Properties

Label 8016.2.a.x.1.1
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $8$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.459587\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.63865 q^{5} +0.604857 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.63865 q^{5} +0.604857 q^{7} +1.00000 q^{9} -4.18757 q^{11} -0.588099 q^{13} +2.63865 q^{15} +3.13576 q^{17} -3.33284 q^{19} -0.604857 q^{21} -2.06079 q^{23} +1.96248 q^{25} -1.00000 q^{27} +10.1474 q^{29} +1.00945 q^{31} +4.18757 q^{33} -1.59601 q^{35} -0.963871 q^{37} +0.588099 q^{39} +10.0988 q^{41} +1.46622 q^{43} -2.63865 q^{45} +3.81359 q^{47} -6.63415 q^{49} -3.13576 q^{51} -2.52878 q^{53} +11.0495 q^{55} +3.33284 q^{57} +3.70087 q^{59} +0.962658 q^{61} +0.604857 q^{63} +1.55179 q^{65} +6.12872 q^{67} +2.06079 q^{69} -4.34451 q^{71} +13.8707 q^{73} -1.96248 q^{75} -2.53288 q^{77} -9.16426 q^{79} +1.00000 q^{81} -15.6712 q^{83} -8.27418 q^{85} -10.1474 q^{87} +5.57911 q^{89} -0.355716 q^{91} -1.00945 q^{93} +8.79421 q^{95} -6.60693 q^{97} -4.18757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.63865 −1.18004 −0.590020 0.807388i \(-0.700880\pi\)
−0.590020 + 0.807388i \(0.700880\pi\)
\(6\) 0 0
\(7\) 0.604857 0.228614 0.114307 0.993445i \(-0.463535\pi\)
0.114307 + 0.993445i \(0.463535\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.18757 −1.26260 −0.631301 0.775538i \(-0.717479\pi\)
−0.631301 + 0.775538i \(0.717479\pi\)
\(12\) 0 0
\(13\) −0.588099 −0.163109 −0.0815547 0.996669i \(-0.525989\pi\)
−0.0815547 + 0.996669i \(0.525989\pi\)
\(14\) 0 0
\(15\) 2.63865 0.681297
\(16\) 0 0
\(17\) 3.13576 0.760534 0.380267 0.924877i \(-0.375832\pi\)
0.380267 + 0.924877i \(0.375832\pi\)
\(18\) 0 0
\(19\) −3.33284 −0.764607 −0.382303 0.924037i \(-0.624869\pi\)
−0.382303 + 0.924037i \(0.624869\pi\)
\(20\) 0 0
\(21\) −0.604857 −0.131991
\(22\) 0 0
\(23\) −2.06079 −0.429705 −0.214852 0.976647i \(-0.568927\pi\)
−0.214852 + 0.976647i \(0.568927\pi\)
\(24\) 0 0
\(25\) 1.96248 0.392495
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.1474 1.88432 0.942162 0.335158i \(-0.108790\pi\)
0.942162 + 0.335158i \(0.108790\pi\)
\(30\) 0 0
\(31\) 1.00945 0.181303 0.0906515 0.995883i \(-0.471105\pi\)
0.0906515 + 0.995883i \(0.471105\pi\)
\(32\) 0 0
\(33\) 4.18757 0.728963
\(34\) 0 0
\(35\) −1.59601 −0.269774
\(36\) 0 0
\(37\) −0.963871 −0.158459 −0.0792297 0.996856i \(-0.525246\pi\)
−0.0792297 + 0.996856i \(0.525246\pi\)
\(38\) 0 0
\(39\) 0.588099 0.0941713
\(40\) 0 0
\(41\) 10.0988 1.57717 0.788585 0.614926i \(-0.210814\pi\)
0.788585 + 0.614926i \(0.210814\pi\)
\(42\) 0 0
\(43\) 1.46622 0.223596 0.111798 0.993731i \(-0.464339\pi\)
0.111798 + 0.993731i \(0.464339\pi\)
\(44\) 0 0
\(45\) −2.63865 −0.393347
\(46\) 0 0
\(47\) 3.81359 0.556270 0.278135 0.960542i \(-0.410284\pi\)
0.278135 + 0.960542i \(0.410284\pi\)
\(48\) 0 0
\(49\) −6.63415 −0.947735
\(50\) 0 0
\(51\) −3.13576 −0.439094
\(52\) 0 0
\(53\) −2.52878 −0.347354 −0.173677 0.984803i \(-0.555565\pi\)
−0.173677 + 0.984803i \(0.555565\pi\)
\(54\) 0 0
\(55\) 11.0495 1.48992
\(56\) 0 0
\(57\) 3.33284 0.441446
\(58\) 0 0
\(59\) 3.70087 0.481812 0.240906 0.970548i \(-0.422555\pi\)
0.240906 + 0.970548i \(0.422555\pi\)
\(60\) 0 0
\(61\) 0.962658 0.123256 0.0616279 0.998099i \(-0.480371\pi\)
0.0616279 + 0.998099i \(0.480371\pi\)
\(62\) 0 0
\(63\) 0.604857 0.0762048
\(64\) 0 0
\(65\) 1.55179 0.192476
\(66\) 0 0
\(67\) 6.12872 0.748742 0.374371 0.927279i \(-0.377859\pi\)
0.374371 + 0.927279i \(0.377859\pi\)
\(68\) 0 0
\(69\) 2.06079 0.248090
\(70\) 0 0
\(71\) −4.34451 −0.515599 −0.257799 0.966198i \(-0.582997\pi\)
−0.257799 + 0.966198i \(0.582997\pi\)
\(72\) 0 0
\(73\) 13.8707 1.62344 0.811721 0.584046i \(-0.198531\pi\)
0.811721 + 0.584046i \(0.198531\pi\)
\(74\) 0 0
\(75\) −1.96248 −0.226607
\(76\) 0 0
\(77\) −2.53288 −0.288649
\(78\) 0 0
\(79\) −9.16426 −1.03106 −0.515530 0.856872i \(-0.672405\pi\)
−0.515530 + 0.856872i \(0.672405\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.6712 −1.72013 −0.860067 0.510182i \(-0.829578\pi\)
−0.860067 + 0.510182i \(0.829578\pi\)
