Properties

Label 4014.2.a.x.1.3
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(0\)
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: \( x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.17253\) of defining polynomial
Character \(\chi\) \(=\) 4014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.585733 q^{5} -1.49689 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.585733 q^{5} -1.49689 q^{7} -1.00000 q^{8} +0.585733 q^{10} -3.18036 q^{11} -6.04779 q^{13} +1.49689 q^{14} +1.00000 q^{16} +4.54325 q^{17} +4.74794 q^{19} -0.585733 q^{20} +3.18036 q^{22} +1.06929 q^{23} -4.65692 q^{25} +6.04779 q^{26} -1.49689 q^{28} +10.4462 q^{29} -6.42013 q^{31} -1.00000 q^{32} -4.54325 q^{34} +0.876775 q^{35} -9.98632 q^{37} -4.74794 q^{38} +0.585733 q^{40} +3.37739 q^{41} -1.53155 q^{43} -3.18036 q^{44} -1.06929 q^{46} -13.0787 q^{47} -4.75933 q^{49} +4.65692 q^{50} -6.04779 q^{52} -7.71455 q^{53} +1.86284 q^{55} +1.49689 q^{56} -10.4462 q^{58} -3.21453 q^{59} +5.85140 q^{61} +6.42013 q^{62} +1.00000 q^{64} +3.54239 q^{65} +11.3418 q^{67} +4.54325 q^{68} -0.876775 q^{70} +15.3383 q^{71} +14.9439 q^{73} +9.98632 q^{74} +4.74794 q^{76} +4.76063 q^{77} +4.50495 q^{79} -0.585733 q^{80} -3.37739 q^{82} +1.09046 q^{83} -2.66113 q^{85} +1.53155 q^{86} +3.18036 q^{88} +4.43131 q^{89} +9.05285 q^{91} +1.06929 q^{92} +13.0787 q^{94} -2.78102 q^{95} -7.60980 q^{97} +4.75933 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.585733 −0.261948 −0.130974 0.991386i \(-0.541810\pi\)
−0.130974 + 0.991386i \(0.541810\pi\)
\(6\) 0 0
\(7\) −1.49689 −0.565770 −0.282885 0.959154i \(-0.591291\pi\)
−0.282885 + 0.959154i \(0.591291\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.585733 0.185225
\(11\) −3.18036 −0.958914 −0.479457 0.877565i \(-0.659166\pi\)
−0.479457 + 0.877565i \(0.659166\pi\)
\(12\) 0 0
\(13\) −6.04779 −1.67735 −0.838677 0.544629i \(-0.816671\pi\)
−0.838677 + 0.544629i \(0.816671\pi\)
\(14\) 1.49689 0.400060
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.54325 1.10190 0.550949 0.834539i \(-0.314266\pi\)
0.550949 + 0.834539i \(0.314266\pi\)
\(18\) 0 0
\(19\) 4.74794 1.08925 0.544626 0.838679i \(-0.316672\pi\)
0.544626 + 0.838679i \(0.316672\pi\)
\(20\) −0.585733 −0.130974
\(21\) 0 0
\(22\) 3.18036 0.678054
\(23\) 1.06929 0.222962 0.111481 0.993767i \(-0.464441\pi\)
0.111481 + 0.993767i \(0.464441\pi\)
\(24\) 0 0
\(25\) −4.65692 −0.931383
\(26\) 6.04779 1.18607
\(27\) 0 0
\(28\) −1.49689 −0.282885
\(29\) 10.4462 1.93980 0.969901 0.243499i \(-0.0782953\pi\)
0.969901 + 0.243499i \(0.0782953\pi\)
\(30\) 0 0
\(31\) −6.42013 −1.15309 −0.576545 0.817066i \(-0.695599\pi\)
−0.576545 + 0.817066i \(0.695599\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.54325 −0.779160
\(35\) 0.876775 0.148202
\(36\) 0 0
\(37\) −9.98632 −1.64174 −0.820871 0.571114i \(-0.806511\pi\)
−0.820871 + 0.571114i \(0.806511\pi\)
\(38\) −4.74794 −0.770217
\(39\) 0 0
\(40\) 0.585733 0.0926125
\(41\) 3.37739 0.527459 0.263730 0.964597i \(-0.415047\pi\)
0.263730 + 0.964597i \(0.415047\pi\)
\(42\) 0 0
\(43\) −1.53155 −0.233559 −0.116779 0.993158i \(-0.537257\pi\)
−0.116779 + 0.993158i \(0.537257\pi\)
\(44\) −3.18036 −0.479457
\(45\) 0 0
\(46\) −1.06929 −0.157658
\(47\) −13.0787 −1.90772 −0.953862 0.300246i \(-0.902931\pi\)
−0.953862 + 0.300246i \(0.902931\pi\)
\(48\) 0 0
\(49\) −4.75933 −0.679904
\(50\) 4.65692 0.658588
\(51\) 0 0
\(52\) −6.04779 −0.838677
\(53\) −7.71455 −1.05967 −0.529837 0.848099i \(-0.677747\pi\)
−0.529837 + 0.848099i \(0.677747\pi\)
\(54\) 0 0
\(55\) 1.86284 0.251185
\(56\) 1.49689 0.200030
\(57\) 0 0
\(58\) −10.4462 −1.37165
\(59\) −3.21453 −0.418496 −0.209248 0.977863i \(-0.567102\pi\)
−0.209248 + 0.977863i \(0.567102\pi\)
\(60\) 0 0
\(61\) 5.85140 0.749195 0.374597 0.927188i \(-0.377781\pi\)
0.374597 + 0.927188i \(0.377781\pi\)
\(62\) 6.42013 0.815357
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.54239 0.439379
\(66\) 0 0
\(67\) 11.3418 1.38562 0.692812 0.721119i \(-0.256372\pi\)
0.692812 + 0.721119i \(0.256372\pi\)
\(68\) 4.54325 0.550949
\(69\) 0 0
\(70\) −0.876775 −0.104795
\(71\) 15.3383 1.82032 0.910161 0.414255i \(-0.135958\pi\)
0.910161 + 0.414255i \(0.135958\pi\)
\(72\) 0 0
\(73\) 14.9439 1.74905 0.874523 0.484983i \(-0.161174\pi\)
0.874523 + 0.484983i \(0.161174\pi\)
\(74\) 9.98632 1.16089
\(75\) 0 0
\(76\) 4.74794 0.544626
\(77\) 4.76063 0.542525
\(78\) 0 0
