Properties

Label 8001.2.a.p.1.11
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $14$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} - 875 x^{5} + 1134 x^{4} + 301 x^{3} - 418 x^{2} - 42 x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(1.86702\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.86702 q^{2} +1.48577 q^{4} +4.06976 q^{5} -1.00000 q^{7} -0.960078 q^{8} +O(q^{10})\) \(q+1.86702 q^{2} +1.48577 q^{4} +4.06976 q^{5} -1.00000 q^{7} -0.960078 q^{8} +7.59833 q^{10} -0.0552813 q^{11} -5.30564 q^{13} -1.86702 q^{14} -4.76403 q^{16} +1.80614 q^{17} +7.67845 q^{19} +6.04673 q^{20} -0.103211 q^{22} +8.46285 q^{23} +11.5630 q^{25} -9.90574 q^{26} -1.48577 q^{28} +2.19120 q^{29} +7.83287 q^{31} -6.97439 q^{32} +3.37211 q^{34} -4.06976 q^{35} +2.03734 q^{37} +14.3358 q^{38} -3.90729 q^{40} -2.50415 q^{41} -9.74934 q^{43} -0.0821353 q^{44} +15.8003 q^{46} -5.74714 q^{47} +1.00000 q^{49} +21.5883 q^{50} -7.88296 q^{52} -3.80018 q^{53} -0.224982 q^{55} +0.960078 q^{56} +4.09103 q^{58} -7.25461 q^{59} +12.8375 q^{61} +14.6241 q^{62} -3.49328 q^{64} -21.5927 q^{65} +1.02309 q^{67} +2.68352 q^{68} -7.59833 q^{70} +4.59524 q^{71} -0.190964 q^{73} +3.80375 q^{74} +11.4084 q^{76} +0.0552813 q^{77} +8.89583 q^{79} -19.3885 q^{80} -4.67530 q^{82} -4.28568 q^{83} +7.35058 q^{85} -18.2022 q^{86} +0.0530743 q^{88} -8.76663 q^{89} +5.30564 q^{91} +12.5739 q^{92} -10.7300 q^{94} +31.2495 q^{95} +12.2594 q^{97} +1.86702 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86702 1.32018 0.660092 0.751185i \(-0.270517\pi\)
0.660092 + 0.751185i \(0.270517\pi\)
\(3\) 0 0
\(4\) 1.48577 0.742885
\(5\) 4.06976 1.82005 0.910026 0.414550i \(-0.136061\pi\)
0.910026 + 0.414550i \(0.136061\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.960078 −0.339439
\(9\) 0 0
\(10\) 7.59833 2.40280
\(11\) −0.0552813 −0.0166679 −0.00833397 0.999965i \(-0.502653\pi\)
−0.00833397 + 0.999965i \(0.502653\pi\)
\(12\) 0 0
\(13\) −5.30564 −1.47152 −0.735760 0.677243i \(-0.763175\pi\)
−0.735760 + 0.677243i \(0.763175\pi\)
\(14\) −1.86702 −0.498983
\(15\) 0 0
\(16\) −4.76403 −1.19101
\(17\) 1.80614 0.438054 0.219027 0.975719i \(-0.429712\pi\)
0.219027 + 0.975719i \(0.429712\pi\)
\(18\) 0 0
\(19\) 7.67845 1.76156 0.880779 0.473528i \(-0.157020\pi\)
0.880779 + 0.473528i \(0.157020\pi\)
\(20\) 6.04673 1.35209
\(21\) 0 0
\(22\) −0.103211 −0.0220047
\(23\) 8.46285 1.76463 0.882313 0.470662i \(-0.155985\pi\)
0.882313 + 0.470662i \(0.155985\pi\)
\(24\) 0 0
\(25\) 11.5630 2.31259
\(26\) −9.90574 −1.94268
\(27\) 0 0
\(28\) −1.48577 −0.280784
\(29\) 2.19120 0.406897 0.203448 0.979086i \(-0.434785\pi\)
0.203448 + 0.979086i \(0.434785\pi\)
\(30\) 0 0
\(31\) 7.83287 1.40683 0.703413 0.710782i \(-0.251659\pi\)
0.703413 + 0.710782i \(0.251659\pi\)
\(32\) −6.97439 −1.23291
\(33\) 0 0
\(34\) 3.37211 0.578312
\(35\) −4.06976 −0.687915
\(36\) 0 0
\(37\) 2.03734 0.334936 0.167468 0.985877i \(-0.446441\pi\)
0.167468 + 0.985877i \(0.446441\pi\)
\(38\) 14.3358 2.32558
\(39\) 0 0
\(40\) −3.90729 −0.617796
\(41\) −2.50415 −0.391082 −0.195541 0.980695i \(-0.562646\pi\)
−0.195541 + 0.980695i \(0.562646\pi\)
\(42\) 0 0
\(43\) −9.74934 −1.48676 −0.743380 0.668869i \(-0.766779\pi\)
−0.743380 + 0.668869i \(0.766779\pi\)
\(44\) −0.0821353 −0.0123824
\(45\) 0 0
\(46\) 15.8003 2.32963
\(47\) −5.74714 −0.838307 −0.419153 0.907915i \(-0.637673\pi\)
−0.419153 + 0.907915i \(0.637673\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 21.5883 3.05305
\(51\) 0 0
\(52\) −7.88296 −1.09317
\(53\) −3.80018 −0.521995 −0.260998 0.965339i \(-0.584051\pi\)
−0.260998 + 0.965339i \(0.584051\pi\)
\(54\) 0 0
\(55\) −0.224982 −0.0303365
\(56\) 0.960078 0.128296
\(57\) 0 0
\(58\) 4.09103 0.537178
\(59\) −7.25461 −0.944470 −0.472235 0.881473i \(-0.656553\pi\)
−0.472235 + 0.881473i \(0.656553\pi\)
\(60\) 0 0
\(61\) 12.8375 1.64367 0.821834 0.569726i \(-0.192951\pi\)
0.821834 + 0.569726i \(0.192951\pi\)
\(62\) 14.6241 1.85727
\(63\) 0 0
\(64\) −3.49328 −0.436660
\(65\) −21.5927 −2.67824
\(66\) 0 0
\(67\) 1.02309 0.124991 0.0624954 0.998045i \(-0.480094\pi\)
0.0624954 + 0.998045i \(0.480094\pi\)
\(68\) 2.68352 0.325424
\(69\) 0 0
\(70\) −7.59833 −0.908175
\(71\) 4.59524 0.545354 0.272677 0.962106i \(-0.412091\pi\)
0.272677 + 0.962106i \(0.412091\pi\)
\(72\) 0 0
\(73\) −0.190964 −0.0223506 −0.0111753 0.999938i \(-0.503557\pi\)
−0.0111753 + 0.999938i \(0.503557\pi\)
\(74\) 3.80375 0.442177
