605\) 9.17226 0.372905
\(606\) 0 0
\(607\) 16.1345 0.654880 0.327440 0.944872i \(-0.393814\pi\)
0.327440 + 0.944872i \(0.393814\pi\)
\(608\) 0 0
\(609\) 8.34452 0.338137
\(610\) 0 0
\(611\) 2.87614 0.116356
\(612\) 0 0
\(613\) −19.3323 −0.780824 −0.390412 0.920640i \(-0.627667\pi\)
−0.390412 + 0.920640i \(0.627667\pi\)
\(614\) 0 0
\(615\) 31.3929 1.26588
\(616\) 0 0
\(617\) 1.01223 0.0407507 0.0203753 0.999792i \(-0.493514\pi\)
0.0203753 + 0.999792i \(0.493514\pi\)
\(618\) 0 0
\(619\) 18.8809 0.758889 0.379445 0.925214i \(-0.376115\pi\)
0.379445 + 0.925214i \(0.376115\pi\)
\(620\) 0 0
\(621\) −36.4562 −1.46294
\(622\) 0 0
\(623\) 2.48061 0.0993835
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.17226 0.166624
\(628\) 0 0
\(629\) −4.87614 −0.194424
\(630\) 0 0
\(631\) 17.4636 0.695214 0.347607 0.937640i \(-0.386994\pi\)
0.347607 + 0.937640i \(0.386994\pi\)
\(632\) 0 0
\(633\) 15.0484 0.598120
\(634\) 0 0
\(635\) 1.79001 0.0710343
\(636\) 0 0
\(637\) −29.2026 −1.15705
\(638\) 0 0
\(639\) 24.9729 0.987914
\(640\) 0 0
\(641\) −22.4051 −0.884950 −0.442475 0.896781i \(-0.645899\pi\)
−0.442475 + 0.896781i \(0.645899\pi\)
\(642\) 0 0
\(643\) −22.6332 −0.892567 −0.446284 0.894892i \(-0.647253\pi\)
−0.446284 + 0.894892i \(0.647253\pi\)
\(644\) 0 0
\(645\) −21.0484 −0.828780
\(646\) 0 0
\(647\) 34.8203 1.36893 0.684464 0.729047i \(-0.260036\pi\)
0.684464 + 0.729047i \(0.260036\pi\)
\(648\) 0 0
\(649\) 13.9852 0.548966
\(650\) 0 0
\(651\) 8.34452 0.327048
\(652\) 0 0
\(653\) −23.1090 −0.904326 −0.452163 0.891935i \(-0.649347\pi\)
−0.452163 + 0.891935i \(0.649347\pi\)
\(654\) 0 0
\(655\) −17.2207 −0.672867
\(656\) 0 0
\(657\) −27.2207 −1.06198
\(658\) 0 0
\(659\) −4.24772 −0.165468 −0.0827339 0.996572i \(-0.526365\pi\)
−0.0827339 + 0.996572i \(0.526365\pi\)
\(660\) 0 0
\(661\) 8.15742 0.317287 0.158644 0.987336i \(-0.449288\pi\)
0.158644 + 0.987336i \(0.449288\pi\)
\(662\) 0 0
\(663\) −27.3929 −1.06385
\(664\) 0 0
\(665\) −0.648061 −0.0251307
\(666\) 0 0
\(667\) −13.9852 −0.541508
\(668\) 0 0
\(669\) −64.3100 −2.48637
\(670\) 0 0
\(671\) 2.86130 0.110459
\(672\) 0 0
\(673\) 39.2239 1.51197 0.755985 0.654589i \(-0.227158\pi\)
0.755985 + 0.654589i \(0.227158\pi\)
\(674\) 0 0
\(675\) −10.8761 −0.418623
\(676\) 0 0
\(677\) 32.4232 1.24613 0.623063 0.782172i \(-0.285888\pi\)
0.623063 + 0.782172i \(0.285888\pi\)
\(678\) 0 0
\(679\) 5.01223 0.192352
\(680\) 0 0