\(79\) 4.50495 0.506846 0.253423 0.967356i \(-0.418444\pi\)
0.253423 + 0.967356i \(0.418444\pi\)
\(80\) −0.585733 −0.0654869
\(81\) 0 0
\(82\) −3.37739 −0.372970
\(83\) 1.09046 0.119694 0.0598469 0.998208i \(-0.480939\pi\)
0.0598469 + 0.998208i \(0.480939\pi\)
\(84\) 0 0
\(85\) −2.66113 −0.288640
\(86\) 1.53155 0.165151
\(87\) 0 0
\(88\) 3.18036 0.339027
\(89\) 4.43131 0.469718 0.234859 0.972029i \(-0.424537\pi\)
0.234859 + 0.972029i \(0.424537\pi\)
\(90\) 0 0
\(91\) 9.05285 0.948997
\(92\) 1.06929 0.111481
\(93\) 0 0
\(94\) 13.0787 1.34896
\(95\) −2.78102 −0.285327
\(96\) 0 0
\(97\) −7.60980 −0.772658 −0.386329 0.922361i \(-0.626257\pi\)
−0.386329 + 0.922361i \(0.626257\pi\)
\(98\) 4.75933 0.480765
\(99\) 0 0
\(100\) −4.65692 −0.465692
\(101\) 11.3248 1.12686 0.563431 0.826163i \(-0.309481\pi\)
0.563431 + 0.826163i \(0.309481\pi\)
\(102\) 0 0
\(103\) −15.8368 −1.56045 −0.780224 0.625501i \(-0.784895\pi\)
−0.780224 + 0.625501i \(0.784895\pi\)
\(104\) 6.04779 0.593034
\(105\) 0 0
\(106\) 7.71455 0.749303
\(107\) 8.73169 0.844125 0.422062 0.906567i \(-0.361306\pi\)
0.422062 + 0.906567i \(0.361306\pi\)
\(108\) 0 0
\(109\) −1.93898 −0.185720 −0.0928601 0.995679i \(-0.529601\pi\)
−0.0928601 + 0.995679i \(0.529601\pi\)
\(110\) −1.86284 −0.177615
\(111\) 0 0
\(112\) −1.49689 −0.141442
\(113\) 11.9224 1.12156 0.560781 0.827964i \(-0.310501\pi\)
0.560781 + 0.827964i \(0.310501\pi\)
\(114\) 0 0
\(115\) −0.626318 −0.0584044
\(116\) 10.4462 0.969901
\(117\) 0 0
\(118\) 3.21453 0.295921
\(119\) −6.80072 −0.623421
\(120\) 0 0
\(121\) −0.885329 −0.0804845
\(122\) −5.85140 −0.529761
\(123\) 0 0
\(124\) −6.42013 −0.576545
\(125\) 5.65637 0.505921
\(126\) 0 0
\(127\) 2.46517 0.218748 0.109374 0.994001i \(-0.465115\pi\)
0.109374 + 0.994001i \(0.465115\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.54239 −0.310688
\(131\) 1.07564 0.0939791 0.0469895 0.998895i \(-0.485037\pi\)
0.0469895 + 0.998895i \(0.485037\pi\)
\(132\) 0 0
\(133\) −7.10712 −0.616266
\(134\) −11.3418 −0.979784
\(135\) 0 0
\(136\) −4.54325 −0.389580
\(137\) −7.49845 −0.640636 −0.320318 0.947310i \(-0.603790\pi\)
−0.320318 + 0.947310i \(0.603790\pi\)
\(138\) 0 0
\(139\) −6.50751 −0.551960 −0.275980 0.961163i \(-0.589002\pi\)
−0.275980 + 0.961163i \(0.589002\pi\)
\(140\) 0.876775 0.0741010
\(141\) 0 0
\(142\) −15.3383 −1.28716
\(143\) 19.2341 1.60844
\(144\) 0 0
\(145\) −6.11865 −0.508127
\(146\) −14.9439 −1.23676
\(147\) 0 0
\(148\) −9.98632 −0.820871
\(149\) 13.2807 1.08800 0.543999 0.839086i \(-0.316909\pi\)
0.543999 + 0.839086i \(0.316909\pi\)
\(150\) 0 0
\(151\) 2.77304 0.225667 0.112833 0.993614i \(-0.464007\pi\)
0.112833 + 0.993614i \(0.464007\pi\)
\(152\) −4.74794 −0.385109
\(153\) 0 0
\(154\) −4.76063 −0.383623
\(155\) 3.76048 0.302049
\(156\) 0 0
\(157\) 3.97992 0.317633 0.158816 0.987308i \(-0.449232\pi\)
0.158816 + 0.987308i \(0.449232\pi\)
\(158\) −4.50495 −0.358394
\(159\) 0 0
\(160\) 0.585733 0.0463062
\(161\) −1.60060 −0.126145
\(162\) 0 0
\(163\) 17.0742 1.33735 0.668677 0.743553i \(-0.266861\pi\)
0.668677 + 0.743553i \(0.266861\pi\)
\(164\) 3.37739 0.263730
\(165\) 0 0
\(166\) −1.09046 −0.0846362
\(167\) 3.88787 0.300853 0.150426 0.988621i \(-0.451935\pi\)
0.150426 + 0.988621i \(0.451935\pi\)
\(168\) 0 0
\(169\) 23.5757 1.81352
\(170\) 2.66113 0.204099
\(171\) 0 0
\(172\) −1.53155 −0.116779
\(173\) 6.81516 0.518147 0.259074 0.965858i \(-0.416583\pi\)
0.259074 + 0.965858i \(0.416583\pi\)
\(174\) 0 0
\(175\) 6.97088 0.526949
\(176\) −3.18036 −0.239728
\(177\) 0 0
\(178\) −4.43131 −0.332141
\(179\) −1.64779 −0.123162 −0.0615808 0.998102i \(-0.519614\pi\)
−0.0615808 + 0.998102i \(0.519614\pi\)
\(180\) 0 0
\(181\) 16.1341 1.19924 0.599621 0.800284i \(-0.295318\pi\)
0.599621 + 0.800284i \(0.295318\pi\)
\(182\) −9.05285 −0.671042
\(183\) 0 0
\(184\) −1.06929 −0.0788291
\(185\) 5.84932 0.430050
\(186\) 0 0
\(187\) −14.4491 −1.05663
\(188\) −13.0787 −0.953862
\(189\) 0 0
\(190\) 2.78102 0.201757
\(191\) −0.988248 −0.0715071 −0.0357536 0.999361i \(-0.511383\pi\)
−0.0357536 + 0.999361i \(0.511383\pi\)
\(192\) 0 0
\(193\) 16.2266 1.16801 0.584006 0.811749i \(-0.301484\pi\)
0.584006 + 0.811749i \(0.301484\pi\)
\(194\) 7.60980 0.546351
\(195\) 0 0
\(196\) −4.75933 −0.339952
\(197\) −21.1776 −1.50884 −0.754422 0.656389i \(-0.772083\pi\)
−0.754422 + 0.656389i \(0.772083\pi\)
\(198\) 0 0
\(199\) −5.43075 −0.384975 −0.192488 0.981299i \(-0.561656\pi\)