\(75\) 0 0
\(76\) 11.4084 1.30864
\(77\) 0.0552813 0.00629989
\(78\) 0 0
\(79\) 8.89583 1.00086 0.500429 0.865777i \(-0.333176\pi\)
0.500429 + 0.865777i \(0.333176\pi\)
\(80\) −19.3885 −2.16770
\(81\) 0 0
\(82\) −4.67530 −0.516301
\(83\) −4.28568 −0.470415 −0.235207 0.971945i \(-0.575577\pi\)
−0.235207 + 0.971945i \(0.575577\pi\)
\(84\) 0 0
\(85\) 7.35058 0.797282
\(86\) −18.2022 −1.96280
\(87\) 0 0
\(88\) 0.0530743 0.00565774
\(89\) −8.76663 −0.929261 −0.464631 0.885505i \(-0.653813\pi\)
−0.464631 + 0.885505i \(0.653813\pi\)
\(90\) 0 0
\(91\) 5.30564 0.556182
\(92\) 12.5739 1.31092
\(93\) 0 0
\(94\) −10.7300 −1.10672
\(95\) 31.2495 3.20613
\(96\) 0 0
\(97\) 12.2594 1.24475 0.622375 0.782719i \(-0.286168\pi\)
0.622375 + 0.782719i \(0.286168\pi\)
\(98\) 1.86702 0.188598
\(99\) 0 0
\(100\) 17.1799 1.71799
\(101\) 14.8833 1.48095 0.740473 0.672086i \(-0.234601\pi\)
0.740473 + 0.672086i \(0.234601\pi\)
\(102\) 0 0
\(103\) −9.81265 −0.966869 −0.483434 0.875381i \(-0.660611\pi\)
−0.483434 + 0.875381i \(0.660611\pi\)
\(104\) 5.09382 0.499491
\(105\) 0 0
\(106\) −7.09502 −0.689130
\(107\) 2.99814 0.289841 0.144921 0.989443i \(-0.453707\pi\)
0.144921 + 0.989443i \(0.453707\pi\)
\(108\) 0 0
\(109\) 14.9832 1.43513 0.717567 0.696489i \(-0.245256\pi\)
0.717567 + 0.696489i \(0.245256\pi\)
\(110\) −0.420046 −0.0400498
\(111\) 0 0
\(112\) 4.76403 0.450158
\(113\) 9.11161 0.857149 0.428574 0.903507i \(-0.359016\pi\)
0.428574 + 0.903507i \(0.359016\pi\)
\(114\) 0 0
\(115\) 34.4418 3.21171
\(116\) 3.25563 0.302277
\(117\) 0 0
\(118\) −13.5445 −1.24687
\(119\) −1.80614 −0.165569
\(120\) 0 0
\(121\) −10.9969 −0.999722
\(122\) 23.9678 2.16995
\(123\) 0 0
\(124\) 11.6379 1.04511
\(125\) 26.7097 2.38899
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 7.42675 0.656438
\(129\) 0 0
\(130\) −40.3140 −3.53577
\(131\) 9.44110 0.824873 0.412436 0.910986i \(-0.364678\pi\)
0.412436 + 0.910986i \(0.364678\pi\)
\(132\) 0 0
\(133\) −7.67845 −0.665806
\(134\) 1.91014 0.165011
\(135\) 0 0
\(136\) −1.73404 −0.148693
\(137\) −12.2081 −1.04301 −0.521506 0.853248i \(-0.674630\pi\)
−0.521506 + 0.853248i \(0.674630\pi\)
\(138\) 0 0
\(139\) −8.66264 −0.734755 −0.367378 0.930072i \(-0.619744\pi\)
−0.367378 + 0.930072i \(0.619744\pi\)
\(140\) −6.04673 −0.511042
\(141\) 0 0
\(142\) 8.57941 0.719968
\(143\) 0.293303 0.0245272
\(144\) 0 0
\(145\) 8.91768 0.740573
\(146\) −0.356534 −0.0295069
\(147\) 0 0
\(148\) 3.02702 0.248819
\(149\) 11.0129 0.902215 0.451108 0.892470i \(-0.351029\pi\)
0.451108 + 0.892470i \(0.351029\pi\)
\(150\) 0 0
\(151\) −20.8494 −1.69670 −0.848348 0.529439i \(-0.822403\pi\)
−0.848348 + 0.529439i \(0.822403\pi\)
\(152\) −7.37191 −0.597941
\(153\) 0 0
\(154\) 0.103211 0.00831701
\(155\) 31.8779 2.56050
\(156\) 0 0
\(157\) 13.2607 1.05832 0.529160 0.848522i \(-0.322507\pi\)
0.529160 + 0.848522i \(0.322507\pi\)
\(158\) 16.6087 1.32132
\(159\) 0 0
\(160\) −28.3841 −2.24396
\(161\) −8.46285 −0.666966
\(162\) 0 0
\(163\) 12.2204 0.957173 0.478587 0.878040i \(-0.341149\pi\)
0.478587 + 0.878040i \(0.341149\pi\)
\(164\) −3.72059 −0.290529
\(165\) 0 0
\(166\) −8.00147 −0.621034
\(167\) −9.53262 −0.737656 −0.368828 0.929498i \(-0.620241\pi\)
−0.368828 + 0.929498i \(0.620241\pi\)
\(168\) 0 0
\(169\) 15.1498 1.16537
\(170\) 13.7237 1.05256
\(171\) 0 0
\(172\) −14.4853 −1.10449
\(173\) −15.7093 −1.19436 −0.597179 0.802108i \(-0.703712\pi\)
−0.597179 + 0.802108i \(0.703712\pi\)
\(174\) 0 0
\(175\) −11.5630 −0.874078
\(176\) 0.263362 0.0198516
\(177\) 0 0
\(178\) −16.3675 −1.22680
\(179\) 14.1899 1.06060 0.530302 0.847809i \(-0.322078\pi\)
0.530302 + 0.847809i \(0.322078\pi\)
\(180\) 0 0
\(181\) −3.68771 −0.274105 −0.137053 0.990564i \(-0.543763\pi\)
−0.137053 + 0.990564i \(0.543763\pi\)
\(182\) 9.90574 0.734263
\(183\) 0 0
\(184\) −8.12500 −0.598983
\(185\) 8.29148 0.609602
\(186\) 0 0
\(187\) −0.0998461 −0.00730147
\(188\) −8.53894 −0.622766
\(189\) 0 0
\(190\) 58.3434 4.23268
\(191\) 5.60888 0.405845 0.202922 0.979195i \(-0.434956\pi\)
0.202922 + 0.979195i \(0.434956\pi\)
\(192\) 0 0
\(193\) 8.02831 0.577890 0.288945 0.957346i \(-0.406695\pi\)
0.288945 + 0.957346i \(0.406695\pi\)
\(194\) 22.8885 1.64330
\(195\) 0 0
\(196\) 1.48577 0.106126
\(197\) 25.1278 1.79028 0.895140 0.445785i \(-0.147075\pi\)
0.895140 + 0.445785i \(0.147075\pi\)
\(198\) 0 0
\(199\) 5.16230 0.365946 0.182973 0.983118i \(-0.441428\pi\)
0.182973 + 0.983118i \(0.441428\pi\)
