Properties

Label 6039.2.a.a.1.1
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $5$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 671)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.369680\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.33536 q^{2} -0.216816 q^{4} -1.70504 q^{5} -3.82358 q^{7} +2.96025 q^{8} +O(q^{10})\) \(q-1.33536 q^{2} -0.216816 q^{4} -1.70504 q^{5} -3.82358 q^{7} +2.96025 q^{8} +2.27684 q^{10} -1.00000 q^{11} -1.41350 q^{13} +5.10585 q^{14} -3.51936 q^{16} +2.70504 q^{17} -2.25521 q^{19} +0.369680 q^{20} +1.33536 q^{22} -2.60270 q^{23} -2.09284 q^{25} +1.88753 q^{26} +0.829015 q^{28} +5.40314 q^{29} +1.94062 q^{31} -1.22088 q^{32} -3.61220 q^{34} +6.51936 q^{35} +2.07879 q^{37} +3.01151 q^{38} -5.04734 q^{40} +1.68627 q^{41} +4.50699 q^{43} +0.216816 q^{44} +3.47554 q^{46} +2.55218 q^{47} +7.61978 q^{49} +2.79469 q^{50} +0.306471 q^{52} +5.43865 q^{53} +1.70504 q^{55} -11.3187 q^{56} -7.21513 q^{58} -4.50124 q^{59} +1.00000 q^{61} -2.59143 q^{62} +8.66904 q^{64} +2.41008 q^{65} +1.34599 q^{67} -0.586497 q^{68} -8.70568 q^{70} +2.13067 q^{71} -12.4393 q^{73} -2.77593 q^{74} +0.488966 q^{76} +3.82358 q^{77} -3.83244 q^{79} +6.00064 q^{80} -2.25178 q^{82} +7.05810 q^{83} -4.61220 q^{85} -6.01845 q^{86} -2.96025 q^{88} +13.7435 q^{89} +5.40465 q^{91} +0.564307 q^{92} -3.40807 q^{94} +3.84522 q^{95} +7.63215 q^{97} -10.1751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 2 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + 2 q^{5} - q^{7} + 6 q^{8} + 5 q^{10} - 5 q^{11} - 10 q^{13} + 3 q^{14} + 2 q^{16} + 3 q^{17} - 13 q^{19} - 2 q^{22} - 15 q^{25} - 9 q^{26} - 12 q^{28} + 7 q^{29} - 13 q^{31} - q^{32} - 3 q^{34} + 13 q^{35} - 6 q^{37} - 9 q^{38} - 5 q^{40} + 9 q^{41} + 2 q^{43} - 7 q^{46} + 3 q^{47} - 6 q^{49} - 9 q^{50} - 12 q^{52} - 3 q^{53} - 2 q^{55} - 28 q^{56} - 15 q^{58} + 14 q^{59} + 5 q^{61} - 31 q^{62} + 6 q^{64} - 9 q^{65} - 5 q^{67} - 5 q^{70} - 3 q^{71} - 4 q^{73} - 2 q^{74} - 19 q^{76} + q^{77} - 27 q^{79} + 2 q^{80} + 11 q^{82} + 3 q^{83} - 8 q^{85} - 5 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 13 q^{92} - 18 q^{94} - 7 q^{95} - 5 q^{97} - 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33536 −0.944241 −0.472121 0.881534i \(-0.656511\pi\)
−0.472121 + 0.881534i \(0.656511\pi\)
\(3\) 0 0
\(4\) −0.216816 −0.108408
\(5\) −1.70504 −0.762517 −0.381258 0.924469i \(-0.624509\pi\)
−0.381258 + 0.924469i \(0.624509\pi\)
\(6\) 0 0
\(7\) −3.82358 −1.44518 −0.722589 0.691278i \(-0.757048\pi\)
−0.722589 + 0.691278i \(0.757048\pi\)
\(8\) 2.96025 1.04660
\(9\) 0 0
\(10\) 2.27684 0.720000
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.41350 −0.392035 −0.196018 0.980600i \(-0.562801\pi\)
−0.196018 + 0.980600i \(0.562801\pi\)
\(14\) 5.10585 1.36460
\(15\) 0 0
\(16\) −3.51936 −0.879840
\(17\) 2.70504 0.656068 0.328034 0.944666i \(-0.393614\pi\)
0.328034 + 0.944666i \(0.393614\pi\)
\(18\) 0 0
\(19\) −2.25521 −0.517380 −0.258690 0.965960i \(-0.583291\pi\)
−0.258690 + 0.965960i \(0.583291\pi\)
\(20\) 0.369680 0.0826630
\(21\) 0 0
\(22\) 1.33536 0.284699
\(23\) −2.60270 −0.542700 −0.271350 0.962481i \(-0.587470\pi\)
−0.271350 + 0.962481i \(0.587470\pi\)
\(24\) 0 0
\(25\) −2.09284 −0.418568
\(26\) 1.88753 0.370176
\(27\) 0 0
\(28\) 0.829015 0.156669
\(29\) 5.40314 1.00334 0.501669 0.865060i \(-0.332720\pi\)
0.501669 + 0.865060i \(0.332720\pi\)
\(30\) 0 0
\(31\) 1.94062 0.348546 0.174273 0.984697i \(-0.444243\pi\)
0.174273 + 0.984697i \(0.444243\pi\)
\(32\) −1.22088 −0.215824
\(33\) 0 0
\(34\) −3.61220 −0.619487
\(35\) 6.51936 1.10197
\(36\) 0 0
\(37\) 2.07879 0.341751 0.170875 0.985293i \(-0.445340\pi\)
0.170875 + 0.985293i \(0.445340\pi\)
\(38\) 3.01151 0.488531
\(39\) 0 0
\(40\) −5.04734 −0.798054
\(41\) 1.68627 0.263352 0.131676 0.991293i \(-0.457964\pi\)
0.131676 + 0.991293i \(0.457964\pi\)
\(42\) 0 0
\(43\) 4.50699 0.687309 0.343655 0.939096i \(-0.388335\pi\)
0.343655 + 0.939096i \(0.388335\pi\)
\(44\) 0.216816 0.0326863
\(45\) 0 0
\(46\) 3.47554 0.512440
\(47\) 2.55218 0.372273 0.186137 0.982524i \(-0.440403\pi\)
0.186137 + 0.982524i \(0.440403\pi\)
\(48\) 0 0
\(49\) 7.61978 1.08854
\(50\) 2.79469 0.395229
\(51\) 0 0
\(52\) 0.306471 0.0424998
\(53\) 5.43865 0.747056 0.373528 0.927619i \(-0.378148\pi\)
0.373528 + 0.927619i \(0.378148\pi\)
\(54\) 0 0
\(55\) 1.70504 0.229907
\(56\) −11.3187 −1.51253
\(57\) 0 0
\(58\) −7.21513 −0.947394
\(59\) −4.50124 −0.586011 −0.293006 0.956111i \(-0.594655\pi\)