\(681\) −78.2494 −2.99852
\(682\) 0 0
\(683\) 35.6539 1.36426 0.682130 0.731231i \(-0.261054\pi\)
0.682130 + 0.731231i \(0.261054\pi\)
\(684\) 0 0
\(685\) −10.1116 −0.386346
\(686\) 0 0
\(687\) −80.3659 −3.06615
\(688\) 0 0
\(689\) 28.5726 1.08853
\(690\) 0 0
\(691\) 44.0261 1.67483 0.837417 0.546565i \(-0.184065\pi\)
0.837417 + 0.546565i \(0.184065\pi\)
\(692\) 0 0
\(693\) −5.71610 −0.217137
\(694\) 0 0
\(695\) −9.69646 −0.367808
\(696\) 0 0
\(697\) 20.3445 0.770604
\(698\) 0 0
\(699\) 46.6284 1.76365
\(700\) 0 0
\(701\) 28.9023 1.09162 0.545812 0.837908i \(-0.316221\pi\)
0.545812 + 0.837908i \(0.316221\pi\)
\(702\) 0 0
\(703\) −2.43807 −0.0919535
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) −0.876139 −0.0329506
\(708\) 0 0
\(709\) −10.6433 −0.399716 −0.199858 0.979825i \(-0.564048\pi\)
−0.199858 + 0.979825i \(0.564048\pi\)
\(710\) 0 0
\(711\) −13.0484 −0.489353
\(712\) 0 0
\(713\) −13.9852 −0.523748
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) −64.7710 −2.41892
\(718\) 0 0
\(719\) 41.4339 1.54522 0.772612 0.634879i \(-0.218950\pi\)
0.772612 + 0.634879i \(0.218950\pi\)
\(720\) 0 0
\(721\) 2.45616 0.0914720
\(722\) 0 0
\(723\) 27.3929 1.01875
\(724\) 0 0
\(725\) −4.17226 −0.154954
\(726\) 0 0
\(727\) 4.72352 0.175186 0.0875929 0.996156i \(-0.472083\pi\)
0.0875929 + 0.996156i \(0.472083\pi\)
\(728\) 0 0
\(729\) −9.40515 −0.348339
\(730\) 0 0
\(731\) −13.6406 −0.504517
\(732\) 0 0
\(733\) 17.6917 0.653456 0.326728 0.945118i \(-0.394054\pi\)
0.326728 + 0.945118i \(0.394054\pi\)
\(734\) 0 0
\(735\) −20.3068 −0.749027
\(736\) 0 0
\(737\) −16.4051 −0.604291
\(738\) 0 0
\(739\) −12.2233 −0.449640 −0.224820 0.974400i \(-0.572179\pi\)
−0.224820 + 0.974400i \(0.572179\pi\)
\(740\) 0 0
\(741\) −13.6965 −0.503152
\(742\) 0 0
\(743\) −19.6635 −0.721385 −0.360693 0.932685i \(-0.617460\pi\)
−0.360693 + 0.932685i \(0.617460\pi\)
\(744\) 0 0
\(745\) −5.88356 −0.215557
\(746\) 0 0
\(747\) −62.1378 −2.27350
\(748\) 0 0
\(749\) 5.54384 0.202568
\(750\) 0 0
\(751\) 41.6768 1.52081 0.760404 0.649450i \(-0.225001\pi\)
0.760404 + 0.649450i \(0.225001\pi\)
\(752\) 0 0
\(753\) 12.6890 0.462414
\(754\) 0 0
\(755\) 18.6284 0.677957
\(756\) 0 0
\(757\) −42.6136 −1.54882 −0.774408 0.632686i \(-0.781952\pi\)
−0.774408 + 0.632686i \(0.781952\pi\)
\(758\) 0 0
\(759\) 13.9852 0.507629
\(760\) 0 0
\(761\) −30.8809 −1.11943 −0.559717 0.828684i \(-0.689090\pi\)
−0.559717 + 0.828684i \(0.689090\pi\)