\(200\) −11.1013 −0.784983
\(201\) 0 0
\(202\) 27.7875 1.95512
\(203\) −2.19120 −0.153792
\(204\) 0 0
\(205\) −10.1913 −0.711791
\(206\) −18.3204 −1.27644
\(207\) 0 0
\(208\) 25.2762 1.75259
\(209\) −0.424475 −0.0293615
\(210\) 0 0
\(211\) 10.6293 0.731750 0.365875 0.930664i \(-0.380770\pi\)
0.365875 + 0.930664i \(0.380770\pi\)
\(212\) −5.64620 −0.387783
\(213\) 0 0
\(214\) 5.59760 0.382644
\(215\) −39.6775 −2.70598
\(216\) 0 0
\(217\) −7.83287 −0.531730
\(218\) 27.9740 1.89464
\(219\) 0 0
\(220\) −0.334271 −0.0225366
\(221\) −9.58275 −0.644606
\(222\) 0 0
\(223\) −0.250705 −0.0167884 −0.00839421 0.999965i \(-0.502672\pi\)
−0.00839421 + 0.999965i \(0.502672\pi\)
\(224\) 6.97439 0.465996
\(225\) 0 0
\(226\) 17.0116 1.13159
\(227\) 20.9788 1.39241 0.696205 0.717843i \(-0.254870\pi\)
0.696205 + 0.717843i \(0.254870\pi\)
\(228\) 0 0
\(229\) −29.3191 −1.93746 −0.968729 0.248122i \(-0.920187\pi\)
−0.968729 + 0.248122i \(0.920187\pi\)
\(230\) 64.3036 4.24005
\(231\) 0 0
\(232\) −2.10373 −0.138116
\(233\) −0.354551 −0.0232274 −0.0116137 0.999933i \(-0.503697\pi\)
−0.0116137 + 0.999933i \(0.503697\pi\)
\(234\) 0 0
\(235\) −23.3895 −1.52576
\(236\) −10.7787 −0.701633
\(237\) 0 0
\(238\) −3.37211 −0.218582
\(239\) 15.3007 0.989717 0.494859 0.868974i \(-0.335220\pi\)
0.494859 + 0.868974i \(0.335220\pi\)
\(240\) 0 0
\(241\) 2.37634 0.153073 0.0765367 0.997067i \(-0.475614\pi\)
0.0765367 + 0.997067i \(0.475614\pi\)
\(242\) −20.5315 −1.31982
\(243\) 0 0
\(244\) 19.0735 1.22106
\(245\) 4.06976 0.260008
\(246\) 0 0
\(247\) −40.7391 −2.59217
\(248\) −7.52017 −0.477531
\(249\) 0 0
\(250\) 49.8676 3.15390
\(251\) 11.8899 0.750485 0.375243 0.926927i \(-0.377559\pi\)
0.375243 + 0.926927i \(0.377559\pi\)
\(252\) 0 0
\(253\) −0.467838 −0.0294127
\(254\) −1.86702 −0.117147
\(255\) 0 0
\(256\) 20.8525 1.30328
\(257\) −15.7227 −0.980757 −0.490378 0.871510i \(-0.663141\pi\)
−0.490378 + 0.871510i \(0.663141\pi\)
\(258\) 0 0
\(259\) −2.03734 −0.126594
\(260\) −32.0818 −1.98963
\(261\) 0 0
\(262\) 17.6267 1.08898
\(263\) −19.8326 −1.22293 −0.611467 0.791270i \(-0.709420\pi\)
−0.611467 + 0.791270i \(0.709420\pi\)
\(264\) 0 0
\(265\) −15.4658 −0.950059
\(266\) −14.3358 −0.878987
\(267\) 0 0
\(268\) 1.52008 0.0928538
\(269\) −1.39070 −0.0847926 −0.0423963 0.999101i \(-0.513499\pi\)
−0.0423963 + 0.999101i \(0.513499\pi\)
\(270\) 0 0
\(271\) 23.6311 1.43549 0.717743 0.696308i \(-0.245175\pi\)
0.717743 + 0.696308i \(0.245175\pi\)
\(272\) −8.60452 −0.521726
\(273\) 0 0
\(274\) −22.7929 −1.37697
\(275\) −0.639216 −0.0385462
\(276\) 0 0
\(277\) −23.4732 −1.41037 −0.705185 0.709023i \(-0.749136\pi\)
−0.705185 + 0.709023i \(0.749136\pi\)
\(278\) −16.1733 −0.970012
\(279\) 0 0
\(280\) 3.90729 0.233505
\(281\) 19.5302 1.16507 0.582536 0.812805i \(-0.302061\pi\)
0.582536 + 0.812805i \(0.302061\pi\)
\(282\) 0 0
\(283\) −4.65763 −0.276867 −0.138434 0.990372i \(-0.544207\pi\)
−0.138434 + 0.990372i \(0.544207\pi\)
\(284\) 6.82747 0.405136
\(285\) 0 0
\(286\) 0.547603 0.0323804
\(287\) 2.50415 0.147815
\(288\) 0 0
\(289\) −13.7378 −0.808108
\(290\) 16.6495 0.977693
\(291\) 0 0
\(292\) −0.283728 −0.0166040
\(293\) −11.0040 −0.642863 −0.321432 0.946933i \(-0.604164\pi\)
−0.321432 + 0.946933i \(0.604164\pi\)
\(294\) 0 0
\(295\) −29.5245 −1.71898
\(296\) −1.95600 −0.113690
\(297\) 0 0
\(298\) 20.5614 1.19109
\(299\) −44.9008 −2.59668
\(300\) 0 0
\(301\) 9.74934 0.561942
\(302\) −38.9262 −2.23995
\(303\) 0 0
\(304\) −36.5804 −2.09803
\(305\) 52.2454 2.99156
\(306\) 0 0
\(307\) −11.9605 −0.682622 −0.341311 0.939950i \(-0.610871\pi\)
−0.341311 + 0.939950i \(0.610871\pi\)
\(308\) 0.0821353 0.00468010
\(309\) 0 0
\(310\) 59.5168 3.38033
\(311\) −3.41715 −0.193769 −0.0968843 0.995296i \(-0.530888\pi\)
−0.0968843 + 0.995296i \(0.530888\pi\)
\(312\) 0 0
\(313\) −22.8014 −1.28881 −0.644405 0.764685i \(-0.722895\pi\)
−0.644405 + 0.764685i \(0.722895\pi\)
\(314\) 24.7580 1.39718
\(315\) 0 0
\(316\) 13.2172 0.743523
\(317\) −30.8873 −1.73480 −0.867401 0.497610i \(-0.834211\pi\)
−0.867401 + 0.497610i \(0.834211\pi\)
\(318\) 0 0
\(319\) −0.121133 −0.00678213
\(320\) −14.2168 −0.794744
\(321\) 0 0
\(322\) −15.8003 −0.880518
\(323\) 13.8684 0.771658
\(324\) 0 0
\(325\) −61.3489 −3.40302
\(326\) 22.8157 1.26364
\(327\) 0 0
\(328\) 2.40418 0.132748
\(329\) 5.74714 0.316850
\(330\) 0 0
\(331\) 8.37607 0.460391 0.230195 0.973144i \(-0.426064\pi\)
0.230195 + 0.973144i \(0.426064\pi\)