−0.192488 + 0.981299i \(0.561656\pi\)
\(200\) 4.65692 0.329294
\(201\) 0 0
\(202\) −11.3248 −0.796812
\(203\) −15.6367 −1.09748
\(204\) 0 0
\(205\) −1.97825 −0.138167
\(206\) 15.8368 1.10340
\(207\) 0 0
\(208\) −6.04779 −0.419339
\(209\) −15.1001 −1.04450
\(210\) 0 0
\(211\) 8.70251 0.599105 0.299553 0.954080i \(-0.403163\pi\)
0.299553 + 0.954080i \(0.403163\pi\)
\(212\) −7.71455 −0.529837
\(213\) 0 0
\(214\) −8.73169 −0.596886
\(215\) 0.897078 0.0611802
\(216\) 0 0
\(217\) 9.61020 0.652383
\(218\) 1.93898 0.131324
\(219\) 0 0
\(220\) 1.86284 0.125593
\(221\) −27.4766 −1.84827
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) 1.49689 0.100015
\(225\) 0 0
\(226\) −11.9224 −0.793064
\(227\) 5.33837 0.354320 0.177160 0.984182i \(-0.443309\pi\)
0.177160 + 0.984182i \(0.443309\pi\)
\(228\) 0 0
\(229\) −16.9242 −1.11838 −0.559191 0.829039i \(-0.688888\pi\)
−0.559191 + 0.829039i \(0.688888\pi\)
\(230\) 0.626318 0.0412982
\(231\) 0 0
\(232\) −10.4462 −0.685824
\(233\) 16.5149 1.08193 0.540963 0.841046i \(-0.318060\pi\)
0.540963 + 0.841046i \(0.318060\pi\)
\(234\) 0 0
\(235\) 7.66062 0.499724
\(236\) −3.21453 −0.209248
\(237\) 0 0
\(238\) 6.80072 0.440825
\(239\) 14.8861 0.962903 0.481451 0.876473i \(-0.340110\pi\)
0.481451 + 0.876473i \(0.340110\pi\)
\(240\) 0 0
\(241\) 1.70313 0.109708 0.0548541 0.998494i \(-0.482531\pi\)
0.0548541 + 0.998494i \(0.482531\pi\)
\(242\) 0.885329 0.0569111
\(243\) 0 0
\(244\) 5.85140 0.374597
\(245\) 2.78770 0.178099
\(246\) 0 0
\(247\) −28.7145 −1.82706
\(248\) 6.42013 0.407679
\(249\) 0 0
\(250\) −5.65637 −0.357740
\(251\) 28.4403 1.79514 0.897569 0.440873i \(-0.145331\pi\)
0.897569 + 0.440873i \(0.145331\pi\)
\(252\) 0 0
\(253\) −3.40072 −0.213802
\(254\) −2.46517 −0.154678
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 17.2851 1.07821 0.539106 0.842238i \(-0.318762\pi\)
0.539106 + 0.842238i \(0.318762\pi\)
\(258\) 0 0
\(259\) 14.9484 0.928848
\(260\) 3.54239 0.219690
\(261\) 0 0
\(262\) −1.07564 −0.0664532
\(263\) −3.84344 −0.236996 −0.118498 0.992954i \(-0.537808\pi\)
−0.118498 + 0.992954i \(0.537808\pi\)
\(264\) 0 0
\(265\) 4.51866 0.277579
\(266\) 7.10712 0.435766
\(267\) 0 0
\(268\) 11.3418 0.692812
\(269\) −1.43863 −0.0877150 −0.0438575 0.999038i \(-0.513965\pi\)
−0.0438575 + 0.999038i \(0.513965\pi\)
\(270\) 0 0
\(271\) −8.44794 −0.513176 −0.256588 0.966521i \(-0.582598\pi\)
−0.256588 + 0.966521i \(0.582598\pi\)
\(272\) 4.54325 0.275475
\(273\) 0 0
\(274\) 7.49845 0.452998
\(275\) 14.8107 0.893116
\(276\) 0 0
\(277\) 20.1492 1.21065 0.605324 0.795979i \(-0.293043\pi\)
0.605324 + 0.795979i \(0.293043\pi\)
\(278\) 6.50751 0.390294
\(279\) 0 0
\(280\) −0.876775 −0.0523973
\(281\) 0.863739 0.0515264 0.0257632 0.999668i \(-0.491798\pi\)
0.0257632 + 0.999668i \(0.491798\pi\)
\(282\) 0 0
\(283\) 14.7643 0.877646 0.438823 0.898573i \(-0.355395\pi\)
0.438823 + 0.898573i \(0.355395\pi\)
\(284\) 15.3383 0.910161
\(285\) 0 0
\(286\) −19.2341 −1.13734
\(287\) −5.05556 −0.298421
\(288\) 0 0
\(289\) 3.64108 0.214181
\(290\) 6.11865 0.359300
\(291\) 0 0
\(292\) 14.9439 0.874523
\(293\) −27.7810 −1.62298 −0.811491 0.584364i \(-0.801344\pi\)
−0.811491 + 0.584364i \(0.801344\pi\)
\(294\) 0 0
\(295\) 1.88285 0.109624
\(296\) 9.98632 0.580443
\(297\) 0 0
\(298\) −13.2807 −0.769331
\(299\) −6.46684 −0.373987
\(300\) 0 0
\(301\) 2.29255 0.132141
\(302\) −2.77304 −0.159570
\(303\) 0 0
\(304\) 4.74794 0.272313
\(305\) −3.42735 −0.196250
\(306\) 0 0
\(307\) −24.5648 −1.40199 −0.700994 0.713167i \(-0.747260\pi\)
−0.700994 + 0.713167i \(0.747260\pi\)
\(308\) 4.76063 0.271262
\(309\) 0 0
\(310\) −3.76048 −0.213581
\(311\) 23.3167 1.32217 0.661083 0.750313i \(-0.270097\pi\)
0.661083 + 0.750313i \(0.270097\pi\)
\(312\) 0 0
\(313\) 33.8040 1.91071 0.955357 0.295453i \(-0.0954704\pi\)
0.955357 + 0.295453i \(0.0954704\pi\)
\(314\) −3.97992 −0.224600
\(315\) 0 0
\(316\) 4.50495 0.253423
\(317\) 28.4355 1.59709 0.798547 0.601932i \(-0.205602\pi\)
0.798547 + 0.601932i \(0.205602\pi\)
\(318\) 0 0
\(319\) −33.2225 −1.86010
\(320\) −0.585733 −0.0327435
\(321\) 0 0
\(322\) 1.60060 0.0891982
\(323\) 21.5710 1.20024
\(324\) 0 0
\(325\) 28.1640 1.56226
\(326\) −17.0742 −0.945652
\(327\) 0 0
\(328\) −3.37739 −0.186485
\(329\) 19.5773 1.07933
\(330\) 0 0
\(331\) −33.1657 −1.82295 −0.911475 0.411356i \(-0.865055\pi\)
−0.911475 + 0.411356i \(0.865055\pi\)
\(332\) 1.09046 0.0598469
\(333\) 0 0
\(334\) −3.88787 −0.212735