\(84\) 0 0
\(85\) −8.27418 −0.897461
\(86\) 0 0
\(87\) −10.1474 −1.08791
\(88\) 0 0
\(89\) 5.57911 0.591384 0.295692 0.955283i \(-0.404450\pi\)
0.295692 + 0.955283i \(0.404450\pi\)
\(90\) 0 0
\(91\) −0.355716 −0.0372892
\(92\) 0 0
\(93\) −1.00945 −0.104675
\(94\) 0 0
\(95\) 8.79421 0.902267
\(96\) 0 0
\(97\) −6.60693 −0.670832 −0.335416 0.942070i \(-0.608877\pi\)
−0.335416 + 0.942070i \(0.608877\pi\)
\(98\) 0 0
\(99\) −4.18757 −0.420867
\(100\) 0 0
\(101\) 0.212823 0.0211767 0.0105883 0.999944i \(-0.496630\pi\)
0.0105883 + 0.999944i \(0.496630\pi\)
\(102\) 0 0
\(103\) 1.50031 0.147830 0.0739150 0.997265i \(-0.476451\pi\)
0.0739150 + 0.997265i \(0.476451\pi\)
\(104\) 0 0
\(105\) 1.59601 0.155754
\(106\) 0 0
\(107\) 7.70835 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(108\) 0 0
\(109\) −19.3144 −1.84998 −0.924991 0.379988i \(-0.875928\pi\)
−0.924991 + 0.379988i \(0.875928\pi\)
\(110\) 0 0
\(111\) 0.963871 0.0914866
\(112\) 0 0
\(113\) 2.91046 0.273793 0.136896 0.990585i \(-0.456287\pi\)
0.136896 + 0.990585i \(0.456287\pi\)
\(114\) 0 0
\(115\) 5.43771 0.507069
\(116\) 0 0
\(117\) −0.588099 −0.0543698
\(118\) 0 0
\(119\) 1.89669 0.173869
\(120\) 0 0
\(121\) 6.53578 0.594162
\(122\) 0 0
\(123\) −10.0988 −0.910580
\(124\) 0 0
\(125\) 8.01496 0.716880
\(126\) 0 0
\(127\) 11.5989 1.02923 0.514617 0.857420i \(-0.327934\pi\)
0.514617 + 0.857420i \(0.327934\pi\)
\(128\) 0 0
\(129\) −1.46622 −0.129093
\(130\) 0 0
\(131\) 8.26258 0.721905 0.360952 0.932584i \(-0.382452\pi\)
0.360952 + 0.932584i \(0.382452\pi\)
\(132\) 0 0
\(133\) −2.01589 −0.174800
\(134\) 0 0
\(135\) 2.63865 0.227099
\(136\) 0 0
\(137\) 6.10912 0.521937 0.260969 0.965347i \(-0.415958\pi\)
0.260969 + 0.965347i \(0.415958\pi\)
\(138\) 0 0
\(139\) −14.8050 −1.25574 −0.627872 0.778317i \(-0.716074\pi\)
−0.627872 + 0.778317i \(0.716074\pi\)
\(140\) 0 0
\(141\) −3.81359 −0.321163
\(142\) 0 0
\(143\) 2.46271 0.205942
\(144\) 0 0
\(145\) −26.7754 −2.22358
\(146\) 0 0
\(147\) 6.63415 0.547175
\(148\) 0 0
\(149\) 7.27551 0.596033 0.298017 0.954561i \(-0.403675\pi\)
0.298017 + 0.954561i \(0.403675\pi\)
\(150\) 0 0
\(151\) 6.77287 0.551168 0.275584 0.961277i \(-0.411129\pi\)
0.275584 + 0.961277i \(0.411129\pi\)
\(152\) 0 0
\(153\) 3.13576 0.253511
\(154\) 0 0
\(155\) −2.66359 −0.213945
\(156\) 0 0
\(157\) −13.7559 −1.09784 −0.548919 0.835875i \(-0.684961\pi\)
−0.548919 + 0.835875i \(0.684961\pi\)
\(158\) 0 0
\(159\) 2.52878 0.200545
\(160\) 0 0
\(161\) −1.24648 −0.0982367
\(162\) 0 0
\(163\) −10.7400 −0.841222 −0.420611 0.907241i \(-0.638184\pi\)
−0.420611 + 0.907241i \(0.638184\pi\)
\(164\) 0 0
\(165\) −11.0495 −0.860206
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −12.6541 −0.973395
\(170\) 0 0
\(171\) −3.33284 −0.254869
\(172\) 0 0
\(173\) 1.63924 0.124629 0.0623145 0.998057i \(-0.480152\pi\)
0.0623145 + 0.998057i \(0.480152\pi\)
\(174\) 0 0
\(175\) 1.18702 0.0897300
\(176\) 0 0
\(177\) −3.70087 −0.278175
\(178\) 0 0
\(179\) 19.0432 1.42336 0.711678 0.702505i \(-0.247935\pi\)
0.711678 + 0.702505i \(0.247935\pi\)
\(180\) 0 0
\(181\) 14.7592 1.09704 0.548520 0.836138i \(-0.315192\pi\)
0.548520 + 0.836138i \(0.315192\pi\)
\(182\) 0 0
\(183\) −0.962658 −0.0711617
\(184\) 0 0
\(185\) 2.54332 0.186989
\(186\) 0 0
\(187\) −13.1312 −0.960251
\(188\) 0 0
\(189\) −0.604857 −0.0439968
\(190\) 0 0
\(191\) 8.24196 0.596367 0.298184 0.954508i \(-0.403619\pi\)
0.298184 + 0.954508i \(0.403619\pi\)
\(192\) 0 0
\(193\) 12.4363 0.895186 0.447593 0.894237i \(-0.352281\pi\)
0.447593 + 0.894237i \(0.352281\pi\)
\(194\) 0 0
\(195\) −1.55179 −0.111126
\(196\) 0 0
\(197\) −18.4826 −1.31683 −0.658415 0.752655i \(-0.728773\pi\)
−0.658415 + 0.752655i \(0.728773\pi\)
\(198\) 0 0
\(199\) −4.83561 −0.342787 −0.171394 0.985203i \(-0.554827\pi\)
−0.171394 + 0.985203i \(0.554827\pi\)
\(200\) 0 0
\(201\) −6.12872 −0.432287
\(202\) 0 0
\(203\) 6.13772 0.430783
\(204\) 0 0
\(205\) −26.6472 −1.86112
\(206\) 0 0
\(207\) −2.06079 −0.143235
\(208\) 0 0
\(209\) 13.9565 0.965394
\(210\) 0 0
\(211\) −25.4148 −1.74963 −0.874814 0.484459i \(-0.839016\pi\)