\(332\) −6.36754 −0.349464
\(333\) 0 0
\(334\) −17.7976 −0.973842
\(335\) 4.16375 0.227490
\(336\) 0 0
\(337\) −24.8021 −1.35105 −0.675527 0.737335i \(-0.736084\pi\)
−0.675527 + 0.737335i \(0.736084\pi\)
\(338\) 28.2850 1.53850
\(339\) 0 0
\(340\) 10.9213 0.592289
\(341\) −0.433012 −0.0234489
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 9.36012 0.504664
\(345\) 0 0
\(346\) −29.3296 −1.57677
\(347\) 0.992249 0.0532667 0.0266333 0.999645i \(-0.491521\pi\)
0.0266333 + 0.999645i \(0.491521\pi\)
\(348\) 0 0
\(349\) −27.1236 −1.45189 −0.725947 0.687751i \(-0.758598\pi\)
−0.725947 + 0.687751i \(0.758598\pi\)
\(350\) −21.5883 −1.15394
\(351\) 0 0
\(352\) 0.385553 0.0205501
\(353\) −18.5948 −0.989700 −0.494850 0.868979i \(-0.664777\pi\)
−0.494850 + 0.868979i \(0.664777\pi\)
\(354\) 0 0
\(355\) 18.7015 0.992574
\(356\) −13.0252 −0.690334
\(357\) 0 0
\(358\) 26.4929 1.40019
\(359\) −5.21516 −0.275246 −0.137623 0.990485i \(-0.543946\pi\)
−0.137623 + 0.990485i \(0.543946\pi\)
\(360\) 0 0
\(361\) 39.9586 2.10309
\(362\) −6.88504 −0.361870
\(363\) 0 0
\(364\) 7.88296 0.413179
\(365\) −0.777177 −0.0406793
\(366\) 0 0
\(367\) 3.94418 0.205885 0.102942 0.994687i \(-0.467174\pi\)
0.102942 + 0.994687i \(0.467174\pi\)
\(368\) −40.3173 −2.10168
\(369\) 0 0
\(370\) 15.4804 0.804786
\(371\) 3.80018 0.197296
\(372\) 0 0
\(373\) −13.5188 −0.699978 −0.349989 0.936754i \(-0.613815\pi\)
−0.349989 + 0.936754i \(0.613815\pi\)
\(374\) −0.186415 −0.00963928
\(375\) 0 0
\(376\) 5.51770 0.284554
\(377\) −11.6257 −0.598756
\(378\) 0 0
\(379\) −16.4645 −0.845727 −0.422863 0.906193i \(-0.638975\pi\)
−0.422863 + 0.906193i \(0.638975\pi\)
\(380\) 46.4295 2.38179
\(381\) 0 0
\(382\) 10.4719 0.535789
\(383\) 2.34362 0.119754 0.0598768 0.998206i \(-0.480929\pi\)
0.0598768 + 0.998206i \(0.480929\pi\)
\(384\) 0 0
\(385\) 0.224982 0.0114661
\(386\) 14.9890 0.762922
\(387\) 0 0
\(388\) 18.2146 0.924707
\(389\) −24.2313 −1.22858 −0.614289 0.789081i \(-0.710557\pi\)
−0.614289 + 0.789081i \(0.710557\pi\)
\(390\) 0 0
\(391\) 15.2851 0.773003
\(392\) −0.960078 −0.0484912
\(393\) 0 0
\(394\) 46.9141 2.36350
\(395\) 36.2039 1.82162
\(396\) 0 0
\(397\) −26.9611 −1.35314 −0.676570 0.736378i \(-0.736534\pi\)
−0.676570 + 0.736378i \(0.736534\pi\)
\(398\) 9.63813 0.483116
\(399\) 0 0
\(400\) −55.0863 −2.75431
\(401\) −4.82431 −0.240915 −0.120457 0.992718i \(-0.538436\pi\)
−0.120457 + 0.992718i \(0.538436\pi\)
\(402\) 0 0
\(403\) −41.5584 −2.07017
\(404\) 22.1132 1.10017
\(405\) 0 0
\(406\) −4.09103 −0.203034
\(407\) −0.112627 −0.00558270
\(408\) 0 0
\(409\) 21.0183 1.03929 0.519645 0.854382i \(-0.326064\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(410\) −19.0274 −0.939694
\(411\) 0 0
\(412\) −14.5793 −0.718273
\(413\) 7.25461 0.356976
\(414\) 0 0
\(415\) −17.4417 −0.856180
\(416\) 37.0036 1.81425
\(417\) 0 0
\(418\) −0.792504 −0.0387626
\(419\) −16.2607 −0.794386 −0.397193 0.917735i \(-0.630016\pi\)
−0.397193 + 0.917735i \(0.630016\pi\)
\(420\) 0 0
\(421\) −10.5008 −0.511779 −0.255889 0.966706i \(-0.582368\pi\)
−0.255889 + 0.966706i \(0.582368\pi\)
\(422\) 19.8451 0.966045
\(423\) 0 0
\(424\) 3.64847 0.177185
\(425\) 20.8844 1.01304
\(426\) 0 0
\(427\) −12.8375 −0.621248
\(428\) 4.45455 0.215319
\(429\) 0 0
\(430\) −74.0787 −3.57239
\(431\) −25.0340 −1.20585 −0.602923 0.797799i \(-0.705997\pi\)
−0.602923 + 0.797799i \(0.705997\pi\)
\(432\) 0 0
\(433\) 22.3517 1.07415 0.537077 0.843533i \(-0.319529\pi\)
0.537077 + 0.843533i \(0.319529\pi\)
\(434\) −14.6241 −0.701981
\(435\) 0 0
\(436\) 22.2617 1.06614
\(437\) 64.9816 3.10849
\(438\) 0 0
\(439\) −0.586878 −0.0280102 −0.0140051 0.999902i \(-0.504458\pi\)
−0.0140051 + 0.999902i \(0.504458\pi\)
\(440\) 0.216000 0.0102974
\(441\) 0 0
\(442\) −17.8912 −0.850998
\(443\) 33.8683 1.60913 0.804565 0.593865i \(-0.202399\pi\)
0.804565 + 0.593865i \(0.202399\pi\)
\(444\) 0 0
\(445\) −35.6781 −1.69130
\(446\) −0.468071 −0.0221638
\(447\) 0 0
\(448\) 3.49328 0.165042
\(449\) −28.6096 −1.35017 −0.675085 0.737740i \(-0.735893\pi\)
−0.675085 + 0.737740i \(0.735893\pi\)
\(450\) 0 0
\(451\) 0.138433 0.00651854
\(452\) 13.5378 0.636763
\(453\) 0 0
\(454\) 39.1678 1.83824
\(455\) 21.5927 1.01228
\(456\) 0 0
\(457\) −22.0367 −1.03084 −0.515418 0.856939i \(-0.672363\pi\)
−0.515418 + 0.856939i \(0.672363\pi\)
\(458\) −54.7393 −2.55780
\(459\) 0 0
\(460\) 51.1726 2.38594
\(461\) 9.68071 0.450876 0.225438 0.974258i \(-0.427619\pi\)