\(335\) −6.64327 −0.362961
\(336\) 0 0
\(337\) 6.40874 0.349107 0.174553 0.984648i \(-0.444152\pi\)
0.174553 + 0.984648i \(0.444152\pi\)
\(338\) −23.5757 −1.28235
\(339\) 0 0
\(340\) −2.66113 −0.144320
\(341\) 20.4183 1.10571
\(342\) 0 0
\(343\) 17.6024 0.950439
\(344\) 1.53155 0.0825756
\(345\) 0 0
\(346\) −6.81516 −0.366385
\(347\) −30.3873 −1.63128 −0.815639 0.578562i \(-0.803614\pi\)
−0.815639 + 0.578562i \(0.803614\pi\)
\(348\) 0 0
\(349\) 13.1951 0.706319 0.353159 0.935563i \(-0.385107\pi\)
0.353159 + 0.935563i \(0.385107\pi\)
\(350\) −6.97088 −0.372609
\(351\) 0 0
\(352\) 3.18036 0.169514
\(353\) 16.0690 0.855265 0.427633 0.903953i \(-0.359348\pi\)
0.427633 + 0.903953i \(0.359348\pi\)
\(354\) 0 0
\(355\) −8.98414 −0.476829
\(356\) 4.43131 0.234859
\(357\) 0 0
\(358\) 1.64779 0.0870884
\(359\) 10.4685 0.552507 0.276253 0.961085i \(-0.410907\pi\)
0.276253 + 0.961085i \(0.410907\pi\)
\(360\) 0 0
\(361\) 3.54291 0.186469
\(362\) −16.1341 −0.847992
\(363\) 0 0
\(364\) 9.05285 0.474498
\(365\) −8.75311 −0.458159
\(366\) 0 0
\(367\) 26.1262 1.36378 0.681888 0.731457i \(-0.261159\pi\)
0.681888 + 0.731457i \(0.261159\pi\)
\(368\) 1.06929 0.0557406
\(369\) 0 0
\(370\) −5.84932 −0.304091
\(371\) 11.5478 0.599532
\(372\) 0 0
\(373\) −26.2059 −1.35689 −0.678446 0.734650i \(-0.737346\pi\)
−0.678446 + 0.734650i \(0.737346\pi\)
\(374\) 14.4491 0.747147
\(375\) 0 0
\(376\) 13.0787 0.674482
\(377\) −63.1761 −3.25374
\(378\) 0 0
\(379\) 32.9269 1.69134 0.845670 0.533706i \(-0.179201\pi\)
0.845670 + 0.533706i \(0.179201\pi\)
\(380\) −2.78102 −0.142663
\(381\) 0 0
\(382\) 0.988248 0.0505632
\(383\) −20.5468 −1.04989 −0.524946 0.851136i \(-0.675914\pi\)
−0.524946 + 0.851136i \(0.675914\pi\)
\(384\) 0 0
\(385\) −2.78846 −0.142113
\(386\) −16.2266 −0.825910
\(387\) 0 0
\(388\) −7.60980 −0.386329
\(389\) −22.5931 −1.14551 −0.572757 0.819725i \(-0.694126\pi\)
−0.572757 + 0.819725i \(0.694126\pi\)
\(390\) 0 0
\(391\) 4.85804 0.245682
\(392\) 4.75933 0.240383
\(393\) 0 0
\(394\) 21.1776 1.06691
\(395\) −2.63870 −0.132767
\(396\) 0 0
\(397\) 7.21751 0.362236 0.181118 0.983461i \(-0.442028\pi\)
0.181118 + 0.983461i \(0.442028\pi\)
\(398\) 5.43075 0.272219
\(399\) 0 0
\(400\) −4.65692 −0.232846
\(401\) 11.3770 0.568143 0.284071 0.958803i \(-0.408315\pi\)
0.284071 + 0.958803i \(0.408315\pi\)
\(402\) 0 0
\(403\) 38.8276 1.93414
\(404\) 11.3248 0.563431
\(405\) 0 0
\(406\) 15.6367 0.776037
\(407\) 31.7601 1.57429
\(408\) 0 0
\(409\) 2.13979 0.105806 0.0529030 0.998600i \(-0.483153\pi\)
0.0529030 + 0.998600i \(0.483153\pi\)
\(410\) 1.97825 0.0976986
\(411\) 0 0
\(412\) −15.8368 −0.780224
\(413\) 4.81178 0.236772
\(414\) 0 0
\(415\) −0.638719 −0.0313535
\(416\) 6.04779 0.296517
\(417\) 0 0
\(418\) 15.1001 0.738572
\(419\) −33.3352 −1.62853 −0.814266 0.580493i \(-0.802860\pi\)
−0.814266 + 0.580493i \(0.802860\pi\)
\(420\) 0 0
\(421\) −25.1838 −1.22738 −0.613691 0.789546i \(-0.710316\pi\)
−0.613691 + 0.789546i \(0.710316\pi\)
\(422\) −8.70251 −0.423631
\(423\) 0 0
\(424\) 7.71455 0.374652
\(425\) −21.1575 −1.02629
\(426\) 0 0
\(427\) −8.75888 −0.423872
\(428\) 8.73169 0.422062
\(429\) 0 0
\(430\) −0.897078 −0.0432609
\(431\) 21.3427 1.02804 0.514020 0.857778i \(-0.328156\pi\)
0.514020 + 0.857778i \(0.328156\pi\)
\(432\) 0 0
\(433\) 31.4723 1.51246 0.756231 0.654305i \(-0.227039\pi\)
0.756231 + 0.654305i \(0.227039\pi\)
\(434\) −9.61020 −0.461305
\(435\) 0 0
\(436\) −1.93898 −0.0928601
\(437\) 5.07692 0.242862
\(438\) 0 0
\(439\) 8.30672 0.396458 0.198229 0.980156i \(-0.436481\pi\)
0.198229 + 0.980156i \(0.436481\pi\)
\(440\) −1.86284 −0.0888074
\(441\) 0 0
\(442\) 27.4766 1.30693
\(443\) −5.18961 −0.246566 −0.123283 0.992372i \(-0.539342\pi\)
−0.123283 + 0.992372i \(0.539342\pi\)
\(444\) 0 0
\(445\) −2.59556 −0.123041
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −1.49689 −0.0707212
\(449\) 12.2820 0.579623 0.289811 0.957084i \(-0.406407\pi\)
0.289811 + 0.957084i \(0.406407\pi\)
\(450\) 0 0
\(451\) −10.7413 −0.505788
\(452\) 11.9224 0.560781
\(453\) 0 0
\(454\) −5.33837 −0.250542
\(455\) −5.30255 −0.248587
\(456\) 0 0
\(457\) 36.6297 1.71347 0.856733 0.515760i \(-0.172490\pi\)
0.856733 + 0.515760i \(0.172490\pi\)
\(458\) 16.9242 0.790815
\(459\) 0 0
\(460\) −0.626318 −0.0292022
\(461\) 23.3611 1.08804 0.544018 0.839074i \(-0.316902\pi\)
0.544018 + 0.839074i \(0.316902\pi\)
\(462\) 0 0