−0.293006 + 0.956111i \(0.594655\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) −2.59143 −0.329111
\(63\) 0 0
\(64\) 8.66904 1.08363
\(65\) 2.41008 0.298933
\(66\) 0 0
\(67\) 1.34599 0.164438 0.0822192 0.996614i \(-0.473799\pi\)
0.0822192 + 0.996614i \(0.473799\pi\)
\(68\) −0.586497 −0.0711232
\(69\) 0 0
\(70\) −8.70568 −1.04053
\(71\) 2.13067 0.252865 0.126432 0.991975i \(-0.459647\pi\)
0.126432 + 0.991975i \(0.459647\pi\)
\(72\) 0 0
\(73\) −12.4393 −1.45591 −0.727955 0.685625i \(-0.759529\pi\)
−0.727955 + 0.685625i \(0.759529\pi\)
\(74\) −2.77593 −0.322695
\(75\) 0 0
\(76\) 0.488966 0.0560882
\(77\) 3.82358 0.435738
\(78\) 0 0
\(79\) −3.83244 −0.431183 −0.215592 0.976484i \(-0.569168\pi\)
−0.215592 + 0.976484i \(0.569168\pi\)
\(80\) 6.00064 0.670892
\(81\) 0 0
\(82\) −2.25178 −0.248668
\(83\) 7.05810 0.774727 0.387364 0.921927i \(-0.373386\pi\)
0.387364 + 0.921927i \(0.373386\pi\)
\(84\) 0 0
\(85\) −4.61220 −0.500263
\(86\) −6.01845 −0.648986
\(87\) 0 0
\(88\) −2.96025 −0.315563
\(89\) 13.7435 1.45681 0.728405 0.685147i \(-0.240262\pi\)
0.728405 + 0.685147i \(0.240262\pi\)
\(90\) 0 0
\(91\) 5.40465 0.566561
\(92\) 0.564307 0.0588331
\(93\) 0 0
\(94\) −3.40807 −0.351516
\(95\) 3.84522 0.394511
\(96\) 0 0
\(97\) 7.63215 0.774927 0.387464 0.921885i \(-0.373351\pi\)
0.387464 + 0.921885i \(0.373351\pi\)
\(98\) −10.1751 −1.02784
\(99\) 0 0
\(100\) 0.453762 0.0453762
\(101\) −0.540109 −0.0537429 −0.0268714 0.999639i \(-0.508554\pi\)
−0.0268714 + 0.999639i \(0.508554\pi\)
\(102\) 0 0
\(103\) 14.3630 1.41523 0.707613 0.706600i \(-0.249772\pi\)
0.707613 + 0.706600i \(0.249772\pi\)
\(104\) −4.18432 −0.410306
\(105\) 0 0
\(106\) −7.26255 −0.705401
\(107\) 2.64826 0.256017 0.128009 0.991773i \(-0.459141\pi\)
0.128009 + 0.991773i \(0.459141\pi\)
\(108\) 0 0
\(109\) −12.8191 −1.22785 −0.613923 0.789366i \(-0.710410\pi\)
−0.613923 + 0.789366i \(0.710410\pi\)
\(110\) −2.27684 −0.217088
\(111\) 0 0
\(112\) 13.4566 1.27152
\(113\) −11.2624 −1.05947 −0.529737 0.848162i \(-0.677710\pi\)
−0.529737 + 0.848162i \(0.677710\pi\)
\(114\) 0 0
\(115\) 4.43770 0.413818
\(116\) −1.17149 −0.108770
\(117\) 0 0
\(118\) 6.01077 0.553336
\(119\) −10.3429 −0.948136
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.33536 −0.120898
\(123\) 0 0
\(124\) −0.420758 −0.0377852
\(125\) 12.0936 1.08168
\(126\) 0 0
\(127\) 21.5844 1.91531 0.957654 0.287922i \(-0.0929645\pi\)
0.957654 + 0.287922i \(0.0929645\pi\)
\(128\) −9.13451 −0.807384
\(129\) 0 0
\(130\) −3.21832 −0.282265
\(131\) 0.639961 0.0559136 0.0279568 0.999609i \(-0.491100\pi\)
0.0279568 + 0.999609i \(0.491100\pi\)
\(132\) 0 0
\(133\) 8.62297 0.747706
\(134\) −1.79738 −0.155270
\(135\) 0 0
\(136\) 8.00758 0.686644
\(137\) −12.9479 −1.10621 −0.553106 0.833111i \(-0.686558\pi\)
−0.553106 + 0.833111i \(0.686558\pi\)
\(138\) 0 0
\(139\) 6.69742 0.568068 0.284034 0.958814i \(-0.408327\pi\)
0.284034 + 0.958814i \(0.408327\pi\)
\(140\) −1.41350 −0.119463
\(141\) 0 0
\(142\) −2.84522 −0.238765
\(143\) 1.41350 0.118203
\(144\) 0 0
\(145\) −9.21257 −0.765062
\(146\) 16.6109 1.37473
\(147\) 0 0
\(148\) −0.450715 −0.0370486
\(149\) 9.11935 0.747086 0.373543 0.927613i \(-0.378143\pi\)
0.373543 + 0.927613i \(0.378143\pi\)
\(150\) 0 0
\(151\) 1.89341 0.154084 0.0770418 0.997028i \(-0.475453\pi\)
0.0770418 + 0.997028i \(0.475453\pi\)
\(152\) −6.67596 −0.541492
\(153\) 0 0
\(154\) −5.10585 −0.411441
\(155\) −3.30883 −0.265772
\(156\) 0 0
\(157\) −18.5119 −1.47741 −0.738705 0.674029i \(-0.764562\pi\)
−0.738705 + 0.674029i \(0.764562\pi\)
\(158\) 5.11768 0.407141
\(159\) 0 0
\(160\) 2.08166 0.164569
\(161\) 9.95163 0.784298
\(162\) 0 0
\(163\) 14.4961 1.13542 0.567712 0.823227i \(-0.307829\pi\)
0.567712 + 0.823227i \(0.307829\pi\)
\(164\) −0.365612 −0.0285495
\(165\) 0 0
\(166\) −9.42510 −0.731530
\(167\) 11.4744 0.887919 0.443959 0.896047i \(-0.353573\pi\)
0.443959 + 0.896047i \(0.353573\pi\)
\(168\) 0 0
\(169\) −11.0020 −0.846308
\(170\) 6.15894 0.472369
\(171\) 0 0
\(172\) −0.977189 −0.0745099
\(173\) −21.0366 −1.59938 −0.799690 0.600413i \(-0.795003\pi\)
−0.799690 + 0.600413i \(0.795003\pi\)
\(174\) 0 0
\(175\) 8.00215 0.604906
\(176\) 3.51936 0.265282
\(177\) 0 0
\(178\) −18.3525 −1.37558
\(179\) 15.9191 1.18985 0.594924 0.803782i \(-0.297182\pi\)
0.594924 + 0.803782i \(0.297182\pi\)
\(180\) 0 0
\(181\) −0.154850 −0.0115099 −0.00575497 0.999983i \(-0.501832\pi\)
−0.00575497 + 0.999983i \(0.501832\pi\)
\(182\) −7.21714 −0.534970
\(183\) 0 0
\(184\) −7.70462 −0.567992
\(185\) −3.54442 −0.260591
\(186\) 0 0
\(187\) −2.70504 −0.197812