\(762\) 0 0
\(763\) −4.56779 −0.165365
\(764\) 0 0
\(765\) −13.0484 −0.471766
\(766\) 0 0
\(767\) −45.9097 −1.65770
\(768\) 0 0
\(769\) −5.12867 −0.184945 −0.0924723 0.995715i \(-0.529477\pi\)
−0.0924723 + 0.995715i \(0.529477\pi\)
\(770\) 0 0
\(771\) 16.5874 0.597382
\(772\) 0 0
\(773\) 42.0787 1.51347 0.756733 0.653724i \(-0.226794\pi\)
0.756733 + 0.653724i \(0.226794\pi\)
\(774\) 0 0
\(775\) −4.17226 −0.149872
\(776\) 0 0
\(777\) 4.87614 0.174931
\(778\) 0 0
\(779\) 10.1723 0.364459
\(780\) 0 0
\(781\) −5.17487 −0.185171
\(782\) 0 0
\(783\) 45.3781 1.62168
\(784\) 0 0
\(785\) 17.3929 0.620780
\(786\) 0 0
\(787\) −43.4307 −1.54814 −0.774068 0.633103i \(-0.781781\pi\)
−0.774068 + 0.633103i \(0.781781\pi\)
\(788\) 0 0
\(789\) −23.4078 −0.833338
\(790\) 0 0
\(791\) 12.0606 0.428826
\(792\) 0 0
\(793\) −9.39292 −0.333552
\(794\) 0 0
\(795\) 19.8687 0.704671
\(796\) 0 0
\(797\) −40.0032 −1.41699 −0.708494 0.705717i \(-0.750625\pi\)
−0.708494 + 0.705717i \(0.750625\pi\)
\(798\) 0 0
\(799\) 1.29612 0.0458535
\(800\) 0 0
\(801\) 24.9729 0.882375
\(802\) 0 0
\(803\) 5.64064 0.199054
\(804\) 0 0
\(805\) −2.17226 −0.0765621
\(806\) 0 0
\(807\) 8.68904 0.305869
\(808\) 0 0
\(809\) −28.7401 −1.01045 −0.505223 0.862989i \(-0.668590\pi\)
−0.505223 + 0.862989i \(0.668590\pi\)
\(810\) 0 0
\(811\) −16.0510 −0.563627 −0.281814 0.959469i \(-0.590936\pi\)
−0.281814 + 0.959469i \(0.590936\pi\)
\(812\) 0 0
\(813\) −51.6768 −1.81239
\(814\) 0 0
\(815\) 1.17968 0.0413223
\(816\) 0 0
\(817\) −6.82032 −0.238613
\(818\) 0 0
\(819\) 18.7645 0.655685
\(820\) 0 0
\(821\) 7.82774 0.273190 0.136595 0.990627i \(-0.456384\pi\)
0.136595 + 0.990627i \(0.456384\pi\)
\(822\) 0 0
\(823\) −9.78676 −0.341145 −0.170572 0.985345i \(-0.554562\pi\)
−0.170572 + 0.985345i \(0.554562\pi\)
\(824\) 0 0
\(825\) 4.17226 0.145259
\(826\) 0 0
\(827\) −50.1707 −1.74461 −0.872303 0.488965i \(-0.837374\pi\)
−0.872303 + 0.488965i \(0.837374\pi\)
\(828\) 0 0
\(829\) −21.2961 −0.739645 −0.369822 0.929102i \(-0.620581\pi\)
−0.369822 + 0.929102i \(0.620581\pi\)
\(830\) 0 0
\(831\) −77.3026 −2.68160
\(832\) 0 0
\(833\) −13.1600 −0.455968
\(834\) 0 0
\(835\) −24.7268 −0.855705
\(836\) 0 0
\(837\) 45.3781 1.56850
\(838\) 0 0
\(839\) −30.9516 −1.06857 −0.534284 0.845305i \(-0.679419\pi\)
−0.534284 + 0.845305i \(0.679419\pi\)
\(840\) 0 0
\(841\) −11.5922 −0.399733
\(842\) 0 0
\(843\) 82.8975 2.85514
\(844\) 0 0