−0.874814 + 0.484459i \(0.839016\pi\)
\(212\) 0 0
\(213\) 4.34451 0.297681
\(214\) 0 0
\(215\) −3.86883 −0.263852
\(216\) 0 0
\(217\) 0.610574 0.0414485
\(218\) 0 0
\(219\) −13.8707 −0.937294
\(220\) 0 0
\(221\) −1.84414 −0.124050
\(222\) 0 0
\(223\) −13.4950 −0.903692 −0.451846 0.892096i \(-0.649234\pi\)
−0.451846 + 0.892096i \(0.649234\pi\)
\(224\) 0 0
\(225\) 1.96248 0.130832
\(226\) 0 0
\(227\) −22.1793 −1.47209 −0.736047 0.676930i \(-0.763310\pi\)
−0.736047 + 0.676930i \(0.763310\pi\)
\(228\) 0 0
\(229\) −8.65236 −0.571764 −0.285882 0.958265i \(-0.592287\pi\)
−0.285882 + 0.958265i \(0.592287\pi\)
\(230\) 0 0
\(231\) 2.53288 0.166651
\(232\) 0 0
\(233\) −15.2852 −1.00136 −0.500682 0.865631i \(-0.666917\pi\)
−0.500682 + 0.865631i \(0.666917\pi\)
\(234\) 0 0
\(235\) −10.0627 −0.656421
\(236\) 0 0
\(237\) 9.16426 0.595282
\(238\) 0 0
\(239\) 29.8813 1.93286 0.966430 0.256930i \(-0.0827111\pi\)
0.966430 + 0.256930i \(0.0827111\pi\)
\(240\) 0 0
\(241\) 17.5171 1.12837 0.564187 0.825647i \(-0.309190\pi\)
0.564187 + 0.825647i \(0.309190\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 17.5052 1.11837
\(246\) 0 0
\(247\) 1.96004 0.124715
\(248\) 0 0
\(249\) 15.6712 0.993119
\(250\) 0 0
\(251\) −11.5902 −0.731567 −0.365784 0.930700i \(-0.619199\pi\)
−0.365784 + 0.930700i \(0.619199\pi\)
\(252\) 0 0
\(253\) 8.62972 0.542546
\(254\) 0 0
\(255\) 8.27418 0.518149
\(256\) 0 0
\(257\) −12.3462 −0.770137 −0.385069 0.922888i \(-0.625822\pi\)
−0.385069 + 0.922888i \(0.625822\pi\)
\(258\) 0 0
\(259\) −0.583004 −0.0362261
\(260\) 0 0
\(261\) 10.1474 0.628108
\(262\) 0 0
\(263\) 10.9134 0.672948 0.336474 0.941693i \(-0.390766\pi\)
0.336474 + 0.941693i \(0.390766\pi\)
\(264\) 0 0
\(265\) 6.67256 0.409892
\(266\) 0 0
\(267\) −5.57911 −0.341436
\(268\) 0 0
\(269\) 2.64700 0.161391 0.0806953 0.996739i \(-0.474286\pi\)
0.0806953 + 0.996739i \(0.474286\pi\)
\(270\) 0 0
\(271\) −15.3534 −0.932651 −0.466326 0.884613i \(-0.654422\pi\)
−0.466326 + 0.884613i \(0.654422\pi\)
\(272\) 0 0
\(273\) 0.355716 0.0215289
\(274\) 0 0
\(275\) −8.21802 −0.495565
\(276\) 0 0
\(277\) 18.8696 1.13376 0.566881 0.823800i \(-0.308150\pi\)
0.566881 + 0.823800i \(0.308150\pi\)
\(278\) 0 0
\(279\) 1.00945 0.0604343
\(280\) 0 0
\(281\) −2.41769 −0.144227 −0.0721136 0.997396i \(-0.522974\pi\)
−0.0721136 + 0.997396i \(0.522974\pi\)
\(282\) 0 0
\(283\) 17.2449 1.02510 0.512551 0.858657i \(-0.328701\pi\)
0.512551 + 0.858657i \(0.328701\pi\)
\(284\) 0 0
\(285\) −8.79421 −0.520924
\(286\) 0 0
\(287\) 6.10834 0.360564
\(288\) 0 0
\(289\) −7.16700 −0.421588
\(290\) 0 0
\(291\) 6.60693 0.387305
\(292\) 0 0
\(293\) −3.14330 −0.183634 −0.0918168 0.995776i \(-0.529267\pi\)
−0.0918168 + 0.995776i \(0.529267\pi\)
\(294\) 0 0
\(295\) −9.76531 −0.568558
\(296\) 0 0
\(297\) 4.18757 0.242988
\(298\) 0 0
\(299\) 1.21195 0.0700889
\(300\) 0 0
\(301\) 0.886850 0.0511172
\(302\) 0 0
\(303\) −0.212823 −0.0122264
\(304\) 0 0
\(305\) −2.54012 −0.145447
\(306\) 0 0
\(307\) −27.5662 −1.57329 −0.786643 0.617409i \(-0.788182\pi\)
−0.786643 + 0.617409i \(0.788182\pi\)
\(308\) 0 0
\(309\) −1.50031 −0.0853497
\(310\) 0 0
\(311\) −3.97921 −0.225640 −0.112820 0.993615i \(-0.535988\pi\)
−0.112820 + 0.993615i \(0.535988\pi\)
\(312\) 0 0
\(313\) −15.9773 −0.903090 −0.451545 0.892248i \(-0.649127\pi\)
−0.451545 + 0.892248i \(0.649127\pi\)
\(314\) 0 0
\(315\) −1.59601 −0.0899247
\(316\) 0 0
\(317\) −25.4618 −1.43008 −0.715038 0.699086i \(-0.753590\pi\)
−0.715038 + 0.699086i \(0.753590\pi\)
\(318\) 0 0
\(319\) −42.4930 −2.37915
\(320\) 0 0
\(321\) −7.70835 −0.430238
\(322\) 0 0
\(323\) −10.4510 −0.581509
\(324\) 0 0
\(325\) −1.15413 −0.0640197
\(326\) 0 0
\(327\) 19.3144 1.06809
\(328\) 0 0
\(329\) 2.30668 0.127171
\(330\) 0 0
\(331\) 24.0840 1.32378 0.661889 0.749602i \(-0.269755\pi\)
0.661889 + 0.749602i \(0.269755\pi\)
\(332\) 0 0
\(333\) −0.963871 −0.0528198
\(334\) 0 0
\(335\) −16.1715 −0.883546
\(336\) 0 0
\(337\) −7.35478 −0.400641 −0.200320 0.979730i \(-0.564198\pi\)
−0.200320 + 0.979730i \(0.564198\pi\)
\(338\) 0 0
\(339\) −2.91046 −0.158074