\(188\) −0.553353 −0.0403574
\(189\) 0 0
\(190\) −5.13474 −0.372513
\(191\) 12.4405 0.900165 0.450083 0.892987i \(-0.351395\pi\)
0.450083 + 0.892987i \(0.351395\pi\)
\(192\) 0 0
\(193\) −10.9260 −0.786470 −0.393235 0.919438i \(-0.628644\pi\)
−0.393235 + 0.919438i \(0.628644\pi\)
\(194\) −10.1917 −0.731719
\(195\) 0 0
\(196\) −1.65209 −0.118007
\(197\) 13.4945 0.961441 0.480720 0.876874i \(-0.340375\pi\)
0.480720 + 0.876874i \(0.340375\pi\)
\(198\) 0 0
\(199\) −10.3413 −0.733074 −0.366537 0.930403i \(-0.619457\pi\)
−0.366537 + 0.930403i \(0.619457\pi\)
\(200\) −6.19532 −0.438075
\(201\) 0 0
\(202\) 0.721240 0.0507463
\(203\) −20.6594 −1.45000
\(204\) 0 0
\(205\) −2.87516 −0.200810
\(206\) −19.1797 −1.33631
\(207\) 0 0
\(208\) 4.97462 0.344928
\(209\) 2.25521 0.155996
\(210\) 0 0
\(211\) −19.4268 −1.33739 −0.668697 0.743535i \(-0.733148\pi\)
−0.668697 + 0.743535i \(0.733148\pi\)
\(212\) −1.17919 −0.0809870
\(213\) 0 0
\(214\) −3.53638 −0.241742
\(215\) −7.68459 −0.524085
\(216\) 0 0
\(217\) −7.42012 −0.503711
\(218\) 17.1181 1.15938
\(219\) 0 0
\(220\) −0.369680 −0.0249238
\(221\) −3.82358 −0.257202
\(222\) 0 0
\(223\) −18.3560 −1.22921 −0.614603 0.788837i \(-0.710684\pi\)
−0.614603 + 0.788837i \(0.710684\pi\)
\(224\) 4.66815 0.311904
\(225\) 0 0
\(226\) 15.0393 1.00040
\(227\) 7.69300 0.510602 0.255301 0.966862i \(-0.417825\pi\)
0.255301 + 0.966862i \(0.417825\pi\)
\(228\) 0 0
\(229\) −23.2490 −1.53634 −0.768169 0.640247i \(-0.778832\pi\)
−0.768169 + 0.640247i \(0.778832\pi\)
\(230\) −5.92592 −0.390744
\(231\) 0 0
\(232\) 15.9946 1.05010
\(233\) 14.1638 0.927898 0.463949 0.885862i \(-0.346432\pi\)
0.463949 + 0.885862i \(0.346432\pi\)
\(234\) 0 0
\(235\) −4.35156 −0.283864
\(236\) 0.975942 0.0635284
\(237\) 0 0
\(238\) 13.8115 0.895269
\(239\) −18.0080 −1.16484 −0.582422 0.812887i \(-0.697895\pi\)
−0.582422 + 0.812887i \(0.697895\pi\)
\(240\) 0 0
\(241\) 18.1562 1.16955 0.584773 0.811197i \(-0.301183\pi\)
0.584773 + 0.811197i \(0.301183\pi\)
\(242\) −1.33536 −0.0858401
\(243\) 0 0
\(244\) −0.216816 −0.0138802
\(245\) −12.9920 −0.830030
\(246\) 0 0
\(247\) 3.18774 0.202831
\(248\) 5.74471 0.364790
\(249\) 0 0
\(250\) −16.1493 −1.02137
\(251\) −2.13114 −0.134517 −0.0672583 0.997736i \(-0.521425\pi\)
−0.0672583 + 0.997736i \(0.521425\pi\)
\(252\) 0 0
\(253\) 2.60270 0.163630
\(254\) −28.8229 −1.80851
\(255\) 0 0
\(256\) −5.14023 −0.321264
\(257\) −5.50360 −0.343305 −0.171653 0.985158i \(-0.554911\pi\)
−0.171653 + 0.985158i \(0.554911\pi\)
\(258\) 0 0
\(259\) −7.94842 −0.493891
\(260\) −0.522544 −0.0324068
\(261\) 0 0
\(262\) −0.854577 −0.0527960
\(263\) 8.52076 0.525413 0.262706 0.964876i \(-0.415385\pi\)
0.262706 + 0.964876i \(0.415385\pi\)
\(264\) 0 0
\(265\) −9.27311 −0.569643
\(266\) −11.5148 −0.706015
\(267\) 0 0
\(268\) −0.291832 −0.0178265
\(269\) 6.75644 0.411948 0.205974 0.978558i \(-0.433964\pi\)
0.205974 + 0.978558i \(0.433964\pi\)
\(270\) 0 0
\(271\) 17.8835 1.08635 0.543174 0.839620i \(-0.317223\pi\)
0.543174 + 0.839620i \(0.317223\pi\)
\(272\) −9.52000 −0.577235
\(273\) 0 0
\(274\) 17.2901 1.04453
\(275\) 2.09284 0.126203
\(276\) 0 0
\(277\) −19.4528 −1.16881 −0.584403 0.811464i \(-0.698671\pi\)
−0.584403 + 0.811464i \(0.698671\pi\)
\(278\) −8.94346 −0.536393
\(279\) 0 0
\(280\) 19.2989 1.15333
\(281\) −4.87363 −0.290736 −0.145368 0.989378i \(-0.546437\pi\)
−0.145368 + 0.989378i \(0.546437\pi\)
\(282\) 0 0
\(283\) −11.7696 −0.699631 −0.349815 0.936819i \(-0.613756\pi\)
−0.349815 + 0.936819i \(0.613756\pi\)
\(284\) −0.461965 −0.0274126
\(285\) 0 0
\(286\) −1.88753 −0.111612
\(287\) −6.44761 −0.380590
\(288\) 0 0
\(289\) −9.68276 −0.569574
\(290\) 12.3021 0.722403
\(291\) 0 0
\(292\) 2.69704 0.157832
\(293\) −6.94596 −0.405787 −0.202894 0.979201i \(-0.565035\pi\)
−0.202894 + 0.979201i \(0.565035\pi\)
\(294\) 0 0
\(295\) 7.67479 0.446843
\(296\) 6.15372 0.357678
\(297\) 0 0
\(298\) −12.1776 −0.705430
\(299\) 3.67892 0.212757
\(300\) 0 0
\(301\) −17.2328 −0.993284
\(302\) −2.52838 −0.145492
\(303\) 0 0
\(304\) 7.93688 0.455211
\(305\) −1.70504 −0.0976303
\(306\) 0 0
\(307\) −11.6737 −0.666256 −0.333128 0.942882i \(-0.608104\pi\)
−0.333128 + 0.942882i \(0.608104\pi\)
\(308\) −0.829015 −0.0472375
\(309\) 0 0
\(310\) 4.41848 0.250953
\(311\) 1.00838 0.0571797 0.0285899 0.999591i \(-0.490898\pi\)
0.0285899 + 0.999591i \(0.490898\pi\)
\(312\) 0 0
\(313\) 2.72938 0.154274 0.0771369 0.997021i \(-0.475422\pi\)
0.0771369 + 0.997021i \(0.475422\pi\)
\(314\) 24.7200 1.39503
\(315\) 0 0
\(316\) 0.830935 0.0467438