0.225438 + 0.974258i \(0.427619\pi\)
\(462\) 0 0
\(463\) −41.8056 −1.94287 −0.971436 0.237301i \(-0.923737\pi\)
−0.971436 + 0.237301i \(0.923737\pi\)
\(464\) −10.4390 −0.484617
\(465\) 0 0
\(466\) −0.661955 −0.0306645
\(467\) 25.8390 1.19569 0.597843 0.801613i \(-0.296025\pi\)
0.597843 + 0.801613i \(0.296025\pi\)
\(468\) 0 0
\(469\) −1.02309 −0.0472421
\(470\) −43.6687 −2.01429
\(471\) 0 0
\(472\) 6.96499 0.320590
\(473\) 0.538956 0.0247812
\(474\) 0 0
\(475\) 88.7856 4.07376
\(476\) −2.68352 −0.122999
\(477\) 0 0
\(478\) 28.5667 1.30661
\(479\) 35.4921 1.62167 0.810837 0.585272i \(-0.199012\pi\)
0.810837 + 0.585272i \(0.199012\pi\)
\(480\) 0 0
\(481\) −10.8094 −0.492865
\(482\) 4.43667 0.202085
\(483\) 0 0
\(484\) −16.3389 −0.742679
\(485\) 49.8927 2.26551
\(486\) 0 0
\(487\) 1.33739 0.0606030 0.0303015 0.999541i \(-0.490353\pi\)
0.0303015 + 0.999541i \(0.490353\pi\)
\(488\) −12.3250 −0.557925
\(489\) 0 0
\(490\) 7.59833 0.343258
\(491\) −23.5590 −1.06320 −0.531602 0.846994i \(-0.678410\pi\)
−0.531602 + 0.846994i \(0.678410\pi\)
\(492\) 0 0
\(493\) 3.95763 0.178243
\(494\) −76.0608 −3.42214
\(495\) 0 0
\(496\) −37.3160 −1.67554
\(497\) −4.59524 −0.206125
\(498\) 0 0
\(499\) −28.1256 −1.25907 −0.629537 0.776971i \(-0.716755\pi\)
−0.629537 + 0.776971i \(0.716755\pi\)
\(500\) 39.6845 1.77474
\(501\) 0 0
\(502\) 22.1987 0.990778
\(503\) −42.3200 −1.88696 −0.943478 0.331436i \(-0.892467\pi\)
−0.943478 + 0.331436i \(0.892467\pi\)
\(504\) 0 0
\(505\) 60.5716 2.69540
\(506\) −0.873463 −0.0388302
\(507\) 0 0
\(508\) −1.48577 −0.0659204
\(509\) −6.94247 −0.307720 −0.153860 0.988093i \(-0.549170\pi\)
−0.153860 + 0.988093i \(0.549170\pi\)
\(510\) 0 0
\(511\) 0.190964 0.00844774
\(512\) 24.0785 1.06413
\(513\) 0 0
\(514\) −29.3547 −1.29478
\(515\) −39.9351 −1.75975
\(516\) 0 0
\(517\) 0.317710 0.0139729
\(518\) −3.80375 −0.167127
\(519\) 0 0
\(520\) 20.7307 0.909099
\(521\) −5.25810 −0.230361 −0.115181 0.993345i \(-0.536745\pi\)
−0.115181 + 0.993345i \(0.536745\pi\)
\(522\) 0 0
\(523\) 24.3487 1.06469 0.532346 0.846527i \(-0.321310\pi\)
0.532346 + 0.846527i \(0.321310\pi\)
\(524\) 14.0273 0.612786
\(525\) 0 0
\(526\) −37.0280 −1.61450
\(527\) 14.1473 0.616266
\(528\) 0 0
\(529\) 48.6199 2.11391
\(530\) −28.8751 −1.25425
\(531\) 0 0
\(532\) −11.4084 −0.494618
\(533\) 13.2861 0.575485
\(534\) 0 0
\(535\) 12.2017 0.527527
\(536\) −0.982249 −0.0424267
\(537\) 0 0
\(538\) −2.59647 −0.111942
\(539\) −0.0552813 −0.00238113
\(540\) 0 0
\(541\) −31.0769 −1.33610 −0.668049 0.744117i \(-0.732871\pi\)
−0.668049 + 0.744117i \(0.732871\pi\)
\(542\) 44.1197 1.89511
\(543\) 0 0
\(544\) −12.5968 −0.540081
\(545\) 60.9782 2.61202
\(546\) 0 0
\(547\) 32.8650 1.40520 0.702602 0.711583i \(-0.252021\pi\)
0.702602 + 0.711583i \(0.252021\pi\)
\(548\) −18.1385 −0.774838
\(549\) 0 0
\(550\) −1.19343 −0.0508880
\(551\) 16.8251 0.716772
\(552\) 0 0
\(553\) −8.89583 −0.378289
\(554\) −43.8251 −1.86195
\(555\) 0 0
\(556\) −12.8707 −0.545839
\(557\) 29.1692 1.23594 0.617970 0.786202i \(-0.287955\pi\)
0.617970 + 0.786202i \(0.287955\pi\)
\(558\) 0 0
\(559\) 51.7265 2.18780
\(560\) 19.3885 0.819312
\(561\) 0 0
\(562\) 36.4632 1.53811
\(563\) −12.0125 −0.506266 −0.253133 0.967431i \(-0.581461\pi\)
−0.253133 + 0.967431i \(0.581461\pi\)
\(564\) 0 0
\(565\) 37.0821 1.56006
\(566\) −8.69590 −0.365516
\(567\) 0 0
\(568\) −4.41179 −0.185114
\(569\) 5.47747 0.229628 0.114814 0.993387i \(-0.463373\pi\)
0.114814 + 0.993387i \(0.463373\pi\)
\(570\) 0 0
\(571\) 15.0828 0.631195 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(572\) 0.435780 0.0182209
\(573\) 0 0
\(574\) 4.67530 0.195143
\(575\) 97.8556 4.08086
\(576\) 0 0
\(577\) 37.1085 1.54485 0.772424 0.635107i \(-0.219044\pi\)
0.772424 + 0.635107i \(0.219044\pi\)
\(578\) −25.6488 −1.06685
\(579\) 0 0
\(580\) 13.2496 0.550161
\(581\) 4.28568 0.177800
\(582\) 0 0
\(583\) 0.210079 0.00870059
\(584\) 0.183340 0.00758667
\(585\) 0 0
\(586\) −20.5448 −0.848697
\(587\) −0.589414 −0.0243277 −0.0121639 0.999926i \(-0.503872\pi\)
−0.0121639 + 0.999926i \(0.503872\pi\)
\(588\) 0 0
\(589\) 60.1443 2.47820
\(590\) −55.1229 −2.26938
\(591\) 0 0
\(592\) −9.70593 −0.398911
\(593\) −29.6443 −1.21735 −0.608673 0.793421i \(-0.708298\pi\)
−0.608673 + 0.793421i \(0.708298\pi\)
\(594\) 0 0
\(595\) −7.35058 −0.301344
\(596\) 16.3627 0.670243
\(597\) 0 0
\(598\) −83.8309 −3.42810
\(599\) −34.5199 −1.41044 −0.705222 0.708987i \(-0.749153\pi\)
