Properties

Label 8025.2.a.bf.1.8
Level $8025$
Weight $2$
Character 8025.1
Self dual yes
Analytic conductor $64.080$
Analytic rank $1$
Dimension $12$
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] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, 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("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} - 529 x^{3} + 267 x^{2} + 15 x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1605)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.155707\) of defining polynomial
Character \(\chi\) \(=\) 8025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.155707 q^{2} -1.00000 q^{3} -1.97576 q^{4} -0.155707 q^{6} -3.91791 q^{7} -0.619053 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.155707 q^{2} -1.00000 q^{3} -1.97576 q^{4} -0.155707 q^{6} -3.91791 q^{7} -0.619053 q^{8} +1.00000 q^{9} -4.20576 q^{11} +1.97576 q^{12} -6.15258 q^{13} -0.610045 q^{14} +3.85512 q^{16} +4.53299 q^{17} +0.155707 q^{18} +6.39031 q^{19} +3.91791 q^{21} -0.654866 q^{22} +1.55096 q^{23} +0.619053 q^{24} -0.957999 q^{26} -1.00000 q^{27} +7.74083 q^{28} -3.34760 q^{29} -4.12813 q^{31} +1.83837 q^{32} +4.20576 q^{33} +0.705817 q^{34} -1.97576 q^{36} -7.16219 q^{37} +0.995015 q^{38} +6.15258 q^{39} +7.18328 q^{41} +0.610045 q^{42} +9.32016 q^{43} +8.30956 q^{44} +0.241495 q^{46} -11.9785 q^{47} -3.85512 q^{48} +8.35000 q^{49} -4.53299 q^{51} +12.1560 q^{52} +12.4671 q^{53} -0.155707 q^{54} +2.42539 q^{56} -6.39031 q^{57} -0.521244 q^{58} +6.02997 q^{59} +9.14686 q^{61} -0.642779 q^{62} -3.91791 q^{63} -7.42399 q^{64} +0.654866 q^{66} -5.70713 q^{67} -8.95607 q^{68} -1.55096 q^{69} +7.85627 q^{71} -0.619053 q^{72} +9.15535 q^{73} -1.11520 q^{74} -12.6257 q^{76} +16.4778 q^{77} +0.957999 q^{78} +3.30344 q^{79} +1.00000 q^{81} +1.11849 q^{82} -3.87613 q^{83} -7.74083 q^{84} +1.45121 q^{86} +3.34760 q^{87} +2.60359 q^{88} -6.68613 q^{89} +24.1052 q^{91} -3.06432 q^{92} +4.12813 q^{93} -1.86514 q^{94} -1.83837 q^{96} +6.56241 q^{97} +1.30015 q^{98} -4.20576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 3 q^{6} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 3 q^{6} - 7 q^{7} - 3 q^{8} + 12 q^{9} + 4 q^{11} - 15 q^{12} - 13 q^{13} + 4 q^{14} + 13 q^{16} + 4 q^{17} - 3 q^{18} + 14 q^{19} + 7 q^{21} - 15 q^{22} - 11 q^{23} + 3 q^{24} - 8 q^{26} - 12 q^{27} - 16 q^{28} - 7 q^{29} + 4 q^{31} - 4 q^{32} - 4 q^{33} + q^{34} + 15 q^{36} - 24 q^{37} + 11 q^{38} + 13 q^{39} + 13 q^{41} - 4 q^{42} - 25 q^{43} + 10 q^{44} - 22 q^{46} - 19 q^{47} - 13 q^{48} + 9 q^{49} - 4 q^{51} - 20 q^{52} - 11 q^{53} + 3 q^{54} - 37 q^{56} - 14 q^{57} + 2 q^{58} + 8 q^{59} + 7 q^{61} + 11 q^{62} - 7 q^{63} - 19 q^{64} + 15 q^{66} - 33 q^{67} + 24 q^{68} + 11 q^{69} - 3 q^{72} - 34 q^{73} - 27 q^{74} - 9 q^{76} + 29 q^{77} + 8 q^{78} + 12 q^{81} - q^{82} + 24 q^{83} + 16 q^{84} - 36 q^{86} + 7 q^{87} + 6 q^{88} - 10 q^{89} + 30 q^{91} + 28 q^{92} - 4 q^{93} - 8 q^{94} + 4 q^{96} - 16 q^{97} + 36 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.155707 0.110101 0.0550507 0.998484i \(-0.482468\pi\)