\(317\) 16.6058 0.932676 0.466338 0.884607i \(-0.345573\pi\)
0.466338 + 0.884607i \(0.345573\pi\)
\(318\) 0 0
\(319\) −5.40314 −0.302518
\(320\) −14.7810 −0.826286
\(321\) 0 0
\(322\) −13.2890 −0.740567
\(323\) −6.10042 −0.339437
\(324\) 0 0
\(325\) 2.95824 0.164093
\(326\) −19.3575 −1.07211
\(327\) 0 0
\(328\) 4.99179 0.275625
\(329\) −9.75845 −0.538001
\(330\) 0 0
\(331\) −20.5759 −1.13095 −0.565476 0.824765i \(-0.691307\pi\)
−0.565476 + 0.824765i \(0.691307\pi\)
\(332\) −1.53031 −0.0839868
\(333\) 0 0
\(334\) −15.3225 −0.838410
\(335\) −2.29496 −0.125387
\(336\) 0 0
\(337\) −18.6890 −1.01806 −0.509028 0.860750i \(-0.669995\pi\)
−0.509028 + 0.860750i \(0.669995\pi\)
\(338\) 14.6916 0.799119
\(339\) 0 0
\(340\) 1.00000 0.0542326
\(341\) −1.94062 −0.105091
\(342\) 0 0
\(343\) −2.36978 −0.127956
\(344\) 13.3418 0.719341
\(345\) 0 0
\(346\) 28.0914 1.51020
\(347\) 2.11153 0.113353 0.0566765 0.998393i \(-0.481950\pi\)
0.0566765 + 0.998393i \(0.481950\pi\)
\(348\) 0 0
\(349\) −21.3375 −1.14217 −0.571084 0.820891i \(-0.693477\pi\)
−0.571084 + 0.820891i \(0.693477\pi\)
\(350\) −10.6857 −0.571177
\(351\) 0 0
\(352\) 1.22088 0.0650734
\(353\) −23.1273 −1.23094 −0.615471 0.788159i \(-0.711034\pi\)
−0.615471 + 0.788159i \(0.711034\pi\)
\(354\) 0 0
\(355\) −3.63288 −0.192813
\(356\) −2.97982 −0.157930
\(357\) 0 0
\(358\) −21.2577 −1.12350
\(359\) −5.99727 −0.316524 −0.158262 0.987397i \(-0.550589\pi\)
−0.158262 + 0.987397i \(0.550589\pi\)
\(360\) 0 0
\(361\) −13.9140 −0.732318
\(362\) 0.206781 0.0108682
\(363\) 0 0
\(364\) −1.17182 −0.0614198
\(365\) 21.2095 1.11016
\(366\) 0 0
\(367\) 15.5421 0.811292 0.405646 0.914030i \(-0.367047\pi\)
0.405646 + 0.914030i \(0.367047\pi\)
\(368\) 9.15982 0.477489
\(369\) 0 0
\(370\) 4.73307 0.246060
\(371\) −20.7951 −1.07963
\(372\) 0 0
\(373\) 32.9020 1.70360 0.851800 0.523868i \(-0.175511\pi\)
0.851800 + 0.523868i \(0.175511\pi\)
\(374\) 3.61220 0.186782
\(375\) 0 0
\(376\) 7.55507 0.389623
\(377\) −7.63736 −0.393344
\(378\) 0 0
\(379\) −16.2472 −0.834561 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(380\) −0.833706 −0.0427682
\(381\) 0 0
\(382\) −16.6126 −0.849973
\(383\) −31.4959 −1.60936 −0.804682 0.593706i \(-0.797664\pi\)
−0.804682 + 0.593706i \(0.797664\pi\)
\(384\) 0 0
\(385\) −6.51936 −0.332257
\(386\) 14.5901 0.742618
\(387\) 0 0
\(388\) −1.65477 −0.0840085
\(389\) −39.1814 −1.98658 −0.993289 0.115662i \(-0.963101\pi\)
−0.993289 + 0.115662i \(0.963101\pi\)
\(390\) 0 0
\(391\) −7.04040 −0.356048
\(392\) 22.5564 1.13927
\(393\) 0 0
\(394\) −18.0200 −0.907832
\(395\) 6.53446 0.328784
\(396\) 0 0
\(397\) 22.2329 1.11584 0.557920 0.829895i \(-0.311600\pi\)
0.557920 + 0.829895i \(0.311600\pi\)
\(398\) 13.8093 0.692199
\(399\) 0 0
\(400\) 7.36546 0.368273
\(401\) 21.6821 1.08275 0.541377 0.840780i \(-0.317903\pi\)
0.541377 + 0.840780i \(0.317903\pi\)
\(402\) 0 0
\(403\) −2.74307 −0.136642
\(404\) 0.117105 0.00582617
\(405\) 0 0
\(406\) 27.5877 1.36915
\(407\) −2.07879 −0.103042
\(408\) 0 0
\(409\) −31.4770 −1.55644 −0.778218 0.627994i \(-0.783876\pi\)
−0.778218 + 0.627994i \(0.783876\pi\)
\(410\) 3.83938 0.189613
\(411\) 0 0
\(412\) −3.11413 −0.153422
\(413\) 17.2108 0.846891
\(414\) 0 0
\(415\) −12.0343 −0.590743
\(416\) 1.72572 0.0846106
\(417\) 0 0
\(418\) −3.01151 −0.147298
\(419\) −10.2586 −0.501167 −0.250583 0.968095i \(-0.580622\pi\)
−0.250583 + 0.968095i \(0.580622\pi\)
\(420\) 0 0
\(421\) 15.1615 0.738928 0.369464 0.929245i \(-0.379541\pi\)
0.369464 + 0.929245i \(0.379541\pi\)
\(422\) 25.9417 1.26282
\(423\) 0 0
\(424\) 16.0997 0.781872
\(425\) −5.66122 −0.274609
\(426\) 0 0
\(427\) −3.82358 −0.185036
\(428\) −0.574187 −0.0277544
\(429\) 0 0
\(430\) 10.2617 0.494863
\(431\) −11.3162 −0.545081 −0.272541 0.962144i \(-0.587864\pi\)
−0.272541 + 0.962144i \(0.587864\pi\)
\(432\) 0 0
\(433\) 23.5749 1.13294 0.566469 0.824083i \(-0.308309\pi\)
0.566469 + 0.824083i \(0.308309\pi\)
\(434\) 9.90853 0.475625
\(435\) 0 0
\(436\) 2.77939 0.133109
\(437\) 5.86962 0.280782
\(438\) 0 0
\(439\) 11.4939 0.548572 0.274286 0.961648i \(-0.411559\pi\)
0.274286 + 0.961648i \(0.411559\pi\)
\(440\) 5.04734 0.240622
\(441\) 0 0
\(442\) 5.10585 0.242861
\(443\) 3.89812 0.185205 0.0926027 0.995703i \(-0.470481\pi\)
0.0926027 + 0.995703i \(0.470481\pi\)
\(444\) 0 0
\(445\) −23.4332 −1.11084
\(446\) 24.5118 1.16067
\(447\) 0 0
\(448\) −33.1468 −1.56604
\(449\) −9.64992 −0.455408 −0.227704 0.973730i \(-0.573122\pi\)
−0.227704 + 0.973730i \(0.573122\pi\)
\(450\) 0 0
\(451\) −1.68627 −0.0794036