\(340\) 0 0
\(341\) −4.22716 −0.228913
\(342\) 0 0
\(343\) −8.24671 −0.445280
\(344\) 0 0
\(345\) −5.43771 −0.292756
\(346\) 0 0
\(347\) −20.8836 −1.12109 −0.560544 0.828125i \(-0.689408\pi\)
−0.560544 + 0.828125i \(0.689408\pi\)
\(348\) 0 0
\(349\) −26.6111 −1.42446 −0.712229 0.701947i \(-0.752314\pi\)
−0.712229 + 0.701947i \(0.752314\pi\)
\(350\) 0 0
\(351\) 0.588099 0.0313904
\(352\) 0 0
\(353\) 2.13148 0.113447 0.0567237 0.998390i \(-0.481935\pi\)
0.0567237 + 0.998390i \(0.481935\pi\)
\(354\) 0 0
\(355\) 11.4637 0.608428
\(356\) 0 0
\(357\) −1.89669 −0.100383
\(358\) 0 0
\(359\) −12.5447 −0.662082 −0.331041 0.943616i \(-0.607400\pi\)
−0.331041 + 0.943616i \(0.607400\pi\)
\(360\) 0 0
\(361\) −7.89215 −0.415376
\(362\) 0 0
\(363\) −6.53578 −0.343040
\(364\) 0 0
\(365\) −36.5999 −1.91573
\(366\) 0 0
\(367\) 18.2125 0.950686 0.475343 0.879800i \(-0.342324\pi\)
0.475343 + 0.879800i \(0.342324\pi\)
\(368\) 0 0
\(369\) 10.0988 0.525723
\(370\) 0 0
\(371\) −1.52955 −0.0794102
\(372\) 0 0
\(373\) 24.0633 1.24595 0.622976 0.782241i \(-0.285923\pi\)
0.622976 + 0.782241i \(0.285923\pi\)
\(374\) 0 0
\(375\) −8.01496 −0.413891
\(376\) 0 0
\(377\) −5.96768 −0.307351
\(378\) 0 0
\(379\) −18.1878 −0.934243 −0.467122 0.884193i \(-0.654709\pi\)
−0.467122 + 0.884193i \(0.654709\pi\)
\(380\) 0 0
\(381\) −11.5989 −0.594228
\(382\) 0 0
\(383\) 21.9159 1.11985 0.559926 0.828542i \(-0.310829\pi\)
0.559926 + 0.828542i \(0.310829\pi\)
\(384\) 0 0
\(385\) 6.68339 0.340617
\(386\) 0 0
\(387\) 1.46622 0.0745319
\(388\) 0 0
\(389\) 6.79088 0.344311 0.172156 0.985070i \(-0.444927\pi\)
0.172156 + 0.985070i \(0.444927\pi\)
\(390\) 0 0
\(391\) −6.46215 −0.326805
\(392\) 0 0
\(393\) −8.26258 −0.416792
\(394\) 0 0
\(395\) 24.1813 1.21669
\(396\) 0 0
\(397\) −36.3338 −1.82354 −0.911770 0.410701i \(-0.865284\pi\)
−0.911770 + 0.410701i \(0.865284\pi\)
\(398\) 0 0
\(399\) 2.01589 0.100921
\(400\) 0 0
\(401\) −26.5238 −1.32453 −0.662267 0.749268i \(-0.730405\pi\)
−0.662267 + 0.749268i \(0.730405\pi\)
\(402\) 0 0
\(403\) −0.593658 −0.0295722
\(404\) 0 0
\(405\) −2.63865 −0.131116
\(406\) 0 0
\(407\) 4.03628 0.200071
\(408\) 0 0
\(409\) 11.5743 0.572313 0.286157 0.958183i \(-0.407622\pi\)
0.286157 + 0.958183i \(0.407622\pi\)
\(410\) 0 0
\(411\) −6.10912 −0.301341
\(412\) 0 0
\(413\) 2.23850 0.110149
\(414\) 0 0
\(415\) 41.3507 2.02983
\(416\) 0 0
\(417\) 14.8050 0.725004
\(418\) 0 0
\(419\) −28.7087 −1.40251 −0.701256 0.712909i \(-0.747377\pi\)
−0.701256 + 0.712909i \(0.747377\pi\)
\(420\) 0 0
\(421\) −31.6417 −1.54212 −0.771061 0.636762i \(-0.780274\pi\)
−0.771061 + 0.636762i \(0.780274\pi\)
\(422\) 0 0
\(423\) 3.81359 0.185423
\(424\) 0 0
\(425\) 6.15386 0.298506
\(426\) 0 0
\(427\) 0.582270 0.0281780
\(428\) 0 0
\(429\) −2.46271 −0.118901
\(430\) 0 0
\(431\) −27.0921 −1.30498 −0.652490 0.757798i \(-0.726275\pi\)
−0.652490 + 0.757798i \(0.726275\pi\)
\(432\) 0 0
\(433\) −37.9931 −1.82583 −0.912915 0.408150i \(-0.866174\pi\)
−0.912915 + 0.408150i \(0.866174\pi\)
\(434\) 0 0
\(435\) 26.7754 1.28378
\(436\) 0 0
\(437\) 6.86830 0.328555
\(438\) 0 0
\(439\) 27.8044 1.32703 0.663516 0.748162i \(-0.269063\pi\)
0.663516 + 0.748162i \(0.269063\pi\)
\(440\) 0 0
\(441\) −6.63415 −0.315912
\(442\) 0 0
\(443\) 7.02886 0.333951 0.166976 0.985961i \(-0.446600\pi\)
0.166976 + 0.985961i \(0.446600\pi\)
\(444\) 0 0
\(445\) −14.7213 −0.697857
\(446\) 0 0
\(447\) −7.27551 −0.344120
\(448\) 0 0
\(449\) 6.19816 0.292509 0.146255 0.989247i \(-0.453278\pi\)
0.146255 + 0.989247i \(0.453278\pi\)
\(450\) 0 0
\(451\) −42.2895 −1.99134
\(452\) 0 0
\(453\) −6.77287 −0.318217
\(454\) 0 0
\(455\) 0.938610 0.0440027
\(456\) 0 0
\(457\) −24.9646 −1.16779 −0.583897 0.811828i \(-0.698473\pi\)
−0.583897 + 0.811828i \(0.698473\pi\)
\(458\) 0 0
\(459\) −3.13576 −0.146365
\(460\) 0 0
\(461\) 33.7554 1.57215 0.786073 0.618134i \(-0.212111\pi\)
0.786073 + 0.618134i \(0.212111\pi\)
\(462\) 0 0
\(463\) −17.1602 −0.797502 −0.398751 0.917059i \(-0.630556\pi\)
−0.398751 + 0.917059i \(0.630556\pi\)
\(464\) 0 0
\(465\) 2.66359 0.123521
\(466\) 0 0