\(463\) −31.6070 −1.46890 −0.734451 0.678661i \(-0.762560\pi\)
−0.734451 + 0.678661i \(0.762560\pi\)
\(464\) 10.4462 0.484951
\(465\) 0 0
\(466\) −16.5149 −0.765037
\(467\) −41.2962 −1.91096 −0.955479 0.295058i \(-0.904661\pi\)
−0.955479 + 0.295058i \(0.904661\pi\)
\(468\) 0 0
\(469\) −16.9774 −0.783944
\(470\) −7.66062 −0.353358
\(471\) 0 0
\(472\) 3.21453 0.147961
\(473\) 4.87087 0.223963
\(474\) 0 0
\(475\) −22.1108 −1.01451
\(476\) −6.80072 −0.311711
\(477\) 0 0
\(478\) −14.8861 −0.680875
\(479\) −18.4305 −0.842112 −0.421056 0.907035i \(-0.638340\pi\)
−0.421056 + 0.907035i \(0.638340\pi\)
\(480\) 0 0
\(481\) 60.3952 2.75378
\(482\) −1.70313 −0.0775754
\(483\) 0 0
\(484\) −0.885329 −0.0402422
\(485\) 4.45731 0.202396
\(486\) 0 0
\(487\) −10.4137 −0.471891 −0.235945 0.971766i \(-0.575819\pi\)
−0.235945 + 0.971766i \(0.575819\pi\)
\(488\) −5.85140 −0.264880
\(489\) 0 0
\(490\) −2.78770 −0.125935
\(491\) −23.7037 −1.06973 −0.534867 0.844936i \(-0.679638\pi\)
−0.534867 + 0.844936i \(0.679638\pi\)
\(492\) 0 0
\(493\) 47.4594 2.13747
\(494\) 28.7145 1.29193
\(495\) 0 0
\(496\) −6.42013 −0.288272
\(497\) −22.9597 −1.02988
\(498\) 0 0
\(499\) −24.2061 −1.08361 −0.541807 0.840503i \(-0.682260\pi\)
−0.541807 + 0.840503i \(0.682260\pi\)
\(500\) 5.65637 0.252961
\(501\) 0 0
\(502\) −28.4403 −1.26935
\(503\) 3.47026 0.154731 0.0773657 0.997003i \(-0.475349\pi\)
0.0773657 + 0.997003i \(0.475349\pi\)
\(504\) 0 0
\(505\) −6.63332 −0.295179
\(506\) 3.40072 0.151181
\(507\) 0 0
\(508\) 2.46517 0.109374
\(509\) 1.53550 0.0680600 0.0340300 0.999421i \(-0.489166\pi\)
0.0340300 + 0.999421i \(0.489166\pi\)
\(510\) 0 0
\(511\) −22.3693 −0.989558
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −17.2851 −0.762411
\(515\) 9.27614 0.408755
\(516\) 0 0
\(517\) 41.5949 1.82934
\(518\) −14.9484 −0.656795
\(519\) 0 0
\(520\) −3.54239 −0.155344
\(521\) −16.9837 −0.744068 −0.372034 0.928219i \(-0.621340\pi\)
−0.372034 + 0.928219i \(0.621340\pi\)
\(522\) 0 0
\(523\) −2.10471 −0.0920326 −0.0460163 0.998941i \(-0.514653\pi\)
−0.0460163 + 0.998941i \(0.514653\pi\)
\(524\) 1.07564 0.0469895
\(525\) 0 0
\(526\) 3.84344 0.167582
\(527\) −29.1682 −1.27059
\(528\) 0 0
\(529\) −21.8566 −0.950288
\(530\) −4.51866 −0.196278
\(531\) 0 0
\(532\) −7.10712 −0.308133
\(533\) −20.4257 −0.884736
\(534\) 0 0
\(535\) −5.11444 −0.221116
\(536\) −11.3418 −0.489892
\(537\) 0 0
\(538\) 1.43863 0.0620239
\(539\) 15.1364 0.651970
\(540\) 0 0
\(541\) 13.3108 0.572277 0.286138 0.958188i \(-0.407628\pi\)
0.286138 + 0.958188i \(0.407628\pi\)
\(542\) 8.44794 0.362870
\(543\) 0 0
\(544\) −4.54325 −0.194790
\(545\) 1.13572 0.0486490
\(546\) 0 0
\(547\) −11.3342 −0.484617 −0.242308 0.970199i \(-0.577905\pi\)
−0.242308 + 0.970199i \(0.577905\pi\)
\(548\) −7.49845 −0.320318
\(549\) 0 0
\(550\) −14.8107 −0.631529
\(551\) 49.5977 2.11293
\(552\) 0 0
\(553\) −6.74340 −0.286758
\(554\) −20.1492 −0.856058
\(555\) 0 0
\(556\) −6.50751 −0.275980
\(557\) 37.7363 1.59894 0.799470 0.600706i \(-0.205114\pi\)
0.799470 + 0.600706i \(0.205114\pi\)
\(558\) 0 0
\(559\) 9.26248 0.391761
\(560\) 0.876775 0.0370505
\(561\) 0 0
\(562\) −0.863739 −0.0364346
\(563\) −18.2752 −0.770209 −0.385105 0.922873i \(-0.625835\pi\)
−0.385105 + 0.922873i \(0.625835\pi\)
\(564\) 0 0
\(565\) −6.98332 −0.293790
\(566\) −14.7643 −0.620590
\(567\) 0 0
\(568\) −15.3383 −0.643581
\(569\) 12.2166 0.512146 0.256073 0.966657i \(-0.417571\pi\)
0.256073 + 0.966657i \(0.417571\pi\)
\(570\) 0 0
\(571\) 0.810246 0.0339077 0.0169539 0.999856i \(-0.494603\pi\)
0.0169539 + 0.999856i \(0.494603\pi\)
\(572\) 19.2341 0.804219
\(573\) 0 0
\(574\) 5.05556 0.211015
\(575\) −4.97959 −0.207663
\(576\) 0 0
\(577\) −19.9944 −0.832376 −0.416188 0.909279i \(-0.636634\pi\)
−0.416188 + 0.909279i \(0.636634\pi\)
\(578\) −3.64108 −0.151449
\(579\) 0 0
\(580\) −6.11865 −0.254063
\(581\) −1.63230 −0.0677191
\(582\) 0 0
\(583\) 24.5350 1.01614
\(584\) −14.9439 −0.618381
\(585\) 0 0
\(586\) 27.7810 1.14762
\(587\) 15.1575 0.625617 0.312808 0.949816i \(-0.398730\pi\)
0.312808 + 0.949816i \(0.398730\pi\)
\(588\) 0 0
\(589\) −30.4824 −1.25600
\(590\) −1.88285 −0.0775159
\(591\) 0 0
\(592\) −9.98632 −0.410435
\(593\) −10.1080 −0.415085 −0.207542 0.978226i \(-0.566546\pi\)
−0.207542 + 0.978226i \(0.566546\pi\)
\(594\) 0 0
\(595\) 3.98341 0.163304
\(596\) 13.2807 0.543999
\(597\) 0 0
\(598\) 6.46684 0.264449
\(599\) 13.7754 0.562849 0.281424 0.959583i \(-0.409193\pi\)