0.0550507 + 0.998484i \(0.482468\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.97576 −0.987878
\(5\) 0 0
\(6\) −0.155707 −0.0635671
\(7\) −3.91791 −1.48083 −0.740415 0.672150i \(-0.765371\pi\)
−0.740415 + 0.672150i \(0.765371\pi\)
\(8\) −0.619053 −0.218868
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.20576 −1.26809 −0.634043 0.773298i \(-0.718606\pi\)
−0.634043 + 0.773298i \(0.718606\pi\)
\(12\) 1.97576 0.570351
\(13\) −6.15258 −1.70642 −0.853209 0.521568i \(-0.825347\pi\)
−0.853209 + 0.521568i \(0.825347\pi\)
\(14\) −0.610045 −0.163041
\(15\) 0 0
\(16\) 3.85512 0.963780
\(17\) 4.53299 1.09941 0.549705 0.835359i \(-0.314740\pi\)
0.549705 + 0.835359i \(0.314740\pi\)
\(18\) 0.155707 0.0367005
\(19\) 6.39031 1.46604 0.733018 0.680209i \(-0.238111\pi\)
0.733018 + 0.680209i \(0.238111\pi\)
\(20\) 0 0
\(21\) 3.91791 0.854958
\(22\) −0.654866 −0.139618
\(23\) 1.55096 0.323397 0.161699 0.986840i \(-0.448303\pi\)
0.161699 + 0.986840i \(0.448303\pi\)
\(24\) 0.619053 0.126364
\(25\) 0 0
\(26\) −0.957999 −0.187879
\(27\) −1.00000 −0.192450
\(28\) 7.74083 1.46288
\(29\) −3.34760 −0.621634 −0.310817 0.950470i \(-0.600603\pi\)
−0.310817 + 0.950470i \(0.600603\pi\)
\(30\) 0 0
\(31\) −4.12813 −0.741434 −0.370717 0.928746i \(-0.620888\pi\)
−0.370717 + 0.928746i \(0.620888\pi\)
\(32\) 1.83837 0.324982
\(33\) 4.20576 0.732129
\(34\) 0.705817 0.121047
\(35\) 0 0
\(36\) −1.97576 −0.329293
\(37\) −7.16219 −1.17746 −0.588728 0.808331i \(-0.700371\pi\)
−0.588728 + 0.808331i \(0.700371\pi\)
\(38\) 0.995015 0.161413
\(39\) 6.15258 0.985201
\(40\) 0 0
\(41\) 7.18328 1.12184 0.560920 0.827870i \(-0.310448\pi\)
0.560920 + 0.827870i \(0.310448\pi\)
\(42\) 0.610045 0.0941320
\(43\) 9.32016 1.42131 0.710656 0.703540i \(-0.248398\pi\)
0.710656 + 0.703540i \(0.248398\pi\)
\(44\) 8.30956 1.25271
\(45\) 0 0
\(46\) 0.241495 0.0356065
\(47\) −11.9785 −1.74725 −0.873623 0.486603i \(-0.838236\pi\)
−0.873623 + 0.486603i \(0.838236\pi\)
\(48\) −3.85512 −0.556439
\(49\) 8.35000 1.19286
\(50\) 0 0
\(51\) −4.53299 −0.634745
\(52\) 12.1560 1.68573
\(53\) 12.4671 1.71249 0.856247 0.516567i \(-0.172790\pi\)
0.856247 + 0.516567i \(0.172790\pi\)
\(54\) −0.155707 −0.0211890
\(55\) 0 0
\(56\) 2.42539 0.324107
\(57\) −6.39031 −0.846417
\(58\) −0.521244 −0.0684428
\(59\) 6.02997 0.785035 0.392518 0.919744i \(-0.371604\pi\)
0.392518 + 0.919744i \(0.371604\pi\)
\(60\) 0 0
\(61\) 9.14686 1.17113 0.585567 0.810624i \(-0.300872\pi\)
0.585567 + 0.810624i \(0.300872\pi\)
\(62\) −0.642779 −0.0816330
\(63\) −3.91791 −0.493610
\(64\) −7.42399 −0.927999
\(65\) 0 0
\(66\) 0.654866 0.0806085
\(67\) −5.70713 −0.697237 −0.348619 0.937265i \(-0.613349\pi\)