\(452\) 2.44187 0.114856
\(453\) 0 0
\(454\) −10.2729 −0.482132
\(455\) −9.21513 −0.432012
\(456\) 0 0
\(457\) 6.14484 0.287443 0.143722 0.989618i \(-0.454093\pi\)
0.143722 + 0.989618i \(0.454093\pi\)
\(458\) 31.0458 1.45067
\(459\) 0 0
\(460\) −0.962166 −0.0448612
\(461\) −29.4655 −1.37235 −0.686173 0.727438i \(-0.740711\pi\)
−0.686173 + 0.727438i \(0.740711\pi\)
\(462\) 0 0
\(463\) −20.0053 −0.929727 −0.464864 0.885382i \(-0.653897\pi\)
−0.464864 + 0.885382i \(0.653897\pi\)
\(464\) −19.0156 −0.882777
\(465\) 0 0
\(466\) −18.9137 −0.876160
\(467\) −9.14972 −0.423399 −0.211699 0.977335i \(-0.567900\pi\)
−0.211699 + 0.977335i \(0.567900\pi\)
\(468\) 0 0
\(469\) −5.14649 −0.237643
\(470\) 5.81089 0.268037
\(471\) 0 0
\(472\) −13.3248 −0.613322
\(473\) −4.50699 −0.207232
\(474\) 0 0
\(475\) 4.71979 0.216559
\(476\) 2.24252 0.102786
\(477\) 0 0
\(478\) 24.0472 1.09989
\(479\) 32.8073 1.49900 0.749501 0.662003i \(-0.230293\pi\)
0.749501 + 0.662003i \(0.230293\pi\)
\(480\) 0 0
\(481\) −2.93837 −0.133978
\(482\) −24.2451 −1.10433
\(483\) 0 0
\(484\) −0.216816 −0.00985529
\(485\) −13.0131 −0.590895
\(486\) 0 0
\(487\) −23.6056 −1.06967 −0.534835 0.844956i \(-0.679626\pi\)
−0.534835 + 0.844956i \(0.679626\pi\)
\(488\) 2.96025 0.134004
\(489\) 0 0
\(490\) 17.3490 0.783749
\(491\) 12.5762 0.567556 0.283778 0.958890i \(-0.408412\pi\)
0.283778 + 0.958890i \(0.408412\pi\)
\(492\) 0 0
\(493\) 14.6157 0.658259
\(494\) −4.25678 −0.191522
\(495\) 0 0
\(496\) −6.82974 −0.306664
\(497\) −8.14681 −0.365434
\(498\) 0 0
\(499\) 25.9798 1.16301 0.581507 0.813542i \(-0.302463\pi\)
0.581507 + 0.813542i \(0.302463\pi\)
\(500\) −2.62208 −0.117263
\(501\) 0 0
\(502\) 2.84584 0.127016
\(503\) 22.2224 0.990846 0.495423 0.868652i \(-0.335013\pi\)
0.495423 + 0.868652i \(0.335013\pi\)
\(504\) 0 0
\(505\) 0.920908 0.0409799
\(506\) −3.47554 −0.154506
\(507\) 0 0
\(508\) −4.67985 −0.207635
\(509\) −13.1445 −0.582620 −0.291310 0.956629i \(-0.594091\pi\)
−0.291310 + 0.956629i \(0.594091\pi\)
\(510\) 0 0
\(511\) 47.5627 2.10405
\(512\) 25.1331 1.11073
\(513\) 0 0
\(514\) 7.34928 0.324163
\(515\) −24.4894 −1.07913
\(516\) 0 0
\(517\) −2.55218 −0.112245
\(518\) 10.6140 0.466352
\(519\) 0 0
\(520\) 7.13443 0.312865
\(521\) −28.1051 −1.23131 −0.615653 0.788017i \(-0.711108\pi\)
−0.615653 + 0.788017i \(0.711108\pi\)
\(522\) 0 0
\(523\) 29.7361 1.30027 0.650134 0.759820i \(-0.274713\pi\)
0.650134 + 0.759820i \(0.274713\pi\)
\(524\) −0.138754 −0.00606149
\(525\) 0 0
\(526\) −11.3783 −0.496117
\(527\) 5.24946 0.228670
\(528\) 0 0
\(529\) −16.2260 −0.705477
\(530\) 12.3829 0.537880
\(531\) 0 0
\(532\) −1.86960 −0.0810574
\(533\) −2.38355 −0.103243
\(534\) 0 0
\(535\) −4.51539 −0.195217
\(536\) 3.98445 0.172102
\(537\) 0 0
\(538\) −9.02228 −0.388978
\(539\) −7.61978 −0.328207
\(540\) 0 0
\(541\) 3.77397 0.162256 0.0811278 0.996704i \(-0.474148\pi\)
0.0811278 + 0.996704i \(0.474148\pi\)
\(542\) −23.8809 −1.02577
\(543\) 0 0
\(544\) −3.30254 −0.141595
\(545\) 21.8571 0.936254
\(546\) 0 0
\(547\) −24.3850 −1.04263 −0.521315 0.853365i \(-0.674558\pi\)
−0.521315 + 0.853365i \(0.674558\pi\)
\(548\) 2.80731 0.119922
\(549\) 0 0
\(550\) −2.79469 −0.119166
\(551\) −12.1852 −0.519107
\(552\) 0 0
\(553\) 14.6536 0.623136
\(554\) 25.9765 1.10363
\(555\) 0 0
\(556\) −1.45211 −0.0615832
\(557\) −40.2900 −1.70714 −0.853571 0.520977i \(-0.825568\pi\)
−0.853571 + 0.520977i \(0.825568\pi\)
\(558\) 0 0
\(559\) −6.37064 −0.269449
\(560\) −22.9440 −0.969559
\(561\) 0 0
\(562\) 6.50804 0.274525
\(563\) 18.6626 0.786534 0.393267 0.919424i \(-0.371345\pi\)
0.393267 + 0.919424i \(0.371345\pi\)
\(564\) 0 0
\(565\) 19.2028 0.807867
\(566\) 15.7167 0.660620
\(567\) 0 0
\(568\) 6.30732 0.264649
\(569\) 24.0420 1.00789 0.503946 0.863735i \(-0.331881\pi\)
0.503946 + 0.863735i \(0.331881\pi\)
\(570\) 0 0
\(571\) −2.78884 −0.116710 −0.0583548 0.998296i \(-0.518585\pi\)
−0.0583548 + 0.998296i \(0.518585\pi\)
\(572\) −0.306471 −0.0128142
\(573\) 0 0
\(574\) 8.60987 0.359369
\(575\) 5.44703 0.227157
\(576\) 0 0
\(577\) 20.3226 0.846040 0.423020 0.906120i \(-0.360970\pi\)
0.423020 + 0.906120i \(0.360970\pi\)
\(578\) 12.9300 0.537816
\(579\) 0 0
\(580\) 1.99744 0.0829390
\(581\) −26.9872 −1.11962
\(582\) 0 0
\(583\) −5.43865 −0.225246
\(584\) −36.8234 −1.52376
\(585\) 0 0
\(586\) 9.27534 0.383161
\(587\) −45.0937 −1.86122 −0.930608 0.366017i \(-0.880721\pi\)
−0.930608 + 0.366017i \(0.880721\pi\)
\(588\) 0 0
\(589\) −4.37650 −0.180331
\(590\) −10.2486 −0.421928
\(591\) 0 0
\(592\) −7.31600 −0.300686