\(845\) −6.69646 −0.230365
\(846\) 0 0
\(847\) −5.94418 −0.204245
\(848\) 0 0
\(849\) −66.4265 −2.27975
\(850\) 0 0
\(851\) −8.17226 −0.280141
\(852\) 0 0
\(853\) 38.9219 1.33266 0.666331 0.745656i \(-0.267864\pi\)
0.666331 + 0.745656i \(0.267864\pi\)
\(854\) 0 0
\(855\) −6.52420 −0.223123
\(856\) 0 0
\(857\) 52.1245 1.78054 0.890270 0.455434i \(-0.150516\pi\)
0.890270 + 0.455434i \(0.150516\pi\)
\(858\) 0 0
\(859\) 4.41998 0.150808 0.0754039 0.997153i \(-0.475975\pi\)
0.0754039 + 0.997153i \(0.475975\pi\)
\(860\) 0 0
\(861\) −20.3445 −0.693339
\(862\) 0 0
\(863\) 12.4184 0.422728 0.211364 0.977407i \(-0.432209\pi\)
0.211364 + 0.977407i \(0.432209\pi\)
\(864\) 0 0
\(865\) −17.2026 −0.584905
\(866\) 0 0
\(867\) 40.1197 1.36254
\(868\) 0 0
\(869\) 2.70388 0.0917228
\(870\) 0 0
\(871\) 53.8539 1.82477
\(872\) 0 0
\(873\) 50.4594 1.70779
\(874\) 0 0
\(875\) −0.648061 −0.0219085
\(876\) 0 0
\(877\) 40.3020 1.36090 0.680451 0.732794i \(-0.261784\pi\)
0.680451 + 0.732794i \(0.261784\pi\)
\(878\) 0 0
\(879\) −57.6210 −1.94351
\(880\) 0 0
\(881\) −4.47580 −0.150794 −0.0753968 0.997154i \(-0.524022\pi\)
−0.0753968 + 0.997154i \(0.524022\pi\)
\(882\) 0 0
\(883\) 27.5455 0.926981 0.463491 0.886102i \(-0.346597\pi\)
0.463491 + 0.886102i \(0.346597\pi\)
\(884\) 0 0
\(885\) −31.9245 −1.07313
\(886\) 0 0
\(887\) 48.5152 1.62898 0.814491 0.580176i \(-0.197016\pi\)
0.814491 + 0.580176i \(0.197016\pi\)
\(888\) 0 0
\(889\) −1.16003 −0.0389063
\(890\) 0 0
\(891\) −18.9171 −0.633747
\(892\) 0 0
\(893\) 0.648061 0.0216865
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) −45.9097 −1.53288
\(898\) 0 0
\(899\) 17.4078 0.580581
\(900\) 0 0
\(901\) 12.8761 0.428966
\(902\) 0 0
\(903\) 13.6406 0.453932
\(904\) 0 0
\(905\) −16.0968 −0.535076
\(906\) 0 0
\(907\) 4.55451 0.151230 0.0756150 0.997137i \(-0.475908\pi\)
0.0756150 + 0.997137i \(0.475908\pi\)
\(908\) 0 0
\(909\) −8.82032 −0.292552
\(910\) 0 0
\(911\) −0.298732 −0.00989745 −0.00494872 0.999988i \(-0.501575\pi\)
−0.00494872 + 0.999988i \(0.501575\pi\)
\(912\) 0 0
\(913\) 12.8761 0.426138
\(914\) 0 0
\(915\) −6.53162 −0.215929
\(916\) 0 0
\(917\) 11.1600 0.368537
\(918\) 0 0
\(919\) 0.0754620 0.00248926 0.00124463 0.999999i \(-0.499604\pi\)
0.00124463 + 0.999999i \(0.499604\pi\)
\(920\) 0 0
\(921\) −31.8081 −1.04811
\(922\) 0 0
\(923\) 16.9878 0.559159
\(924\) 0 0
\(925\) −2.43807 −0.0801632
\(926\) 0 0
\(927\) 24.7268 0.812134