−0.348619 + 0.937265i \(0.613349\pi\)
\(68\) −8.95607 −1.08608
\(69\) −1.55096 −0.186714
\(70\) 0 0
\(71\) 7.85627 0.932368 0.466184 0.884688i \(-0.345628\pi\)
0.466184 + 0.884688i \(0.345628\pi\)
\(72\) −0.619053 −0.0729561
\(73\) 9.15535 1.07155 0.535776 0.844360i \(-0.320019\pi\)
0.535776 + 0.844360i \(0.320019\pi\)
\(74\) −1.11520 −0.129640
\(75\) 0 0
\(76\) −12.6257 −1.44827
\(77\) 16.4778 1.87782
\(78\) 0.957999 0.108472
\(79\) 3.30344 0.371666 0.185833 0.982581i \(-0.440502\pi\)
0.185833 + 0.982581i \(0.440502\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.11849 0.123516
\(83\) −3.87613 −0.425460 −0.212730 0.977111i \(-0.568236\pi\)
−0.212730 + 0.977111i \(0.568236\pi\)
\(84\) −7.74083 −0.844594
\(85\) 0 0
\(86\) 1.45121 0.156488
\(87\) 3.34760 0.358900
\(88\) 2.60359 0.277543
\(89\) −6.68613 −0.708728 −0.354364 0.935107i \(-0.615303\pi\)
−0.354364 + 0.935107i \(0.615303\pi\)
\(90\) 0 0
\(91\) 24.1052 2.52692
\(92\) −3.06432 −0.319477
\(93\) 4.12813 0.428067
\(94\) −1.86514 −0.192374
\(95\) 0 0
\(96\) −1.83837 −0.187628
\(97\) 6.56241 0.666312 0.333156 0.942872i \(-0.391886\pi\)
0.333156 + 0.942872i \(0.391886\pi\)
\(98\) 1.30015 0.131335
\(99\) −4.20576 −0.422695
\(100\) 0 0
\(101\) 1.61730 0.160927 0.0804637 0.996758i \(-0.474360\pi\)
0.0804637 + 0.996758i \(0.474360\pi\)
\(102\) −0.705817 −0.0698863
\(103\) 2.47404 0.243775 0.121887 0.992544i \(-0.461105\pi\)
0.121887 + 0.992544i \(0.461105\pi\)
\(104\) 3.80877 0.373481
\(105\) 0 0
\(106\) 1.94122 0.188548
\(107\) −1.00000 −0.0966736
\(108\) 1.97576 0.190117
\(109\) 16.1850 1.55024 0.775121 0.631812i \(-0.217689\pi\)
0.775121 + 0.631812i \(0.217689\pi\)
\(110\) 0 0
\(111\) 7.16219 0.679805
\(112\) −15.1040 −1.42719
\(113\) −16.5476 −1.55667 −0.778334 0.627851i \(-0.783935\pi\)
−0.778334 + 0.627851i \(0.783935\pi\)
\(114\) −0.995015 −0.0931917
\(115\) 0 0
\(116\) 6.61404 0.614098
\(117\) −6.15258 −0.568806
\(118\) 0.938908 0.0864335
\(119\) −17.7598 −1.62804
\(120\) 0 0
\(121\) 6.68844 0.608040
\(122\) 1.42423 0.128944
\(123\) −7.18328 −0.647694
\(124\) 8.15618 0.732447
\(125\) 0 0
\(126\) −0.610045 −0.0543472
\(127\) −21.3131 −1.89123 −0.945615 0.325288i \(-0.894539\pi\)
−0.945615 + 0.325288i \(0.894539\pi\)
\(128\) −4.83272 −0.427156
\(129\) −9.32016 −0.820595
\(130\) 0 0
\(131\) 7.55127 0.659758 0.329879 0.944023i \(-0.392992\pi\)
0.329879 + 0.944023i \(0.392992\pi\)
\(132\) −8.30956 −0.723254
\(133\) −25.0366 −2.17095
\(134\) −0.888640 −0.0767668
\(135\) 0 0
\(136\) −2.80616 −0.240626
\(137\) 13.2009 1.12783 0.563915 0.825833i \(-0.309294\pi\)
0.563915 + 0.825833i \(0.309294\pi\)
\(138\) −0.241495 −0.0205574
\(139\) −13.2352 −1.12260 −0.561298 0.827614i \(-0.689698\pi\)
−0.561298 + 0.827614i \(0.689698\pi\)
\(140\) 0 0
\(141\) 11.9785 1.00877
\(142\) 1.22328 0.102655
\(143\) 25.8763 2.16388
\(144\) 3.85512 0.321260
\(145\) 0 0