−0.705222 + 0.708987i \(0.749153\pi\)
\(600\) 0 0
\(601\) 33.3294 1.35953 0.679767 0.733428i \(-0.262081\pi\)
0.679767 + 0.733428i \(0.262081\pi\)
\(602\) 18.2022 0.741867
\(603\) 0 0
\(604\) −30.9774 −1.26045
\(605\) −44.7549 −1.81955
\(606\) 0 0
\(607\) 4.90428 0.199059 0.0995294 0.995035i \(-0.468266\pi\)
0.0995294 + 0.995035i \(0.468266\pi\)
\(608\) −53.5525 −2.17184
\(609\) 0 0
\(610\) 97.5433 3.94941
\(611\) 30.4923 1.23359
\(612\) 0 0
\(613\) −34.5427 −1.39517 −0.697584 0.716503i \(-0.745741\pi\)
−0.697584 + 0.716503i \(0.745741\pi\)
\(614\) −22.3305 −0.901187
\(615\) 0 0
\(616\) −0.0530743 −0.00213843
\(617\) −35.8258 −1.44229 −0.721147 0.692782i \(-0.756385\pi\)
−0.721147 + 0.692782i \(0.756385\pi\)
\(618\) 0 0
\(619\) 2.37973 0.0956495 0.0478248 0.998856i \(-0.484771\pi\)
0.0478248 + 0.998856i \(0.484771\pi\)
\(620\) 47.3633 1.90216
\(621\) 0 0
\(622\) −6.37989 −0.255810
\(623\) 8.76663 0.351228
\(624\) 0 0
\(625\) 50.8872 2.03549
\(626\) −42.5706 −1.70147
\(627\) 0 0
\(628\) 19.7024 0.786210
\(629\) 3.67973 0.146720
\(630\) 0 0
\(631\) 0.370334 0.0147428 0.00737138 0.999973i \(-0.497654\pi\)
0.00737138 + 0.999973i \(0.497654\pi\)
\(632\) −8.54068 −0.339730
\(633\) 0 0
\(634\) −57.6672 −2.29026
\(635\) −4.06976 −0.161504
\(636\) 0 0
\(637\) −5.30564 −0.210217
\(638\) −0.226157 −0.00895366
\(639\) 0 0
\(640\) 30.2251 1.19475
\(641\) 6.08684 0.240416 0.120208 0.992749i \(-0.461644\pi\)
0.120208 + 0.992749i \(0.461644\pi\)
\(642\) 0 0
\(643\) −10.2301 −0.403434 −0.201717 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(644\) −12.5739 −0.495479
\(645\) 0 0
\(646\) 25.8926 1.01873
\(647\) 8.43033 0.331431 0.165715 0.986174i \(-0.447007\pi\)
0.165715 + 0.986174i \(0.447007\pi\)
\(648\) 0 0
\(649\) 0.401044 0.0157424
\(650\) −114.540 −4.49262
\(651\) 0 0
\(652\) 18.1567 0.711070
\(653\) −28.7265 −1.12415 −0.562077 0.827085i \(-0.689998\pi\)
−0.562077 + 0.827085i \(0.689998\pi\)
\(654\) 0 0
\(655\) 38.4230 1.50131
\(656\) 11.9298 0.465782
\(657\) 0 0
\(658\) 10.7300 0.418301
\(659\) 16.3064 0.635208 0.317604 0.948223i \(-0.397122\pi\)
0.317604 + 0.948223i \(0.397122\pi\)
\(660\) 0 0
\(661\) −45.9195 −1.78606 −0.893030 0.449997i \(-0.851425\pi\)
−0.893030 + 0.449997i \(0.851425\pi\)
\(662\) 15.6383 0.607800
\(663\) 0 0
\(664\) 4.11459 0.159677
\(665\) −31.2495 −1.21180
\(666\) 0 0
\(667\) 18.5438 0.718021
\(668\) −14.1633 −0.547994
\(669\) 0 0
\(670\) 7.77380 0.300328
\(671\) −0.709672 −0.0273966
\(672\) 0 0
\(673\) 33.4208 1.28828 0.644138 0.764909i \(-0.277216\pi\)
0.644138 + 0.764909i \(0.277216\pi\)
\(674\) −46.3060 −1.78364
\(675\) 0 0
\(676\) 22.5091 0.865736
\(677\) 18.6310 0.716049 0.358024 0.933712i \(-0.383450\pi\)
0.358024 + 0.933712i \(0.383450\pi\)
\(678\) 0 0
\(679\) −12.2594 −0.470471
\(680\) −7.05713 −0.270628
\(681\) 0 0
\(682\) −0.808442 −0.0309568
\(683\) −46.4334 −1.77672 −0.888362 0.459144i \(-0.848156\pi\)
−0.888362 + 0.459144i \(0.848156\pi\)
\(684\) 0 0
\(685\) −49.6842 −1.89834
\(686\) −1.86702 −0.0712832
\(687\) 0 0
\(688\) 46.4461 1.77074
\(689\) 20.1624 0.768126
\(690\) 0 0
\(691\) 12.1069 0.460569 0.230284 0.973123i \(-0.426034\pi\)
0.230284 + 0.973123i \(0.426034\pi\)
\(692\) −23.3404 −0.887271
\(693\) 0 0
\(694\) 1.85255 0.0703218
\(695\) −35.2549 −1.33729
\(696\) 0 0
\(697\) −4.52286 −0.171315
\(698\) −50.6404 −1.91677
\(699\) 0 0
\(700\) −17.1799 −0.649339
\(701\) 21.1179 0.797610 0.398805 0.917036i \(-0.369425\pi\)
0.398805 + 0.917036i \(0.369425\pi\)
\(702\) 0 0
\(703\) 15.6436 0.590010
\(704\) 0.193113 0.00727822
\(705\) 0 0
\(706\) −34.7168 −1.30659
\(707\) −14.8833 −0.559745
\(708\) 0 0
\(709\) −3.36394 −0.126335 −0.0631676 0.998003i \(-0.520120\pi\)
−0.0631676 + 0.998003i \(0.520120\pi\)
\(710\) 34.9162 1.31038
\(711\) 0 0
\(712\) 8.41665 0.315427
\(713\) 66.2885 2.48252
\(714\) 0 0
\(715\) 1.19367 0.0446408
\(716\) 21.0830 0.787908
\(717\) 0 0
\(718\) −9.73682 −0.363375
\(719\) −19.7953 −0.738239 −0.369120 0.929382i \(-0.620341\pi\)
−0.369120 + 0.929382i \(0.620341\pi\)
\(720\) 0 0
\(721\) 9.81265 0.365442
\(722\) 74.6036 2.77646
\(723\) 0 0
\(724\) −5.47909 −0.203629
\(725\) 25.3368 0.940986
\(726\) 0 0
\(727\) −9.79554 −0.363297 −0.181648 0.983364i \(-0.558143\pi\)
−0.181648 + 0.983364i \(0.558143\pi\)
\(728\) −5.09382 −0.188790
\(729\) 0 0
\(730\) −1.45101 −0.0537042
\(731\) −17.6087 −0.651282
\(732\) 0 0