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.58002 −0.215651
\(932\) 0 0
\(933\) −38.2691 −1.25287
\(934\) 0 0
\(935\) 2.70388 0.0884263
\(936\) 0 0
\(937\) 42.9581 1.40338 0.701690 0.712482i \(-0.252429\pi\)
0.701690 + 0.712482i \(0.252429\pi\)
\(938\) 0 0
\(939\) 84.7252 2.76490
\(940\) 0 0
\(941\) 20.3297 0.662729 0.331364 0.943503i \(-0.392491\pi\)
0.331364 + 0.943503i \(0.392491\pi\)
\(942\) 0 0
\(943\) 34.0968 1.11034
\(944\) 0 0
\(945\) 7.04840 0.229284
\(946\) 0 0
\(947\) −57.0745 −1.85467 −0.927337 0.374228i \(-0.877908\pi\)
−0.927337 + 0.374228i \(0.877908\pi\)
\(948\) 0 0
\(949\) −18.5168 −0.601080
\(950\) 0 0
\(951\) 105.283 3.41403
\(952\) 0 0
\(953\) 44.4594 1.44018 0.720091 0.693880i \(-0.244100\pi\)
0.720091 + 0.693880i \(0.244100\pi\)
\(954\) 0 0
\(955\) 0.111635 0.00361243
\(956\) 0 0
\(957\) −17.4078 −0.562713
\(958\) 0 0
\(959\) 6.55295 0.211606
\(960\) 0 0
\(961\) −13.5922 −0.438459
\(962\) 0 0
\(963\) 55.8113 1.79850
\(964\) 0 0
\(965\) 6.01809 0.193729
\(966\) 0 0
\(967\) −38.1016 −1.22527 −0.612633 0.790368i \(-0.709889\pi\)
−0.612633 + 0.790368i \(0.709889\pi\)
\(968\) 0 0
\(969\) −6.17226 −0.198282
\(970\) 0 0
\(971\) 15.3716 0.493298 0.246649 0.969105i \(-0.420671\pi\)
0.246649 + 0.969105i \(0.420671\pi\)
\(972\) 0 0
\(973\) 6.28390 0.201452
\(974\) 0 0
\(975\) −13.6965 −0.438638
\(976\) 0 0
\(977\) 25.1271 0.803888 0.401944 0.915664i \(-0.368335\pi\)
0.401944 + 0.915664i \(0.368335\pi\)
\(978\) 0 0
\(979\) −5.17487 −0.165390
\(980\) 0 0
\(981\) −45.9852 −1.46819
\(982\) 0 0
\(983\) −27.3042 −0.870868 −0.435434 0.900221i \(-0.643405\pi\)
−0.435434 + 0.900221i \(0.643405\pi\)
\(984\) 0 0
\(985\) −14.7645 −0.470436
\(986\) 0 0
\(987\) −1.29612 −0.0412560
\(988\) 0 0
\(989\) −22.8613 −0.726947
\(990\) 0 0
\(991\) 57.0697 1.81288 0.906440 0.422335i \(-0.138789\pi\)
0.906440 + 0.422335i \(0.138789\pi\)
\(992\) 0 0
\(993\) −50.2084 −1.59332
\(994\) 0 0
\(995\) 9.75228 0.309168
\(996\) 0 0
\(997\) 18.1213 0.573906 0.286953 0.957945i \(-0.407358\pi\)
0.286953 + 0.957945i \(0.407358\pi\)
\(998\) 0 0
\(999\) 26.5168 0.838954
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 3040.2.a.k.1.1 3
4.3 odd 2 3040.2.a.n.1.3 yes 3
8.3 odd 2 6080.2.a.bp.1.1 3
8.5 even 2 6080.2.a.bz.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.1 3 1.1 even 1 trivial
3040.2.a.n.1.3 yes 3 4.3 odd 2
6080.2.a.bp.1.1 3 8.3 odd 2
6080.2.a.bz.1.3 3 8.5 even 2