\(593\) 15.7514 0.646833 0.323417 0.946257i \(-0.395168\pi\)
0.323417 + 0.946257i \(0.395168\pi\)
\(594\) 0 0
\(595\) 17.6351 0.722969
\(596\) −1.97722 −0.0809903
\(597\) 0 0
\(598\) −4.91268 −0.200894
\(599\) 6.67520 0.272741 0.136371 0.990658i \(-0.456456\pi\)
0.136371 + 0.990658i \(0.456456\pi\)
\(600\) 0 0
\(601\) −5.63719 −0.229946 −0.114973 0.993369i \(-0.536678\pi\)
−0.114973 + 0.993369i \(0.536678\pi\)
\(602\) 23.0120 0.937900
\(603\) 0 0
\(604\) −0.410522 −0.0167039
\(605\) −1.70504 −0.0693197
\(606\) 0 0
\(607\) 24.1058 0.978423 0.489212 0.872165i \(-0.337285\pi\)
0.489212 + 0.872165i \(0.337285\pi\)
\(608\) 2.75335 0.111663
\(609\) 0 0
\(610\) 2.27684 0.0921865
\(611\) −3.60751 −0.145944
\(612\) 0 0
\(613\) −32.8312 −1.32604 −0.663019 0.748603i \(-0.730725\pi\)
−0.663019 + 0.748603i \(0.730725\pi\)
\(614\) 15.5886 0.629107
\(615\) 0 0
\(616\) 11.3187 0.456045
\(617\) 25.0263 1.00752 0.503760 0.863844i \(-0.331950\pi\)
0.503760 + 0.863844i \(0.331950\pi\)
\(618\) 0 0
\(619\) −18.4385 −0.741106 −0.370553 0.928811i \(-0.620832\pi\)
−0.370553 + 0.928811i \(0.620832\pi\)
\(620\) 0.717409 0.0288119
\(621\) 0 0
\(622\) −1.34654 −0.0539915
\(623\) −52.5495 −2.10535
\(624\) 0 0
\(625\) −10.1558 −0.406233
\(626\) −3.64471 −0.145672
\(627\) 0 0
\(628\) 4.01368 0.160163
\(629\) 5.62320 0.224212
\(630\) 0 0
\(631\) 12.0576 0.480004 0.240002 0.970772i \(-0.422852\pi\)
0.240002 + 0.970772i \(0.422852\pi\)
\(632\) −11.3450 −0.451278
\(633\) 0 0
\(634\) −22.1747 −0.880671
\(635\) −36.8023 −1.46045
\(636\) 0 0
\(637\) −10.7706 −0.426746
\(638\) 7.21513 0.285650
\(639\) 0 0
\(640\) 15.5747 0.615644
\(641\) −32.4028 −1.27983 −0.639917 0.768444i \(-0.721031\pi\)
−0.639917 + 0.768444i \(0.721031\pi\)
\(642\) 0 0
\(643\) 1.25921 0.0496584 0.0248292 0.999692i \(-0.492096\pi\)
0.0248292 + 0.999692i \(0.492096\pi\)
\(644\) −2.15768 −0.0850243
\(645\) 0 0
\(646\) 8.14625 0.320510
\(647\) 9.53579 0.374890 0.187445 0.982275i \(-0.439979\pi\)
0.187445 + 0.982275i \(0.439979\pi\)
\(648\) 0 0
\(649\) 4.50124 0.176689
\(650\) −3.95031 −0.154944
\(651\) 0 0
\(652\) −3.14300 −0.123089
\(653\) −23.6093 −0.923904 −0.461952 0.886905i \(-0.652851\pi\)
−0.461952 + 0.886905i \(0.652851\pi\)
\(654\) 0 0
\(655\) −1.09116 −0.0426351
\(656\) −5.93460 −0.231707
\(657\) 0 0
\(658\) 13.0310 0.508003
\(659\) 32.2394 1.25587 0.627934 0.778267i \(-0.283901\pi\)
0.627934 + 0.778267i \(0.283901\pi\)
\(660\) 0 0
\(661\) 30.2262 1.17566 0.587831 0.808984i \(-0.299982\pi\)
0.587831 + 0.808984i \(0.299982\pi\)
\(662\) 27.4762 1.06789
\(663\) 0 0
\(664\) 20.8937 0.810834
\(665\) −14.7025 −0.570138
\(666\) 0 0
\(667\) −14.0627 −0.544512
\(668\) −2.48785 −0.0962576
\(669\) 0 0
\(670\) 3.06460 0.118396
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) −41.0579 −1.58267 −0.791333 0.611385i \(-0.790613\pi\)
−0.791333 + 0.611385i \(0.790613\pi\)
\(674\) 24.9566 0.961291
\(675\) 0 0
\(676\) 2.38542 0.0917467
\(677\) −23.1892 −0.891232 −0.445616 0.895224i \(-0.647015\pi\)
−0.445616 + 0.895224i \(0.647015\pi\)
\(678\) 0 0
\(679\) −29.1822 −1.11991
\(680\) −13.6532 −0.523578
\(681\) 0 0
\(682\) 2.59143 0.0992308
\(683\) −33.4783 −1.28101 −0.640506 0.767953i \(-0.721276\pi\)
−0.640506 + 0.767953i \(0.721276\pi\)
\(684\) 0 0
\(685\) 22.0767 0.843506
\(686\) 3.16451 0.120821
\(687\) 0 0
\(688\) −15.8617 −0.604722
\(689\) −7.68755 −0.292872
\(690\) 0 0
\(691\) 10.4751 0.398493 0.199247 0.979949i \(-0.436150\pi\)
0.199247 + 0.979949i \(0.436150\pi\)
\(692\) 4.56107 0.173386
\(693\) 0 0
\(694\) −2.81965 −0.107033
\(695\) −11.4194 −0.433161
\(696\) 0 0
\(697\) 4.56144 0.172777
\(698\) 28.4932 1.07848
\(699\) 0 0
\(700\) −1.73500 −0.0655767
\(701\) 6.90150 0.260666 0.130333 0.991470i \(-0.458395\pi\)
0.130333 + 0.991470i \(0.458395\pi\)
\(702\) 0 0
\(703\) −4.68810 −0.176815
\(704\) −8.66904 −0.326727
\(705\) 0 0
\(706\) 30.8833 1.16231
\(707\) 2.06515 0.0776680
\(708\) 0 0
\(709\) −25.7827 −0.968290 −0.484145 0.874988i \(-0.660869\pi\)
−0.484145 + 0.874988i \(0.660869\pi\)
\(710\) 4.85120 0.182062
\(711\) 0 0
\(712\) 40.6842 1.52470
\(713\) −5.05085 −0.189156
\(714\) 0 0
\(715\) −2.41008 −0.0901318
\(716\) −3.45152 −0.128989
\(717\) 0 0
\(718\) 8.00851 0.298875
\(719\) −22.4484 −0.837182 −0.418591 0.908175i \(-0.637476\pi\)
−0.418591 + 0.908175i \(0.637476\pi\)
\(720\) 0 0
\(721\) −54.9180 −2.04525
\(722\) 18.5802 0.691485
\(723\) 0 0
\(724\) 0.0335741 0.00124777
\(725\) −11.3079 −0.419965
\(726\) 0 0
\(727\) 17.2835 0.641011 0.320505 0.947247i \(-0.396147\pi\)