\(146\) 1.42555 0.117979
\(147\) −8.35000 −0.688697
\(148\) 14.1507 1.16318
\(149\) −0.813693 −0.0666603 −0.0333302 0.999444i \(-0.510611\pi\)
−0.0333302 + 0.999444i \(0.510611\pi\)
\(150\) 0 0
\(151\) −5.50264 −0.447799 −0.223899 0.974612i \(-0.571879\pi\)
−0.223899 + 0.974612i \(0.571879\pi\)
\(152\) −3.95594 −0.320869
\(153\) 4.53299 0.366470
\(154\) 2.56571 0.206750
\(155\) 0 0
\(156\) −12.1560 −0.973258
\(157\) 0.601189 0.0479801 0.0239900 0.999712i \(-0.492363\pi\)
0.0239900 + 0.999712i \(0.492363\pi\)
\(158\) 0.514369 0.0409210
\(159\) −12.4671 −0.988709
\(160\) 0 0
\(161\) −6.07652 −0.478897
\(162\) 0.155707 0.0122335
\(163\) 21.6686 1.69721 0.848607 0.529025i \(-0.177442\pi\)
0.848607 + 0.529025i \(0.177442\pi\)
\(164\) −14.1924 −1.10824
\(165\) 0 0
\(166\) −0.603540 −0.0468438
\(167\) −6.89307 −0.533402 −0.266701 0.963779i \(-0.585934\pi\)
−0.266701 + 0.963779i \(0.585934\pi\)
\(168\) −2.42539 −0.187123
\(169\) 24.8543 1.91187
\(170\) 0 0
\(171\) 6.39031 0.488679
\(172\) −18.4144 −1.40408
\(173\) −13.4977 −1.02621 −0.513107 0.858325i \(-0.671506\pi\)
−0.513107 + 0.858325i \(0.671506\pi\)
\(174\) 0.521244 0.0395154
\(175\) 0 0
\(176\) −16.2137 −1.22216
\(177\) −6.02997 −0.453240
\(178\) −1.04108 −0.0780320
\(179\) −23.0759 −1.72478 −0.862389 0.506247i \(-0.831032\pi\)
−0.862389 + 0.506247i \(0.831032\pi\)
\(180\) 0 0
\(181\) 3.23219 0.240247 0.120123 0.992759i \(-0.461671\pi\)
0.120123 + 0.992759i \(0.461671\pi\)
\(182\) 3.75335 0.278217
\(183\) −9.14686 −0.676155
\(184\) −0.960125 −0.0707814
\(185\) 0 0
\(186\) 0.642779 0.0471308
\(187\) −19.0647 −1.39415
\(188\) 23.6666 1.72607
\(189\) 3.91791 0.284986
\(190\) 0 0
\(191\) 7.48523 0.541612 0.270806 0.962634i \(-0.412710\pi\)
0.270806 + 0.962634i \(0.412710\pi\)
\(192\) 7.42399 0.535780
\(193\) 23.6594 1.70304 0.851519 0.524324i \(-0.175682\pi\)
0.851519 + 0.524324i \(0.175682\pi\)
\(194\) 1.02181 0.0733619
\(195\) 0 0
\(196\) −16.4976 −1.17840
\(197\) 0.502435 0.0357970 0.0178985 0.999840i \(-0.494302\pi\)
0.0178985 + 0.999840i \(0.494302\pi\)
\(198\) −0.654866 −0.0465393
\(199\) 12.0176 0.851906 0.425953 0.904745i \(-0.359939\pi\)
0.425953 + 0.904745i \(0.359939\pi\)
\(200\) 0 0
\(201\) 5.70713 0.402550
\(202\) 0.251825 0.0177183
\(203\) 13.1156 0.920534
\(204\) 8.95607 0.627051
\(205\) 0 0
\(206\) 0.385226 0.0268399
\(207\) 1.55096 0.107799
\(208\) −23.7189 −1.64461
\(209\) −26.8761 −1.85906
\(210\) 0 0
\(211\) −22.5450 −1.55206 −0.776032 0.630694i \(-0.782770\pi\)
−0.776032 + 0.630694i \(0.782770\pi\)
\(212\) −24.6320 −1.69173
\(213\) −7.85627 −0.538303
\(214\) −0.155707 −0.0106439
\(215\) 0 0
\(216\) 0.619053 0.0421212
\(217\) 16.1736 1.09794
\(218\) 2.52012 0.170684
\(219\) −9.15535 −0.618661
\(220\) 0 0
\(221\) −27.8896 −1.87606
\(222\) 1.11520 0.0748475
\(223\) 11.2208 0.751399 0.375700 0.926741i \(-0.377402\pi\)
0.375700 + 0.926741i \(0.377402\pi\)