\(467\) −23.4137 −1.08346 −0.541728 0.840554i \(-0.682230\pi\)
−0.541728 + 0.840554i \(0.682230\pi\)
\(468\) 0 0
\(469\) 3.70700 0.171173
\(470\) 0 0
\(471\) 13.7559 0.633838
\(472\) 0 0
\(473\) −6.13989 −0.282312
\(474\) 0 0
\(475\) −6.54063 −0.300105
\(476\) 0 0
\(477\) −2.52878 −0.115785
\(478\) 0 0
\(479\) −12.8820 −0.588594 −0.294297 0.955714i \(-0.595086\pi\)
−0.294297 + 0.955714i \(0.595086\pi\)
\(480\) 0 0
\(481\) 0.566852 0.0258462
\(482\) 0 0
\(483\) 1.24648 0.0567170
\(484\) 0 0
\(485\) 17.4334 0.791609
\(486\) 0 0
\(487\) 37.7019 1.70844 0.854218 0.519915i \(-0.174037\pi\)
0.854218 + 0.519915i \(0.174037\pi\)
\(488\) 0 0
\(489\) 10.7400 0.485680
\(490\) 0 0
\(491\) 12.3330 0.556581 0.278290 0.960497i \(-0.410232\pi\)
0.278290 + 0.960497i \(0.410232\pi\)
\(492\) 0 0
\(493\) 31.8198 1.43309
\(494\) 0 0
\(495\) 11.0495 0.496640
\(496\) 0 0
\(497\) −2.62781 −0.117873
\(498\) 0 0
\(499\) −11.1753 −0.500274 −0.250137 0.968210i \(-0.580476\pi\)
−0.250137 + 0.968210i \(0.580476\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 27.2408 1.21461 0.607303 0.794470i \(-0.292252\pi\)
0.607303 + 0.794470i \(0.292252\pi\)
\(504\) 0 0
\(505\) −0.561565 −0.0249893
\(506\) 0 0
\(507\) 12.6541 0.561990
\(508\) 0 0
\(509\) 8.83882 0.391774 0.195887 0.980627i \(-0.437241\pi\)
0.195887 + 0.980627i \(0.437241\pi\)
\(510\) 0 0
\(511\) 8.38978 0.371142
\(512\) 0 0
\(513\) 3.33284 0.147149
\(514\) 0 0
\(515\) −3.95879 −0.174445
\(516\) 0 0
\(517\) −15.9697 −0.702347
\(518\) 0 0
\(519\) −1.63924 −0.0719545
\(520\) 0 0
\(521\) −35.2225 −1.54312 −0.771562 0.636154i \(-0.780524\pi\)
−0.771562 + 0.636154i \(0.780524\pi\)
\(522\) 0 0
\(523\) −0.251562 −0.0110000 −0.00550001 0.999985i \(-0.501751\pi\)
−0.00550001 + 0.999985i \(0.501751\pi\)
\(524\) 0 0
\(525\) −1.18702 −0.0518057
\(526\) 0 0
\(527\) 3.16540 0.137887
\(528\) 0 0
\(529\) −18.7531 −0.815354
\(530\) 0 0
\(531\) 3.70087 0.160604
\(532\) 0 0
\(533\) −5.93911 −0.257251
\(534\) 0 0
\(535\) −20.3396 −0.879359
\(536\) 0 0
\(537\) −19.0432 −0.821775
\(538\) 0 0
\(539\) 27.7810 1.19661
\(540\) 0 0
\(541\) 30.5799 1.31473 0.657366 0.753572i \(-0.271671\pi\)
0.657366 + 0.753572i \(0.271671\pi\)
\(542\) 0 0
\(543\) −14.7592 −0.633376
\(544\) 0 0
\(545\) 50.9639 2.18305
\(546\) 0 0
\(547\) −20.4268 −0.873388 −0.436694 0.899610i \(-0.643851\pi\)
−0.436694 + 0.899610i \(0.643851\pi\)
\(548\) 0 0
\(549\) 0.962658 0.0410853
\(550\) 0 0
\(551\) −33.8197 −1.44077
\(552\) 0 0
\(553\) −5.54306 −0.235715
\(554\) 0 0
\(555\) −2.54332 −0.107958
\(556\) 0 0
\(557\) −4.28570 −0.181591 −0.0907956 0.995870i \(-0.528941\pi\)
−0.0907956 + 0.995870i \(0.528941\pi\)
\(558\) 0 0
\(559\) −0.862281 −0.0364706
\(560\) 0 0
\(561\) 13.1312 0.554401
\(562\) 0 0
\(563\) 34.7819 1.46588 0.732942 0.680292i \(-0.238147\pi\)
0.732942 + 0.680292i \(0.238147\pi\)
\(564\) 0 0
\(565\) −7.67968 −0.323087
\(566\) 0 0
\(567\) 0.604857 0.0254016
\(568\) 0 0
\(569\) 1.01037 0.0423571 0.0211785 0.999776i \(-0.493258\pi\)
0.0211785 + 0.999776i \(0.493258\pi\)
\(570\) 0 0
\(571\) −4.17041 −0.174526 −0.0872632 0.996185i \(-0.527812\pi\)
−0.0872632 + 0.996185i \(0.527812\pi\)
\(572\) 0 0
\(573\) −8.24196 −0.344313
\(574\) 0 0
\(575\) −4.04426 −0.168657
\(576\) 0 0
\(577\) −22.0426 −0.917647 −0.458824 0.888527i \(-0.651729\pi\)
−0.458824 + 0.888527i \(0.651729\pi\)
\(578\) 0 0
\(579\) −12.4363 −0.516836
\(580\) 0 0
\(581\) −9.47881 −0.393247
\(582\) 0 0
\(583\) 10.5894 0.438570
\(584\) 0 0
\(585\) 1.55179 0.0641586
\(586\) 0 0
\(587\) −2.82344 −0.116536 −0.0582678 0.998301i \(-0.518558\pi\)
−0.0582678 + 0.998301i \(0.518558\pi\)
\(588\) 0 0
\(589\) −3.36435 −0.138626
\(590\) 0 0
\(591\) 18.4826 0.760272
\(592\) 0 0
\(593\) −45.6709 −1.87548 −0.937740 0.347339i \(-0.887085\pi\)
−0.937740 + 0.347339i \(0.887085\pi\)
\(594\) 0 0
\(595\) −5.00469 −0.205172
\(596\) 0 0
\(597\) 4.83561 0.197908
\(598\) 0 0
\(599\) −37.8313 −1.54575 −0.772873 0.634561i \(-0.781181\pi\)
−0.772873 + 0.634561i \(0.781181\pi\)
\(600\) 0 0
\(601\) −6.83325 −0.278734 −0.139367 0.990241i \(-0.544507\pi\)
−0.139367 + 0.990241i \(0.544507\pi\)