\(733\) −5.06258 −0.186991 −0.0934953 0.995620i \(-0.529804\pi\)
−0.0934953 + 0.995620i \(0.529804\pi\)
\(734\) 7.36387 0.271805
\(735\) 0 0
\(736\) −59.0232 −2.17562
\(737\) −0.0565579 −0.00208334
\(738\) 0 0
\(739\) 31.5209 1.15952 0.579758 0.814789i \(-0.303147\pi\)
0.579758 + 0.814789i \(0.303147\pi\)
\(740\) 12.3192 0.452864
\(741\) 0 0
\(742\) 7.09502 0.260467
\(743\) −27.1607 −0.996430 −0.498215 0.867053i \(-0.666011\pi\)
−0.498215 + 0.867053i \(0.666011\pi\)
\(744\) 0 0
\(745\) 44.8201 1.64208
\(746\) −25.2399 −0.924100
\(747\) 0 0
\(748\) −0.148348 −0.00542415
\(749\) −2.99814 −0.109550
\(750\) 0 0
\(751\) −21.8959 −0.798992 −0.399496 0.916735i \(-0.630815\pi\)
−0.399496 + 0.916735i \(0.630815\pi\)
\(752\) 27.3795 0.998429
\(753\) 0 0
\(754\) −21.7055 −0.790468
\(755\) −84.8519 −3.08808
\(756\) 0 0
\(757\) −45.6416 −1.65887 −0.829437 0.558600i \(-0.811339\pi\)
−0.829437 + 0.558600i \(0.811339\pi\)
\(758\) −30.7397 −1.11651
\(759\) 0 0
\(760\) −30.0019 −1.08828
\(761\) 7.09246 0.257101 0.128551 0.991703i \(-0.458967\pi\)
0.128551 + 0.991703i \(0.458967\pi\)
\(762\) 0 0
\(763\) −14.9832 −0.542430
\(764\) 8.33351 0.301496
\(765\) 0 0
\(766\) 4.37560 0.158097
\(767\) 38.4903 1.38981
\(768\) 0 0
\(769\) −21.6365 −0.780232 −0.390116 0.920766i \(-0.627565\pi\)
−0.390116 + 0.920766i \(0.627565\pi\)
\(770\) 0.420046 0.0151374
\(771\) 0 0
\(772\) 11.9282 0.429306
\(773\) 23.7001 0.852434 0.426217 0.904621i \(-0.359846\pi\)
0.426217 + 0.904621i \(0.359846\pi\)
\(774\) 0 0
\(775\) 90.5712 3.25341
\(776\) −11.7699 −0.422516
\(777\) 0 0
\(778\) −45.2404 −1.62195
\(779\) −19.2280 −0.688914
\(780\) 0 0
\(781\) −0.254031 −0.00908994
\(782\) 28.5377 1.02051
\(783\) 0 0
\(784\) −4.76403 −0.170144
\(785\) 53.9679 1.92620
\(786\) 0 0
\(787\) 27.9099 0.994880 0.497440 0.867498i \(-0.334274\pi\)
0.497440 + 0.867498i \(0.334274\pi\)
\(788\) 37.3341 1.32997
\(789\) 0 0
\(790\) 67.5935 2.40487
\(791\) −9.11161 −0.323972
\(792\) 0 0
\(793\) −68.1109 −2.41869
\(794\) −50.3370 −1.78639
\(795\) 0 0
\(796\) 7.66999 0.271856
\(797\) −10.7267 −0.379959 −0.189979 0.981788i \(-0.560842\pi\)
−0.189979 + 0.981788i \(0.560842\pi\)
\(798\) 0 0
\(799\) −10.3802 −0.367224
\(800\) −80.6446 −2.85122
\(801\) 0 0
\(802\) −9.00710 −0.318052
\(803\) 0.0105567 0.000372539 0
\(804\) 0 0
\(805\) −34.4418 −1.21391
\(806\) −77.5904 −2.73301
\(807\) 0 0
\(808\) −14.2892 −0.502691
\(809\) 3.45583 0.121500 0.0607502 0.998153i \(-0.480651\pi\)
0.0607502 + 0.998153i \(0.480651\pi\)
\(810\) 0 0
\(811\) −7.65266 −0.268721 −0.134361 0.990932i \(-0.542898\pi\)
−0.134361 + 0.990932i \(0.542898\pi\)
\(812\) −3.25563 −0.114250
\(813\) 0 0
\(814\) −0.210277 −0.00737019
\(815\) 49.7340 1.74211
\(816\) 0 0
\(817\) −74.8598 −2.61901
\(818\) 39.2417 1.37205
\(819\) 0 0
\(820\) −15.1419 −0.528779
\(821\) −40.9623 −1.42959 −0.714797 0.699332i \(-0.753481\pi\)
−0.714797 + 0.699332i \(0.753481\pi\)
\(822\) 0 0
\(823\) −22.5752 −0.786923 −0.393462 0.919341i \(-0.628723\pi\)
−0.393462 + 0.919341i \(0.628723\pi\)
\(824\) 9.42090 0.328193
\(825\) 0 0
\(826\) 13.5445 0.471274
\(827\) −21.4059 −0.744356 −0.372178 0.928161i \(-0.621389\pi\)
−0.372178 + 0.928161i \(0.621389\pi\)
\(828\) 0 0
\(829\) −49.8305 −1.73069 −0.865343 0.501180i \(-0.832899\pi\)
−0.865343 + 0.501180i \(0.832899\pi\)
\(830\) −32.5641 −1.13032
\(831\) 0 0
\(832\) 18.5341 0.642554
\(833\) 1.80614 0.0625792
\(834\) 0 0
\(835\) −38.7955 −1.34257
\(836\) −0.630672 −0.0218123
\(837\) 0 0
\(838\) −30.3590 −1.04874
\(839\) 27.9367 0.964481 0.482241 0.876039i \(-0.339823\pi\)
0.482241 + 0.876039i \(0.339823\pi\)
\(840\) 0 0
\(841\) −24.1986 −0.834435
\(842\) −19.6053 −0.675642
\(843\) 0 0
\(844\) 15.7927 0.543607
\(845\) 61.6561 2.12103
\(846\) 0 0
\(847\) 10.9969 0.377859
\(848\) 18.1042 0.621700
\(849\) 0 0
\(850\) 38.9916 1.33740
\(851\) 17.2417 0.591038
\(852\) 0 0
\(853\) −20.3478 −0.696696 −0.348348 0.937365i \(-0.613257\pi\)
−0.348348 + 0.937365i \(0.613257\pi\)
\(854\) −23.9678 −0.820162
\(855\) 0 0
\(856\) −2.87845 −0.0983834
\(857\) −4.57410 −0.156248 −0.0781241 0.996944i \(-0.524893\pi\)
−0.0781241 + 0.996944i \(0.524893\pi\)
\(858\) 0 0
\(859\) −16.7050 −0.569968 −0.284984 0.958532i \(-0.591988\pi\)
−0.284984 + 0.958532i \(0.591988\pi\)
\(860\) −58.9516 −2.01023
\(861\) 0 0
\(862\) −46.7390 −1.59194
\(863\) 49.1110 1.67176 0.835879 0.548913i \(-0.184958\pi\)
0.835879 + 0.548913i \(0.184958\pi\)
\(864\) 0 0
\(865\) −63.9332 −2.17379