0.281424 + 0.959583i \(0.409193\pi\)
\(600\) 0 0
\(601\) −8.92779 −0.364172 −0.182086 0.983283i \(-0.558285\pi\)
−0.182086 + 0.983283i \(0.558285\pi\)
\(602\) −2.29255 −0.0934375
\(603\) 0 0
\(604\) 2.77304 0.112833
\(605\) 0.518566 0.0210827
\(606\) 0 0
\(607\) 12.3237 0.500204 0.250102 0.968219i \(-0.419536\pi\)
0.250102 + 0.968219i \(0.419536\pi\)
\(608\) −4.74794 −0.192554
\(609\) 0 0
\(610\) 3.42735 0.138770
\(611\) 79.0972 3.19993
\(612\) 0 0
\(613\) −26.9032 −1.08661 −0.543306 0.839535i \(-0.682828\pi\)
−0.543306 + 0.839535i \(0.682828\pi\)
\(614\) 24.5648 0.991355
\(615\) 0 0
\(616\) −4.76063 −0.191811
\(617\) 1.35067 0.0543758 0.0271879 0.999630i \(-0.491345\pi\)
0.0271879 + 0.999630i \(0.491345\pi\)
\(618\) 0 0
\(619\) 9.41979 0.378613 0.189307 0.981918i \(-0.439376\pi\)
0.189307 + 0.981918i \(0.439376\pi\)
\(620\) 3.76048 0.151024
\(621\) 0 0
\(622\) −23.3167 −0.934913
\(623\) −6.63317 −0.265752
\(624\) 0 0
\(625\) 19.9715 0.798859
\(626\) −33.8040 −1.35108
\(627\) 0 0
\(628\) 3.97992 0.158816
\(629\) −45.3703 −1.80903
\(630\) 0 0
\(631\) −32.1621 −1.28035 −0.640177 0.768228i \(-0.721139\pi\)
−0.640177 + 0.768228i \(0.721139\pi\)
\(632\) −4.50495 −0.179197
\(633\) 0 0
\(634\) −28.4355 −1.12932
\(635\) −1.44393 −0.0573006
\(636\) 0 0
\(637\) 28.7834 1.14044
\(638\) 33.2225 1.31529
\(639\) 0 0
\(640\) 0.585733 0.0231531
\(641\) 10.5434 0.416441 0.208220 0.978082i \(-0.433233\pi\)
0.208220 + 0.978082i \(0.433233\pi\)
\(642\) 0 0
\(643\) 42.5228 1.67694 0.838468 0.544951i \(-0.183452\pi\)
0.838468 + 0.544951i \(0.183452\pi\)
\(644\) −1.60060 −0.0630727
\(645\) 0 0
\(646\) −21.5710 −0.848701
\(647\) 5.84286 0.229707 0.114853 0.993382i \(-0.463360\pi\)
0.114853 + 0.993382i \(0.463360\pi\)
\(648\) 0 0
\(649\) 10.2233 0.401301
\(650\) −28.1640 −1.10468
\(651\) 0 0
\(652\) 17.0742 0.668677
\(653\) 5.00429 0.195833 0.0979164 0.995195i \(-0.468782\pi\)
0.0979164 + 0.995195i \(0.468782\pi\)
\(654\) 0 0
\(655\) −0.630037 −0.0246176
\(656\) 3.37739 0.131865
\(657\) 0 0
\(658\) −19.5773 −0.763203
\(659\) 31.8454 1.24052 0.620261 0.784395i \(-0.287027\pi\)
0.620261 + 0.784395i \(0.287027\pi\)
\(660\) 0 0
\(661\) 18.9484 0.737005 0.368503 0.929627i \(-0.379871\pi\)
0.368503 + 0.929627i \(0.379871\pi\)
\(662\) 33.1657 1.28902
\(663\) 0 0
\(664\) −1.09046 −0.0423181
\(665\) 4.16287 0.161429
\(666\) 0 0
\(667\) 11.1700 0.432503
\(668\) 3.88787 0.150426
\(669\) 0 0
\(670\) 6.64327 0.256652
\(671\) −18.6095 −0.718413
\(672\) 0 0
\(673\) 25.9693 1.00104 0.500521 0.865725i \(-0.333142\pi\)
0.500521 + 0.865725i \(0.333142\pi\)
\(674\) −6.40874 −0.246856
\(675\) 0 0
\(676\) 23.5757 0.906759
\(677\) −42.6732 −1.64006 −0.820032 0.572317i \(-0.806045\pi\)
−0.820032 + 0.572317i \(0.806045\pi\)
\(678\) 0 0
\(679\) 11.3910 0.437146
\(680\) 2.66113 0.102050
\(681\) 0 0
\(682\) −20.4183 −0.781857
\(683\) −43.2949 −1.65663 −0.828317 0.560260i \(-0.810701\pi\)
−0.828317 + 0.560260i \(0.810701\pi\)
\(684\) 0 0
\(685\) 4.39209 0.167813
\(686\) −17.6024 −0.672062
\(687\) 0 0
\(688\) −1.53155 −0.0583897
\(689\) 46.6559 1.77745
\(690\) 0 0
\(691\) −19.8538 −0.755273 −0.377636 0.925954i \(-0.623263\pi\)
−0.377636 + 0.925954i \(0.623263\pi\)
\(692\) 6.81516 0.259074
\(693\) 0 0
\(694\) 30.3873 1.15349
\(695\) 3.81166 0.144584
\(696\) 0 0
\(697\) 15.3443 0.581207
\(698\) −13.1951 −0.499443
\(699\) 0 0
\(700\) 6.97088 0.263474
\(701\) 27.2877 1.03064 0.515321 0.856997i \(-0.327673\pi\)
0.515321 + 0.856997i \(0.327673\pi\)
\(702\) 0 0
\(703\) −47.4144 −1.78827
\(704\) −3.18036 −0.119864
\(705\) 0 0
\(706\) −16.0690 −0.604764
\(707\) −16.9520 −0.637545
\(708\) 0 0
\(709\) −24.4005 −0.916382 −0.458191 0.888854i \(-0.651502\pi\)
−0.458191 + 0.888854i \(0.651502\pi\)
\(710\) 8.98414 0.337169
\(711\) 0 0
\(712\) −4.43131 −0.166070
\(713\) −6.86498 −0.257095
\(714\) 0 0
\(715\) −11.2661 −0.421327
\(716\) −1.64779 −0.0615808
\(717\) 0 0
\(718\) −10.4685 −0.390681
\(719\) 5.73870 0.214018 0.107009 0.994258i \(-0.465873\pi\)
0.107009 + 0.994258i \(0.465873\pi\)
\(720\) 0 0
\(721\) 23.7059 0.882854
\(722\) −3.54291 −0.131853
\(723\) 0 0
\(724\) 16.1341 0.599621
\(725\) −48.6469 −1.80670
\(726\) 0 0
\(727\) 3.25441 0.120699 0.0603497 0.998177i \(-0.480778\pi\)
0.0603497 + 0.998177i \(0.480778\pi\)
\(728\) −9.05285 −0.335521
\(729\) 0 0
\(730\) 8.75311 0.323967
\(731\) −6.95820 −0.257358
\(732\) 0 0
\(733\) 39.2895 1.45119 0.725595 0.688122i \(-0.241564\pi\)