\(602\) 0 0
\(603\) 6.12872 0.249581
\(604\) 0 0
\(605\) −17.2456 −0.701135
\(606\) 0 0
\(607\) 22.9026 0.929587 0.464794 0.885419i \(-0.346128\pi\)
0.464794 + 0.885419i \(0.346128\pi\)
\(608\) 0 0
\(609\) −6.13772 −0.248713
\(610\) 0 0
\(611\) −2.24277 −0.0907329
\(612\) 0 0
\(613\) −5.01302 −0.202474 −0.101237 0.994862i \(-0.532280\pi\)
−0.101237 + 0.994862i \(0.532280\pi\)
\(614\) 0 0
\(615\) 26.6472 1.07452
\(616\) 0 0
\(617\) −21.2112 −0.853931 −0.426966 0.904268i \(-0.640417\pi\)
−0.426966 + 0.904268i \(0.640417\pi\)
\(618\) 0 0
\(619\) −12.4720 −0.501290 −0.250645 0.968079i \(-0.580643\pi\)
−0.250645 + 0.968079i \(0.580643\pi\)
\(620\) 0 0
\(621\) 2.06079 0.0826967
\(622\) 0 0
\(623\) 3.37456 0.135199
\(624\) 0 0
\(625\) −30.9611 −1.23844
\(626\) 0 0
\(627\) −13.9565 −0.557370
\(628\) 0 0
\(629\) −3.02247 −0.120514
\(630\) 0 0
\(631\) −43.7516 −1.74172 −0.870862 0.491528i \(-0.836439\pi\)
−0.870862 + 0.491528i \(0.836439\pi\)
\(632\) 0 0
\(633\) 25.4148 1.01015
\(634\) 0 0
\(635\) −30.6054 −1.21454
\(636\) 0 0
\(637\) 3.90154 0.154585
\(638\) 0 0
\(639\) −4.34451 −0.171866
\(640\) 0 0
\(641\) 14.2570 0.563116 0.281558 0.959544i \(-0.409149\pi\)
0.281558 + 0.959544i \(0.409149\pi\)
\(642\) 0 0
\(643\) 32.7973 1.29340 0.646699 0.762745i \(-0.276149\pi\)
0.646699 + 0.762745i \(0.276149\pi\)
\(644\) 0 0
\(645\) 3.86883 0.152335
\(646\) 0 0
\(647\) 2.71497 0.106736 0.0533682 0.998575i \(-0.483004\pi\)
0.0533682 + 0.998575i \(0.483004\pi\)
\(648\) 0 0
\(649\) −15.4977 −0.608337
\(650\) 0 0
\(651\) −0.610574 −0.0239303
\(652\) 0 0
\(653\) 0.282536 0.0110565 0.00552825 0.999985i \(-0.498240\pi\)
0.00552825 + 0.999985i \(0.498240\pi\)
\(654\) 0 0
\(655\) −21.8021 −0.851877
\(656\) 0 0
\(657\) 13.8707 0.541147
\(658\) 0 0
\(659\) −2.66474 −0.103804 −0.0519018 0.998652i \(-0.516528\pi\)
−0.0519018 + 0.998652i \(0.516528\pi\)
\(660\) 0 0
\(661\) 22.1445 0.861321 0.430660 0.902514i \(-0.358281\pi\)
0.430660 + 0.902514i \(0.358281\pi\)
\(662\) 0 0
\(663\) 1.84414 0.0716204
\(664\) 0 0
\(665\) 5.31924 0.206271
\(666\) 0 0
\(667\) −20.9117 −0.809703
\(668\) 0 0
\(669\) 13.4950 0.521747
\(670\) 0 0
\(671\) −4.03120 −0.155623
\(672\) 0 0
\(673\) −13.6686 −0.526886 −0.263443 0.964675i \(-0.584858\pi\)
−0.263443 + 0.964675i \(0.584858\pi\)
\(674\) 0 0
\(675\) −1.96248 −0.0755358
\(676\) 0 0
\(677\) −5.36889 −0.206343 −0.103172 0.994664i \(-0.532899\pi\)
−0.103172 + 0.994664i \(0.532899\pi\)
\(678\) 0 0
\(679\) −3.99625 −0.153362
\(680\) 0 0
\(681\) 22.1793 0.849914
\(682\) 0 0
\(683\) 35.4099 1.35492 0.677460 0.735559i \(-0.263081\pi\)
0.677460 + 0.735559i \(0.263081\pi\)
\(684\) 0 0
\(685\) −16.1198 −0.615907
\(686\) 0 0
\(687\) 8.65236 0.330108
\(688\) 0 0
\(689\) 1.48717 0.0566567
\(690\) 0 0
\(691\) −2.30779 −0.0877924 −0.0438962 0.999036i \(-0.513977\pi\)
−0.0438962 + 0.999036i \(0.513977\pi\)
\(692\) 0 0
\(693\) −2.53288 −0.0962163
\(694\) 0 0
\(695\) 39.0652 1.48183
\(696\) 0 0
\(697\) 31.6675 1.19949
\(698\) 0 0
\(699\) 15.2852 0.578138
\(700\) 0 0
\(701\) −18.6519 −0.704474 −0.352237 0.935911i \(-0.614579\pi\)
−0.352237 + 0.935911i \(0.614579\pi\)
\(702\) 0 0
\(703\) 3.21243 0.121159
\(704\) 0 0
\(705\) 10.0627 0.378985
\(706\) 0 0
\(707\) 0.128727 0.00484129
\(708\) 0 0
\(709\) −8.83611 −0.331847 −0.165924 0.986139i \(-0.553061\pi\)
−0.165924 + 0.986139i \(0.553061\pi\)
\(710\) 0 0
\(711\) −9.16426 −0.343686
\(712\) 0 0
\(713\) −2.08027 −0.0779068
\(714\) 0 0
\(715\) −6.49823 −0.243020
\(716\) 0 0
\(717\) −29.8813 −1.11594
\(718\) 0 0
\(719\) −17.8566 −0.665940 −0.332970 0.942937i \(-0.608051\pi\)
−0.332970 + 0.942937i \(0.608051\pi\)
\(720\) 0 0
\(721\) 0.907473 0.0337960
\(722\) 0 0
\(723\) −17.5171 −0.651466
\(724\) 0 0
\(725\) 19.9140 0.739588
\(726\) 0 0
\(727\) 20.2602 0.751409 0.375704 0.926740i \(-0.377401\pi\)
0.375704 + 0.926740i \(0.377401\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.59770 0.170052
\(732\) 0 0
\(733\) −16.9048 −0.624394 −0.312197 0.950017i \(-0.601065\pi\)
−0.312197 + 0.950017i \(0.601065\pi\)
\(734\) 0 0
\(735\) −17.5052 −0.645689