\(866\) 41.7311 1.41808
\(867\) 0 0
\(868\) −11.6379 −0.395014
\(869\) −0.491773 −0.0166823
\(870\) 0 0
\(871\) −5.42816 −0.183926
\(872\) −14.3851 −0.487140
\(873\) 0 0
\(874\) 121.322 4.10378
\(875\) −26.7097 −0.902952
\(876\) 0 0
\(877\) 17.8961 0.604309 0.302155 0.953259i \(-0.402294\pi\)
0.302155 + 0.953259i \(0.402294\pi\)
\(878\) −1.09571 −0.0369786
\(879\) 0 0
\(880\) 1.07182 0.0361310
\(881\) 35.6129 1.19983 0.599915 0.800064i \(-0.295201\pi\)
0.599915 + 0.800064i \(0.295201\pi\)
\(882\) 0 0
\(883\) 48.4740 1.63128 0.815641 0.578559i \(-0.196385\pi\)
0.815641 + 0.578559i \(0.196385\pi\)
\(884\) −14.2378 −0.478868
\(885\) 0 0
\(886\) 63.2328 2.12435
\(887\) 39.7389 1.33430 0.667151 0.744922i \(-0.267513\pi\)
0.667151 + 0.744922i \(0.267513\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −66.6118 −2.23283
\(891\) 0 0
\(892\) −0.372489 −0.0124719
\(893\) −44.1292 −1.47673
\(894\) 0 0
\(895\) 57.7496 1.93036
\(896\) −7.42675 −0.248110
\(897\) 0 0
\(898\) −53.4147 −1.78247
\(899\) 17.1634 0.572432
\(900\) 0 0
\(901\) −6.86368 −0.228662
\(902\) 0.258457 0.00860567
\(903\) 0 0
\(904\) −8.74786 −0.290949
\(905\) −15.0081 −0.498886
\(906\) 0 0
\(907\) −29.3855 −0.975731 −0.487865 0.872919i \(-0.662224\pi\)
−0.487865 + 0.872919i \(0.662224\pi\)
\(908\) 31.1696 1.03440
\(909\) 0 0
\(910\) 40.3140 1.33640
\(911\) −50.3431 −1.66794 −0.833971 0.551808i \(-0.813938\pi\)
−0.833971 + 0.551808i \(0.813938\pi\)
\(912\) 0 0
\(913\) 0.236918 0.00784085
\(914\) −41.1431 −1.36089
\(915\) 0 0
\(916\) −43.5614 −1.43931
\(917\) −9.44110 −0.311773
\(918\) 0 0
\(919\) 8.03780 0.265143 0.132571 0.991173i \(-0.457677\pi\)
0.132571 + 0.991173i \(0.457677\pi\)
\(920\) −33.0668 −1.09018
\(921\) 0 0
\(922\) 18.0741 0.595239
\(923\) −24.3807 −0.802500
\(924\) 0 0
\(925\) 23.5577 0.774571
\(926\) −78.0520 −2.56495
\(927\) 0 0
\(928\) −15.2823 −0.501666
\(929\) 9.67839 0.317538 0.158769 0.987316i \(-0.449248\pi\)
0.158769 + 0.987316i \(0.449248\pi\)
\(930\) 0 0
\(931\) 7.67845 0.251651
\(932\) −0.526782 −0.0172553
\(933\) 0 0
\(934\) 48.2420 1.57853
\(935\) −0.406350 −0.0132891
\(936\) 0 0
\(937\) −3.14789 −0.102837 −0.0514185 0.998677i \(-0.516374\pi\)
−0.0514185 + 0.998677i \(0.516374\pi\)
\(938\) −1.91014 −0.0623682
\(939\) 0 0
\(940\) −34.7514 −1.13347
\(941\) 38.8319 1.26589 0.632943 0.774199i \(-0.281847\pi\)
0.632943 + 0.774199i \(0.281847\pi\)
\(942\) 0 0
\(943\) −21.1922 −0.690115
\(944\) 34.5612 1.12487
\(945\) 0 0
\(946\) 1.00624 0.0327158
\(947\) 1.96494 0.0638521 0.0319260 0.999490i \(-0.489836\pi\)
0.0319260 + 0.999490i \(0.489836\pi\)
\(948\) 0 0
\(949\) 1.01319 0.0328894
\(950\) 165.765 5.37812
\(951\) 0 0
\(952\) 1.73404 0.0562005
\(953\) −48.9601 −1.58597 −0.792986 0.609240i \(-0.791475\pi\)
−0.792986 + 0.609240i \(0.791475\pi\)
\(954\) 0 0
\(955\) 22.8268 0.738659
\(956\) 22.7333 0.735246
\(957\) 0 0
\(958\) 66.2645 2.14091
\(959\) 12.2081 0.394222
\(960\) 0 0
\(961\) 30.3539 0.979158
\(962\) −20.1813 −0.650673
\(963\) 0 0
\(964\) 3.53069 0.113716
\(965\) 32.6733 1.05179
\(966\) 0 0
\(967\) 4.40139 0.141539 0.0707695 0.997493i \(-0.477455\pi\)
0.0707695 + 0.997493i \(0.477455\pi\)
\(968\) 10.5579 0.339344
\(969\) 0 0
\(970\) 93.1508 2.99089
\(971\) −41.7382 −1.33944 −0.669720 0.742613i \(-0.733586\pi\)
−0.669720 + 0.742613i \(0.733586\pi\)
\(972\) 0 0
\(973\) 8.66264 0.277711
\(974\) 2.49694 0.0800071
\(975\) 0 0
\(976\) −61.1580 −1.95762
\(977\) 0.788999 0.0252423 0.0126212 0.999920i \(-0.495982\pi\)
0.0126212 + 0.999920i \(0.495982\pi\)
\(978\) 0 0
\(979\) 0.484631 0.0154889
\(980\) 6.04673 0.193156
\(981\) 0 0
\(982\) −43.9852 −1.40362
\(983\) 17.9857 0.573656 0.286828 0.957982i \(-0.407399\pi\)
0.286828 + 0.957982i \(0.407399\pi\)
\(984\) 0 0
\(985\) 102.264 3.25840
\(986\) 7.38899 0.235313
\(987\) 0 0
\(988\) −60.5289 −1.92568
\(989\) −82.5072 −2.62358
\(990\) 0 0
\(991\) −57.9431 −1.84062 −0.920311 0.391188i \(-0.872064\pi\)
−0.920311 + 0.391188i \(0.872064\pi\)
\(992\) −54.6295 −1.73449
\(993\) 0 0
\(994\) −8.57941 −0.272122
\(995\) 21.0093 0.666041
\(996\) 0 0
\(997\) −52.4955 −1.66255 −0.831274 0.555862i \(-0.812388\pi\)
−0.831274 + 0.555862i \(0.812388\pi\)
\(998\) −52.5111 −1.66221
\(999\) 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 8001.2.a.p.1.11 14
3.2 odd 2 2667.2.a.m.1.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.m.1.4 14 3.2 odd 2
8001.2.a.p.1.11 14 1.1 even 1 trivial