0.725595 + 0.688122i \(0.241564\pi\)
\(734\) −26.1262 −0.964335
\(735\) 0 0
\(736\) −1.06929 −0.0394145
\(737\) −36.0710 −1.32869
\(738\) 0 0
\(739\) 22.5580 0.829809 0.414904 0.909865i \(-0.363815\pi\)
0.414904 + 0.909865i \(0.363815\pi\)
\(740\) 5.84932 0.215025
\(741\) 0 0
\(742\) −11.5478 −0.423933
\(743\) 16.1152 0.591210 0.295605 0.955310i \(-0.404479\pi\)
0.295605 + 0.955310i \(0.404479\pi\)
\(744\) 0 0
\(745\) −7.77895 −0.284999
\(746\) 26.2059 0.959467
\(747\) 0 0
\(748\) −14.4491 −0.528313
\(749\) −13.0704 −0.477580
\(750\) 0 0
\(751\) 11.6657 0.425688 0.212844 0.977086i \(-0.431727\pi\)
0.212844 + 0.977086i \(0.431727\pi\)
\(752\) −13.0787 −0.476931
\(753\) 0 0
\(754\) 63.1761 2.30074
\(755\) −1.62426 −0.0591128
\(756\) 0 0
\(757\) 44.1726 1.60548 0.802741 0.596328i \(-0.203374\pi\)
0.802741 + 0.596328i \(0.203374\pi\)
\(758\) −32.9269 −1.19596
\(759\) 0 0
\(760\) 2.78102 0.100878
\(761\) −4.55042 −0.164953 −0.0824763 0.996593i \(-0.526283\pi\)
−0.0824763 + 0.996593i \(0.526283\pi\)
\(762\) 0 0
\(763\) 2.90243 0.105075
\(764\) −0.988248 −0.0357536
\(765\) 0 0
\(766\) 20.5468 0.742386
\(767\) 19.4408 0.701966
\(768\) 0 0
\(769\) 12.5915 0.454063 0.227031 0.973887i \(-0.427098\pi\)
0.227031 + 0.973887i \(0.427098\pi\)
\(770\) 2.78846 0.100489
\(771\) 0 0
\(772\) 16.2266 0.584006
\(773\) 37.3896 1.34481 0.672405 0.740183i \(-0.265261\pi\)
0.672405 + 0.740183i \(0.265261\pi\)
\(774\) 0 0
\(775\) 29.8980 1.07397
\(776\) 7.60980 0.273176
\(777\) 0 0
\(778\) 22.5931 0.810001
\(779\) 16.0356 0.574536
\(780\) 0 0
\(781\) −48.7813 −1.74553
\(782\) −4.85804 −0.173723
\(783\) 0 0
\(784\) −4.75933 −0.169976
\(785\) −2.33117 −0.0832031
\(786\) 0 0
\(787\) −20.6138 −0.734803 −0.367402 0.930062i \(-0.619753\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(788\) −21.1776 −0.754422
\(789\) 0 0
\(790\) 2.63870 0.0938806
\(791\) −17.8464 −0.634546
\(792\) 0 0
\(793\) −35.3880 −1.25667
\(794\) −7.21751 −0.256140
\(795\) 0 0
\(796\) −5.43075 −0.192488
\(797\) −23.8712 −0.845560 −0.422780 0.906232i \(-0.638946\pi\)
−0.422780 + 0.906232i \(0.638946\pi\)
\(798\) 0 0
\(799\) −59.4197 −2.10212
\(800\) 4.65692 0.164647
\(801\) 0 0
\(802\) −11.3770 −0.401738
\(803\) −47.5268 −1.67718
\(804\) 0 0
\(805\) 0.937527 0.0330435
\(806\) −38.8276 −1.36764
\(807\) 0 0
\(808\) −11.3248 −0.398406
\(809\) 12.5712 0.441978 0.220989 0.975276i \(-0.429071\pi\)
0.220989 + 0.975276i \(0.429071\pi\)
\(810\) 0 0
\(811\) 31.4170 1.10320 0.551600 0.834109i \(-0.314018\pi\)
0.551600 + 0.834109i \(0.314018\pi\)
\(812\) −15.6367 −0.548741
\(813\) 0 0
\(814\) −31.7601 −1.11319
\(815\) −10.0009 −0.350316
\(816\) 0 0
\(817\) −7.27170 −0.254404
\(818\) −2.13979 −0.0748162
\(819\) 0 0
\(820\) −1.97825 −0.0690833
\(821\) −34.9424 −1.21950 −0.609748 0.792595i \(-0.708730\pi\)
−0.609748 + 0.792595i \(0.708730\pi\)
\(822\) 0 0
\(823\) −16.4458 −0.573263 −0.286632 0.958041i \(-0.592536\pi\)
−0.286632 + 0.958041i \(0.592536\pi\)
\(824\) 15.8368 0.551701
\(825\) 0 0
\(826\) −4.81178 −0.167423
\(827\) −5.00885 −0.174175 −0.0870874 0.996201i \(-0.527756\pi\)
−0.0870874 + 0.996201i \(0.527756\pi\)
\(828\) 0 0
\(829\) −12.1491 −0.421954 −0.210977 0.977491i \(-0.567665\pi\)
−0.210977 + 0.977491i \(0.567665\pi\)
\(830\) 0.638719 0.0221703
\(831\) 0 0
\(832\) −6.04779 −0.209669
\(833\) −21.6228 −0.749186
\(834\) 0 0
\(835\) −2.27726 −0.0788077
\(836\) −15.1001 −0.522249
\(837\) 0 0
\(838\) 33.3352 1.15155
\(839\) −28.3298 −0.978053 −0.489026 0.872269i \(-0.662648\pi\)
−0.489026 + 0.872269i \(0.662648\pi\)
\(840\) 0 0
\(841\) 80.1222 2.76283
\(842\) 25.1838 0.867890
\(843\) 0 0
\(844\) 8.70251 0.299553
\(845\) −13.8091 −0.475047
\(846\) 0 0
\(847\) 1.32524 0.0455357
\(848\) −7.71455 −0.264919
\(849\) 0 0
\(850\) 21.1575 0.725697
\(851\) −10.6783 −0.366046
\(852\) 0 0
\(853\) −34.7850 −1.19101 −0.595507 0.803350i \(-0.703049\pi\)
−0.595507 + 0.803350i \(0.703049\pi\)
\(854\) 8.75888 0.299723
\(855\) 0 0
\(856\) −8.73169 −0.298443
\(857\) −29.5698 −1.01008 −0.505042 0.863095i \(-0.668523\pi\)
−0.505042 + 0.863095i \(0.668523\pi\)
\(858\) 0 0
\(859\) 13.1666 0.449240 0.224620 0.974446i \(-0.427886\pi\)
0.224620 + 0.974446i \(0.427886\pi\)
\(860\) 0.897078 0.0305901
\(861\) 0 0
\(862\) −21.3427 −0.726934
\(863\) −26.5705 −0.904469 −0.452235 0.891899i \(-0.649373\pi\)
−0.452235 + 0.891899i \(0.649373\pi\)
\(864\) 0 0
\(865\) −3.99186 −0.135727