\(736\) 0 0
\(737\) −25.6645 −0.945363
\(738\) 0 0
\(739\) 35.1818 1.29418 0.647092 0.762412i \(-0.275985\pi\)
0.647092 + 0.762412i \(0.275985\pi\)
\(740\) 0 0
\(741\) −1.96004 −0.0720040
\(742\) 0 0
\(743\) −29.8016 −1.09331 −0.546657 0.837356i \(-0.684100\pi\)
−0.546657 + 0.837356i \(0.684100\pi\)
\(744\) 0 0
\(745\) −19.1975 −0.703343
\(746\) 0 0
\(747\) −15.6712 −0.573378
\(748\) 0 0
\(749\) 4.66245 0.170362
\(750\) 0 0
\(751\) 41.8463 1.52699 0.763497 0.645812i \(-0.223481\pi\)
0.763497 + 0.645812i \(0.223481\pi\)
\(752\) 0 0
\(753\) 11.5902 0.422371
\(754\) 0 0
\(755\) −17.8712 −0.650401
\(756\) 0 0
\(757\) 20.1667 0.732973 0.366486 0.930423i \(-0.380561\pi\)
0.366486 + 0.930423i \(0.380561\pi\)
\(758\) 0 0
\(759\) −8.62972 −0.313239
\(760\) 0 0
\(761\) −47.8942 −1.73616 −0.868082 0.496421i \(-0.834647\pi\)
−0.868082 + 0.496421i \(0.834647\pi\)
\(762\) 0 0
\(763\) −11.6824 −0.422933
\(764\) 0 0
\(765\) −8.27418 −0.299154
\(766\) 0 0
\(767\) −2.17648 −0.0785881
\(768\) 0 0
\(769\) 12.1859 0.439433 0.219717 0.975564i \(-0.429487\pi\)
0.219717 + 0.975564i \(0.429487\pi\)
\(770\) 0 0
\(771\) 12.3462 0.444639
\(772\) 0 0
\(773\) 47.6169 1.71266 0.856331 0.516427i \(-0.172738\pi\)
0.856331 + 0.516427i \(0.172738\pi\)
\(774\) 0 0
\(775\) 1.98103 0.0711606
\(776\) 0 0
\(777\) 0.583004 0.0209151
\(778\) 0 0
\(779\) −33.6578 −1.20592
\(780\) 0 0
\(781\) 18.1930 0.650996
\(782\) 0 0
\(783\) −10.1474 −0.362638
\(784\) 0 0
\(785\) 36.2970 1.29549
\(786\) 0 0
\(787\) 25.4329 0.906586 0.453293 0.891361i \(-0.350249\pi\)
0.453293 + 0.891361i \(0.350249\pi\)
\(788\) 0 0
\(789\) −10.9134 −0.388527
\(790\) 0 0
\(791\) 1.76041 0.0625930
\(792\) 0 0
\(793\) −0.566139 −0.0201042
\(794\) 0 0
\(795\) −6.67256 −0.236651
\(796\) 0 0
\(797\) 13.4752 0.477315 0.238658 0.971104i \(-0.423293\pi\)
0.238658 + 0.971104i \(0.423293\pi\)
\(798\) 0 0
\(799\) 11.9585 0.423062
\(800\) 0 0
\(801\) 5.57911 0.197128
\(802\) 0 0
\(803\) −58.0845 −2.04976
\(804\) 0 0
\(805\) 3.28904 0.115923
\(806\) 0 0
\(807\) −2.64700 −0.0931789
\(808\) 0 0
\(809\) −17.0866 −0.600732 −0.300366 0.953824i \(-0.597109\pi\)
−0.300366 + 0.953824i \(0.597109\pi\)
\(810\) 0 0
\(811\) −18.7124 −0.657080 −0.328540 0.944490i \(-0.606557\pi\)
−0.328540 + 0.944490i \(0.606557\pi\)
\(812\) 0 0
\(813\) 15.3534 0.538466
\(814\) 0 0
\(815\) 28.3391 0.992676
\(816\) 0 0
\(817\) −4.88667 −0.170963
\(818\) 0 0
\(819\) −0.355716 −0.0124297
\(820\) 0 0
\(821\) 43.7257 1.52604 0.763019 0.646377i \(-0.223716\pi\)
0.763019 + 0.646377i \(0.223716\pi\)
\(822\) 0 0
\(823\) 20.6687 0.720466 0.360233 0.932862i \(-0.382697\pi\)
0.360233 + 0.932862i \(0.382697\pi\)
\(824\) 0 0
\(825\) 8.21802 0.286115
\(826\) 0 0
\(827\) −53.5678 −1.86274 −0.931368 0.364080i \(-0.881383\pi\)
−0.931368 + 0.364080i \(0.881383\pi\)
\(828\) 0 0
\(829\) 8.09767 0.281244 0.140622 0.990063i \(-0.455090\pi\)
0.140622 + 0.990063i \(0.455090\pi\)
\(830\) 0 0
\(831\) −18.8696 −0.654578
\(832\) 0 0
\(833\) −20.8031 −0.720785
\(834\) 0 0
\(835\) −2.63865 −0.0913143
\(836\) 0 0
\(837\) −1.00945 −0.0348918
\(838\) 0 0
\(839\) −35.9455 −1.24098 −0.620489 0.784215i \(-0.713066\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(840\) 0 0
\(841\) 73.9696 2.55068
\(842\) 0 0
\(843\) 2.41769 0.0832696
\(844\) 0 0
\(845\) 33.3899 1.14865
\(846\) 0 0
\(847\) 3.95321 0.135834
\(848\) 0 0
\(849\) −17.2449 −0.591842
\(850\) 0 0
\(851\) 1.98634 0.0680908
\(852\) 0 0
\(853\) −43.1857 −1.47865 −0.739325 0.673348i \(-0.764855\pi\)
−0.739325 + 0.673348i \(0.764855\pi\)
\(854\) 0 0
\(855\) 8.79421 0.300756
\(856\) 0 0
\(857\) 26.1566 0.893491 0.446746 0.894661i \(-0.352583\pi\)
0.446746 + 0.894661i \(0.352583\pi\)
\(858\) 0 0
\(859\) −15.1284 −0.516174 −0.258087 0.966122i \(-0.583092\pi\)
−0.258087 + 0.966122i \(0.583092\pi\)
\(860\) 0 0
\(861\) −6.10834 −0.208172
\(862\) 0 0
\(863\) −15.0990 −0.513977 −0.256989 0.966414i \(-0.582730\pi\)
−0.256989 + 0.966414i \(0.582730\pi\)
\(864\) 0 0
\(865\) −4.32538 −0.147067
\(866\) 0 0
\(867\) 7.16700 0.243404
\(868\) 0 0
\(869\) 38.3760 1.30182