0.320505 + 0.947247i \(0.396147\pi\)
\(728\) 15.9991 0.592965
\(729\) 0 0
\(730\) −28.3223 −1.04825
\(731\) 12.1916 0.450922
\(732\) 0 0
\(733\) −14.8357 −0.547969 −0.273984 0.961734i \(-0.588342\pi\)
−0.273984 + 0.961734i \(0.588342\pi\)
\(734\) −20.7543 −0.766055
\(735\) 0 0
\(736\) 3.17759 0.117128
\(737\) −1.34599 −0.0495801
\(738\) 0 0
\(739\) 44.5903 1.64028 0.820141 0.572162i \(-0.193895\pi\)
0.820141 + 0.572162i \(0.193895\pi\)
\(740\) 0.768487 0.0282502
\(741\) 0 0
\(742\) 27.7690 1.01943
\(743\) 7.79582 0.286001 0.143000 0.989723i \(-0.454325\pi\)
0.143000 + 0.989723i \(0.454325\pi\)
\(744\) 0 0
\(745\) −15.5489 −0.569666
\(746\) −43.9359 −1.60861
\(747\) 0 0
\(748\) 0.586497 0.0214444
\(749\) −10.1258 −0.369990
\(750\) 0 0
\(751\) 27.4136 1.00034 0.500169 0.865928i \(-0.333271\pi\)
0.500169 + 0.865928i \(0.333271\pi\)
\(752\) −8.98202 −0.327541
\(753\) 0 0
\(754\) 10.1986 0.371412
\(755\) −3.22834 −0.117491
\(756\) 0 0
\(757\) −17.2330 −0.626345 −0.313172 0.949696i \(-0.601392\pi\)
−0.313172 + 0.949696i \(0.601392\pi\)
\(758\) 21.6958 0.788028
\(759\) 0 0
\(760\) 11.3828 0.412897
\(761\) −48.2896 −1.75050 −0.875248 0.483674i \(-0.839302\pi\)
−0.875248 + 0.483674i \(0.839302\pi\)
\(762\) 0 0
\(763\) 49.0149 1.77446
\(764\) −2.69731 −0.0975853
\(765\) 0 0
\(766\) 42.0583 1.51963
\(767\) 6.36251 0.229737
\(768\) 0 0
\(769\) −4.13048 −0.148949 −0.0744745 0.997223i \(-0.523728\pi\)
−0.0744745 + 0.997223i \(0.523728\pi\)
\(770\) 8.70568 0.313731
\(771\) 0 0
\(772\) 2.36893 0.0852598
\(773\) 2.03774 0.0732926 0.0366463 0.999328i \(-0.488333\pi\)
0.0366463 + 0.999328i \(0.488333\pi\)
\(774\) 0 0
\(775\) −4.06141 −0.145890
\(776\) 22.5930 0.811043
\(777\) 0 0
\(778\) 52.3213 1.87581
\(779\) −3.80290 −0.136253
\(780\) 0 0
\(781\) −2.13067 −0.0762415
\(782\) 9.40146 0.336196
\(783\) 0 0
\(784\) −26.8167 −0.957740
\(785\) 31.5635 1.12655
\(786\) 0 0
\(787\) 31.5037 1.12299 0.561494 0.827481i \(-0.310227\pi\)
0.561494 + 0.827481i \(0.310227\pi\)
\(788\) −2.92582 −0.104228
\(789\) 0 0
\(790\) −8.72585 −0.310452
\(791\) 43.0626 1.53113
\(792\) 0 0
\(793\) −1.41350 −0.0501950
\(794\) −29.6890 −1.05362
\(795\) 0 0
\(796\) 2.24216 0.0794712
\(797\) 40.5246 1.43545 0.717727 0.696325i \(-0.245183\pi\)
0.717727 + 0.696325i \(0.245183\pi\)
\(798\) 0 0
\(799\) 6.90373 0.244237
\(800\) 2.55512 0.0903370
\(801\) 0 0
\(802\) −28.9534 −1.02238
\(803\) 12.4393 0.438973
\(804\) 0 0
\(805\) −16.9679 −0.598040
\(806\) 3.66299 0.129023
\(807\) 0 0
\(808\) −1.59886 −0.0562476
\(809\) −45.9846 −1.61673 −0.808367 0.588679i \(-0.799648\pi\)
−0.808367 + 0.588679i \(0.799648\pi\)
\(810\) 0 0
\(811\) −21.5553 −0.756909 −0.378454 0.925620i \(-0.623544\pi\)
−0.378454 + 0.925620i \(0.623544\pi\)
\(812\) 4.47929 0.157192
\(813\) 0 0
\(814\) 2.77593 0.0972963
\(815\) −24.7165 −0.865780
\(816\) 0 0
\(817\) −10.1642 −0.355600
\(818\) 42.0331 1.46965
\(819\) 0 0
\(820\) 0.623383 0.0217695
\(821\) 31.9758 1.11596 0.557982 0.829853i \(-0.311576\pi\)
0.557982 + 0.829853i \(0.311576\pi\)
\(822\) 0 0
\(823\) 4.65642 0.162313 0.0811563 0.996701i \(-0.474139\pi\)
0.0811563 + 0.996701i \(0.474139\pi\)
\(824\) 42.5179 1.48118
\(825\) 0 0
\(826\) −22.9827 −0.799669
\(827\) −48.3361 −1.68081 −0.840405 0.541959i \(-0.817683\pi\)
−0.840405 + 0.541959i \(0.817683\pi\)
\(828\) 0 0
\(829\) −22.7090 −0.788717 −0.394358 0.918957i \(-0.629033\pi\)
−0.394358 + 0.918957i \(0.629033\pi\)
\(830\) 16.0702 0.557804
\(831\) 0 0
\(832\) −12.2537 −0.424821
\(833\) 20.6118 0.714157
\(834\) 0 0
\(835\) −19.5644 −0.677053
\(836\) −0.488966 −0.0169112
\(837\) 0 0
\(838\) 13.6990 0.473222
\(839\) −7.55301 −0.260759 −0.130379 0.991464i \(-0.541620\pi\)
−0.130379 + 0.991464i \(0.541620\pi\)
\(840\) 0 0
\(841\) 0.193941 0.00668761
\(842\) −20.2461 −0.697727
\(843\) 0 0
\(844\) 4.21204 0.144985
\(845\) 18.7589 0.645324
\(846\) 0 0
\(847\) −3.82358 −0.131380
\(848\) −19.1406 −0.657289
\(849\) 0 0
\(850\) 7.55976 0.259298
\(851\) −5.41046 −0.185468
\(852\) 0 0
\(853\) −29.3846 −1.00611 −0.503054 0.864255i \(-0.667790\pi\)
−0.503054 + 0.864255i \(0.667790\pi\)
\(854\) 5.10585 0.174719
\(855\) 0 0
\(856\) 7.83951 0.267949
\(857\) 7.21276 0.246383 0.123192 0.992383i \(-0.460687\pi\)
0.123192 + 0.992383i \(0.460687\pi\)
\(858\) 0 0
\(859\) 26.9437 0.919308 0.459654 0.888098i \(-0.347973\pi\)
0.459654 + 0.888098i \(0.347973\pi\)
\(860\) 1.66615 0.0568151
\(861\) 0 0
\(862\) 15.1112 0.514689
\(863\) −0.507609 −0.0172792 −0.00863961 0.999963i \(-0.502750\pi\)
−0.00863961 + 0.999963i \(0.502750\pi\)