\(866\) −31.4723 −1.06947
\(867\) 0 0
\(868\) 9.61020 0.326192
\(869\) −14.3273 −0.486022
\(870\) 0 0
\(871\) −68.5929 −2.32418
\(872\) 1.93898 0.0656620
\(873\) 0 0
\(874\) −5.07692 −0.171729
\(875\) −8.46695 −0.286235
\(876\) 0 0
\(877\) −53.8177 −1.81730 −0.908648 0.417563i \(-0.862884\pi\)
−0.908648 + 0.417563i \(0.862884\pi\)
\(878\) −8.30672 −0.280338
\(879\) 0 0
\(880\) 1.86284 0.0627963
\(881\) 16.5945 0.559082 0.279541 0.960134i \(-0.409818\pi\)
0.279541 + 0.960134i \(0.409818\pi\)
\(882\) 0 0
\(883\) −51.7445 −1.74134 −0.870671 0.491866i \(-0.836315\pi\)
−0.870671 + 0.491866i \(0.836315\pi\)
\(884\) −27.4766 −0.924137
\(885\) 0 0
\(886\) 5.18961 0.174348
\(887\) 39.9001 1.33972 0.669858 0.742489i \(-0.266355\pi\)
0.669858 + 0.742489i \(0.266355\pi\)
\(888\) 0 0
\(889\) −3.69007 −0.123761
\(890\) 2.59556 0.0870034
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) −62.0968 −2.07799
\(894\) 0 0
\(895\) 0.965164 0.0322619
\(896\) 1.49689 0.0500075
\(897\) 0 0
\(898\) −12.2820 −0.409855
\(899\) −67.0657 −2.23677
\(900\) 0 0
\(901\) −35.0491 −1.16765
\(902\) 10.7413 0.357646
\(903\) 0 0
\(904\) −11.9224 −0.396532
\(905\) −9.45030 −0.314138
\(906\) 0 0
\(907\) 26.9406 0.894549 0.447275 0.894397i \(-0.352395\pi\)
0.447275 + 0.894397i \(0.352395\pi\)
\(908\) 5.33837 0.177160
\(909\) 0 0
\(910\) 5.30255 0.175778
\(911\) −19.2901 −0.639111 −0.319555 0.947568i \(-0.603534\pi\)
−0.319555 + 0.947568i \(0.603534\pi\)
\(912\) 0 0
\(913\) −3.46806 −0.114776
\(914\) −36.6297 −1.21160
\(915\) 0 0
\(916\) −16.9242 −0.559191
\(917\) −1.61011 −0.0531705
\(918\) 0 0
\(919\) 46.0123 1.51781 0.758903 0.651204i \(-0.225736\pi\)
0.758903 + 0.651204i \(0.225736\pi\)
\(920\) 0.626318 0.0206491
\(921\) 0 0
\(922\) −23.3611 −0.769358
\(923\) −92.7628 −3.05332
\(924\) 0 0
\(925\) 46.5055 1.52909
\(926\) 31.6070 1.03867
\(927\) 0 0
\(928\) −10.4462 −0.342912
\(929\) 40.9071 1.34212 0.671059 0.741404i \(-0.265840\pi\)
0.671059 + 0.741404i \(0.265840\pi\)
\(930\) 0 0
\(931\) −22.5970 −0.740587
\(932\) 16.5149 0.540963
\(933\) 0 0
\(934\) 41.2962 1.35125
\(935\) 8.46333 0.276781
\(936\) 0 0
\(937\) −34.4636 −1.12588 −0.562938 0.826499i \(-0.690329\pi\)
−0.562938 + 0.826499i \(0.690329\pi\)
\(938\) 16.9774 0.554332
\(939\) 0 0
\(940\) 7.66062 0.249862
\(941\) −12.9895 −0.423445 −0.211723 0.977330i \(-0.567907\pi\)
−0.211723 + 0.977330i \(0.567907\pi\)
\(942\) 0 0
\(943\) 3.61140 0.117604
\(944\) −3.21453 −0.104624
\(945\) 0 0
\(946\) −4.87087 −0.158366
\(947\) 1.02790 0.0334022 0.0167011 0.999861i \(-0.494684\pi\)
0.0167011 + 0.999861i \(0.494684\pi\)
\(948\) 0 0
\(949\) −90.3773 −2.93377
\(950\) 22.1108 0.717367
\(951\) 0 0
\(952\) 6.80072 0.220413
\(953\) 55.1750 1.78729 0.893647 0.448770i \(-0.148138\pi\)
0.893647 + 0.448770i \(0.148138\pi\)
\(954\) 0 0
\(955\) 0.578849 0.0187311
\(956\) 14.8861 0.481451
\(957\) 0 0
\(958\) 18.4305 0.595463
\(959\) 11.2243 0.362452
\(960\) 0 0
\(961\) 10.2181 0.329615
\(962\) −60.3952 −1.94722
\(963\) 0 0
\(964\) 1.70313 0.0548541
\(965\) −9.50442 −0.305958
\(966\) 0 0
\(967\) −55.2258 −1.77594 −0.887971 0.459899i \(-0.847886\pi\)
−0.887971 + 0.459899i \(0.847886\pi\)
\(968\) 0.885329 0.0284556
\(969\) 0 0
\(970\) −4.45731 −0.143115
\(971\) 52.7978 1.69436 0.847180 0.531305i \(-0.178298\pi\)
0.847180 + 0.531305i \(0.178298\pi\)
\(972\) 0 0
\(973\) 9.74100 0.312282
\(974\) 10.4137 0.333677
\(975\) 0 0
\(976\) 5.85140 0.187299
\(977\) 24.8695 0.795647 0.397824 0.917462i \(-0.369766\pi\)
0.397824 + 0.917462i \(0.369766\pi\)
\(978\) 0 0
\(979\) −14.0931 −0.450419
\(980\) 2.78770 0.0890497
\(981\) 0 0
\(982\) 23.7037 0.756417
\(983\) −29.7887 −0.950111 −0.475056 0.879956i \(-0.657572\pi\)
−0.475056 + 0.879956i \(0.657572\pi\)
\(984\) 0 0
\(985\) 12.4044 0.395238
\(986\) −47.4594 −1.51142
\(987\) 0 0
\(988\) −28.7145 −0.913530
\(989\) −1.63767 −0.0520748
\(990\) 0 0
\(991\) 15.9480 0.506604 0.253302 0.967387i \(-0.418483\pi\)
0.253302 + 0.967387i \(0.418483\pi\)
\(992\) 6.42013 0.203839
\(993\) 0 0
\(994\) 22.9597 0.728237
\(995\) 3.18097 0.100843
\(996\) 0 0
\(997\) 25.9344 0.821352 0.410676 0.911781i \(-0.365293\pi\)
0.410676 + 0.911781i \(0.365293\pi\)
\(998\) 24.2061 0.766231
\(999\) 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 4014.2.a.x.1.3 8
3.2 odd 2 4014.2.a.y.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.a.x.1.3 8 1.1 even 1 trivial
4014.2.a.y.1.6 yes 8 3.2 odd 2