\(870\) 0 0
\(871\) −3.60430 −0.122127
\(872\) 0 0
\(873\) −6.60693 −0.223611
\(874\) 0 0
\(875\) 4.84790 0.163889
\(876\) 0 0
\(877\) −43.4986 −1.46884 −0.734422 0.678693i \(-0.762547\pi\)
−0.734422 + 0.678693i \(0.762547\pi\)
\(878\) 0 0
\(879\) 3.14330 0.106021
\(880\) 0 0
\(881\) 13.7969 0.464828 0.232414 0.972617i \(-0.425338\pi\)
0.232414 + 0.972617i \(0.425338\pi\)
\(882\) 0 0
\(883\) −2.77424 −0.0933607 −0.0466804 0.998910i \(-0.514864\pi\)
−0.0466804 + 0.998910i \(0.514864\pi\)
\(884\) 0 0
\(885\) 9.76531 0.328257
\(886\) 0 0
\(887\) −42.3348 −1.42146 −0.710732 0.703462i \(-0.751636\pi\)
−0.710732 + 0.703462i \(0.751636\pi\)
\(888\) 0 0
\(889\) 7.01565 0.235298
\(890\) 0 0
\(891\) −4.18757 −0.140289
\(892\) 0 0
\(893\) −12.7101 −0.425328
\(894\) 0 0
\(895\) −50.2484 −1.67962
\(896\) 0 0
\(897\) −1.21195 −0.0404659
\(898\) 0 0
\(899\) 10.2433 0.341634
\(900\) 0 0
\(901\) −7.92964 −0.264175
\(902\) 0 0
\(903\) −0.886850 −0.0295125
\(904\) 0 0
\(905\) −38.9443 −1.29455
\(906\) 0 0
\(907\) 7.94687 0.263871 0.131936 0.991258i \(-0.457881\pi\)
0.131936 + 0.991258i \(0.457881\pi\)
\(908\) 0 0
\(909\) 0.212823 0.00705889
\(910\) 0 0
\(911\) 58.6923 1.94456 0.972281 0.233814i \(-0.0751208\pi\)
0.972281 + 0.233814i \(0.0751208\pi\)
\(912\) 0 0
\(913\) 65.6242 2.17184
\(914\) 0 0
\(915\) 2.54012 0.0839737
\(916\) 0 0
\(917\) 4.99768 0.165038
\(918\) 0 0
\(919\) −38.5200 −1.27066 −0.635328 0.772242i \(-0.719135\pi\)
−0.635328 + 0.772242i \(0.719135\pi\)
\(920\) 0 0
\(921\) 27.5662 0.908337
\(922\) 0 0
\(923\) 2.55501 0.0840990
\(924\) 0 0
\(925\) −1.89157 −0.0621946
\(926\) 0 0
\(927\) 1.50031 0.0492767
\(928\) 0 0
\(929\) 38.6862 1.26925 0.634627 0.772818i \(-0.281154\pi\)
0.634627 + 0.772818i \(0.281154\pi\)
\(930\) 0 0
\(931\) 22.1106 0.724645
\(932\) 0 0
\(933\) 3.97921 0.130274
\(934\) 0 0
\(935\) 34.6487 1.13313
\(936\) 0 0
\(937\) −50.9642 −1.66493 −0.832463 0.554080i \(-0.813070\pi\)
−0.832463 + 0.554080i \(0.813070\pi\)
\(938\) 0 0
\(939\) 15.9773 0.521399
\(940\) 0 0
\(941\) −44.6968 −1.45708 −0.728538 0.685006i \(-0.759800\pi\)
−0.728538 + 0.685006i \(0.759800\pi\)
\(942\) 0 0
\(943\) −20.8116 −0.677718
\(944\) 0 0
\(945\) 1.59601 0.0519181
\(946\) 0 0
\(947\) 9.32767 0.303109 0.151554 0.988449i \(-0.451572\pi\)
0.151554 + 0.988449i \(0.451572\pi\)
\(948\) 0 0
\(949\) −8.15734 −0.264799
\(950\) 0 0
\(951\) 25.4618 0.825654
\(952\) 0 0
\(953\) −4.95569 −0.160530 −0.0802652 0.996774i \(-0.525577\pi\)
−0.0802652 + 0.996774i \(0.525577\pi\)
\(954\) 0 0
\(955\) −21.7477 −0.703738
\(956\) 0 0
\(957\) 42.4930 1.37360
\(958\) 0 0
\(959\) 3.69514 0.119322
\(960\) 0 0
\(961\) −29.9810 −0.967129
\(962\) 0 0
\(963\) 7.70835 0.248398
\(964\) 0 0
\(965\) −32.8151 −1.05636
\(966\) 0 0
\(967\) 31.4033 1.00986 0.504931 0.863159i \(-0.331518\pi\)
0.504931 + 0.863159i \(0.331518\pi\)
\(968\) 0 0
\(969\) 10.4510 0.335735
\(970\) 0 0
\(971\) 38.8079 1.24540 0.622702 0.782459i \(-0.286035\pi\)
0.622702 + 0.782459i \(0.286035\pi\)
\(972\) 0 0
\(973\) −8.95490 −0.287081
\(974\) 0 0
\(975\) 1.15413 0.0369618
\(976\) 0 0
\(977\) 36.8665 1.17946 0.589732 0.807599i \(-0.299233\pi\)
0.589732 + 0.807599i \(0.299233\pi\)
\(978\) 0 0
\(979\) −23.3629 −0.746682
\(980\) 0 0
\(981\) −19.3144 −0.616661
\(982\) 0 0
\(983\) 14.6521 0.467330 0.233665 0.972317i \(-0.424928\pi\)
0.233665 + 0.972317i \(0.424928\pi\)
\(984\) 0 0
\(985\) 48.7691 1.55391
\(986\) 0 0
\(987\) −2.30668 −0.0734224
\(988\) 0 0
\(989\) −3.02157 −0.0960802
\(990\) 0 0
\(991\) 22.2752 0.707595 0.353797 0.935322i \(-0.384890\pi\)
0.353797 + 0.935322i \(0.384890\pi\)
\(992\) 0 0
\(993\) −24.0840 −0.764283
\(994\) 0 0
\(995\) 12.7595 0.404503
\(996\) 0 0
\(997\) 6.53346 0.206917 0.103458 0.994634i \(-0.467009\pi\)
0.103458 + 0.994634i \(0.467009\pi\)
\(998\) 0 0
\(999\) 0.963871 0.0304955
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 8016.2.a.x.1.1 8
4.3 odd 2 501.2.a.e.1.3 8
12.11 even 2 1503.2.a.e.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.e.1.3 8 4.3 odd 2
1503.2.a.e.1.6 8 12.11 even 2
8016.2.a.x.1.1 8 1.1 even 1 trivial