\(864\) 0 0
\(865\) 35.8682 1.21955
\(866\) −31.4810 −1.06977
\(867\) 0 0
\(868\) 1.60880 0.0546064
\(869\) 3.83244 0.130007
\(870\) 0 0
\(871\) −1.90256 −0.0644657
\(872\) −37.9477 −1.28507
\(873\) 0 0
\(874\) −7.83805 −0.265126
\(875\) −46.2408 −1.56322
\(876\) 0 0
\(877\) 43.3897 1.46517 0.732583 0.680677i \(-0.238314\pi\)
0.732583 + 0.680677i \(0.238314\pi\)
\(878\) −15.3484 −0.517984
\(879\) 0 0
\(880\) −6.00064 −0.202282
\(881\) 28.8083 0.970575 0.485288 0.874355i \(-0.338715\pi\)
0.485288 + 0.874355i \(0.338715\pi\)
\(882\) 0 0
\(883\) −33.4794 −1.12667 −0.563336 0.826228i \(-0.690482\pi\)
−0.563336 + 0.826228i \(0.690482\pi\)
\(884\) 0.829015 0.0278828
\(885\) 0 0
\(886\) −5.20539 −0.174879
\(887\) −40.9969 −1.37654 −0.688271 0.725453i \(-0.741630\pi\)
−0.688271 + 0.725453i \(0.741630\pi\)
\(888\) 0 0
\(889\) −82.5298 −2.76796
\(890\) 31.2918 1.04890
\(891\) 0 0
\(892\) 3.97987 0.133256
\(893\) −5.75568 −0.192607
\(894\) 0 0
\(895\) −27.1427 −0.907279
\(896\) 34.9265 1.16681
\(897\) 0 0
\(898\) 12.8861 0.430015
\(899\) 10.4854 0.349709
\(900\) 0 0
\(901\) 14.7118 0.490120
\(902\) 2.25178 0.0749761
\(903\) 0 0
\(904\) −33.3394 −1.10885
\(905\) 0.264026 0.00877652
\(906\) 0 0
\(907\) −9.25129 −0.307184 −0.153592 0.988134i \(-0.549084\pi\)
−0.153592 + 0.988134i \(0.549084\pi\)
\(908\) −1.66797 −0.0553534
\(909\) 0 0
\(910\) 12.3055 0.407924
\(911\) 45.2640 1.49966 0.749831 0.661629i \(-0.230135\pi\)
0.749831 + 0.661629i \(0.230135\pi\)
\(912\) 0 0
\(913\) −7.05810 −0.233589
\(914\) −8.20556 −0.271416
\(915\) 0 0
\(916\) 5.04077 0.166552
\(917\) −2.44694 −0.0808052
\(918\) 0 0
\(919\) −55.4379 −1.82873 −0.914364 0.404892i \(-0.867309\pi\)
−0.914364 + 0.404892i \(0.867309\pi\)
\(920\) 13.1367 0.433104
\(921\) 0 0
\(922\) 39.3470 1.29583
\(923\) −3.01172 −0.0991318
\(924\) 0 0
\(925\) −4.35057 −0.143046
\(926\) 26.7143 0.877887
\(927\) 0 0
\(928\) −6.59661 −0.216544
\(929\) 18.7740 0.615955 0.307977 0.951394i \(-0.400348\pi\)
0.307977 + 0.951394i \(0.400348\pi\)
\(930\) 0 0
\(931\) −17.1842 −0.563189
\(932\) −3.07093 −0.100592
\(933\) 0 0
\(934\) 12.2182 0.399791
\(935\) 4.61220 0.150835
\(936\) 0 0
\(937\) −18.9016 −0.617488 −0.308744 0.951145i \(-0.599909\pi\)
−0.308744 + 0.951145i \(0.599909\pi\)
\(938\) 6.87241 0.224392
\(939\) 0 0
\(940\) 0.943489 0.0307732
\(941\) 40.7528 1.32850 0.664252 0.747509i \(-0.268750\pi\)
0.664252 + 0.747509i \(0.268750\pi\)
\(942\) 0 0
\(943\) −4.38886 −0.142921
\(944\) 15.8415 0.515596
\(945\) 0 0
\(946\) 6.01845 0.195677
\(947\) −45.2880 −1.47166 −0.735831 0.677166i \(-0.763208\pi\)
−0.735831 + 0.677166i \(0.763208\pi\)
\(948\) 0 0
\(949\) 17.5830 0.570768
\(950\) −6.30261 −0.204484
\(951\) 0 0
\(952\) −30.6176 −0.992324
\(953\) 28.1077 0.910498 0.455249 0.890364i \(-0.349550\pi\)
0.455249 + 0.890364i \(0.349550\pi\)
\(954\) 0 0
\(955\) −21.2116 −0.686391
\(956\) 3.90444 0.126279
\(957\) 0 0
\(958\) −43.8095 −1.41542
\(959\) 49.5073 1.59867
\(960\) 0 0
\(961\) −27.2340 −0.878516
\(962\) 3.92378 0.126508
\(963\) 0 0
\(964\) −3.93657 −0.126788
\(965\) 18.6292 0.599697
\(966\) 0 0
\(967\) −29.7895 −0.957966 −0.478983 0.877824i \(-0.658994\pi\)
−0.478983 + 0.877824i \(0.658994\pi\)
\(968\) 2.96025 0.0951459
\(969\) 0 0
\(970\) 17.3772 0.557948
\(971\) −28.0668 −0.900706 −0.450353 0.892851i \(-0.648702\pi\)
−0.450353 + 0.892851i \(0.648702\pi\)
\(972\) 0 0
\(973\) −25.6081 −0.820959
\(974\) 31.5219 1.01003
\(975\) 0 0
\(976\) −3.51936 −0.112652
\(977\) 21.3268 0.682306 0.341153 0.940008i \(-0.389183\pi\)
0.341153 + 0.940008i \(0.389183\pi\)
\(978\) 0 0
\(979\) −13.7435 −0.439245
\(980\) 2.81688 0.0899820
\(981\) 0 0
\(982\) −16.7938 −0.535910
\(983\) −53.8387 −1.71719 −0.858594 0.512657i \(-0.828661\pi\)
−0.858594 + 0.512657i \(0.828661\pi\)
\(984\) 0 0
\(985\) −23.0086 −0.733115
\(986\) −19.5172 −0.621555
\(987\) 0 0
\(988\) −0.691154 −0.0219886
\(989\) −11.7303 −0.373003
\(990\) 0 0
\(991\) 24.4123 0.775484 0.387742 0.921768i \(-0.373255\pi\)
0.387742 + 0.921768i \(0.373255\pi\)
\(992\) −2.36927 −0.0752245
\(993\) 0 0
\(994\) 10.8789 0.345058
\(995\) 17.6323 0.558981
\(996\) 0 0
\(997\) 44.4541 1.40788 0.703938 0.710262i \(-0.251423\pi\)
0.703938 + 0.710262i \(0.251423\pi\)
\(998\) −34.6923 −1.09817
\(999\) 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 6039.2.a.a.1.1 5
3.2 odd 2 671.2.a.a.1.5 5
33.32 even 2 7381.2.a.g.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
671.2.a.a.1.5 5 3.2 odd 2
6039.2.a.a.1.1 5 1.1 even 1 trivial
7381.2.a.g.1.1 5 33.32 even 2