\(224\) −7.20258 −0.481243
\(225\) 0 0
\(226\) −2.57658 −0.171391
\(227\) −3.79410 −0.251823 −0.125912 0.992041i \(-0.540186\pi\)
−0.125912 + 0.992041i \(0.540186\pi\)
\(228\) 12.6257 0.836156
\(229\) −21.8196 −1.44188 −0.720941 0.692996i \(-0.756290\pi\)
−0.720941 + 0.692996i \(0.756290\pi\)
\(230\) 0 0
\(231\) −16.4778 −1.08416
\(232\) 2.07234 0.136056
\(233\) −2.52926 −0.165698 −0.0828488 0.996562i \(-0.526402\pi\)
−0.0828488 + 0.996562i \(0.526402\pi\)
\(234\) −0.957999 −0.0626264
\(235\) 0 0
\(236\) −11.9137 −0.775519
\(237\) −3.30344 −0.214582
\(238\) −2.76533 −0.179250
\(239\) −0.0952535 −0.00616144 −0.00308072 0.999995i \(-0.500981\pi\)
−0.00308072 + 0.999995i \(0.500981\pi\)
\(240\) 0 0
\(241\) −19.8084 −1.27597 −0.637984 0.770050i \(-0.720231\pi\)
−0.637984 + 0.770050i \(0.720231\pi\)
\(242\) 1.04144 0.0669461
\(243\) −1.00000 −0.0641500
\(244\) −18.0719 −1.15694
\(245\) 0 0
\(246\) −1.11849 −0.0713121
\(247\) −39.3169 −2.50167
\(248\) 2.55553 0.162276
\(249\) 3.87613 0.245640
\(250\) 0 0
\(251\) −6.15527 −0.388517 −0.194259 0.980950i \(-0.562230\pi\)
−0.194259 + 0.980950i \(0.562230\pi\)
\(252\) 7.74083 0.487626
\(253\) −6.52297 −0.410095
\(254\) −3.31859 −0.208227
\(255\) 0 0
\(256\) 14.0955 0.880969
\(257\) −11.0517 −0.689384 −0.344692 0.938716i \(-0.612017\pi\)
−0.344692 + 0.938716i \(0.612017\pi\)
\(258\) −1.45121 −0.0903486
\(259\) 28.0608 1.74361
\(260\) 0 0
\(261\) −3.34760 −0.207211
\(262\) 1.17579 0.0726403
\(263\) −17.1372 −1.05672 −0.528361 0.849020i \(-0.677193\pi\)
−0.528361 + 0.849020i \(0.677193\pi\)
\(264\) −2.60359 −0.160240
\(265\) 0 0
\(266\) −3.89838 −0.239025
\(267\) 6.68613 0.409185
\(268\) 11.2759 0.688785
\(269\) −2.71376 −0.165461 −0.0827303 0.996572i \(-0.526364\pi\)
−0.0827303 + 0.996572i \(0.526364\pi\)
\(270\) 0 0
\(271\) −8.31668 −0.505202 −0.252601 0.967571i \(-0.581286\pi\)
−0.252601 + 0.967571i \(0.581286\pi\)
\(272\) 17.4752 1.05959
\(273\) −24.1052 −1.45892
\(274\) 2.05547 0.124176
\(275\) 0 0
\(276\) 3.06432 0.184450
\(277\) −0.169617 −0.0101913 −0.00509565 0.999987i \(-0.501622\pi\)
−0.00509565 + 0.999987i \(0.501622\pi\)
\(278\) −2.06081 −0.123599
\(279\) −4.12813 −0.247145
\(280\) 0 0
\(281\) −29.6929 −1.77133 −0.885665 0.464324i \(-0.846297\pi\)
−0.885665 + 0.464324i \(0.846297\pi\)
\(282\) 1.86514 0.111067
\(283\) 11.3215 0.672995 0.336497 0.941684i \(-0.390758\pi\)
0.336497 + 0.941684i \(0.390758\pi\)
\(284\) −15.5221 −0.921066
\(285\) 0 0
\(286\) 4.02912 0.238247
\(287\) −28.1434 −1.66125
\(288\) 1.83837 0.108327
\(289\) 3.54797 0.208704
\(290\) 0 0
\(291\) −6.56241 −0.384695
\(292\) −18.0887 −1.05856
\(293\) 32.0481 1.87227 0.936134 0.351642i \(-0.114377\pi\)
0.936134 + 0.351642i \(0.114377\pi\)
\(294\) −1.30015 −0.0758265
\(295\) 0 0
\(296\) 4.43377 0.257708
\(297\) 4.20576 0.244043
\(298\) −0.126698 −0.00733940
\(299\) −9.54240 −0.551851
\(300\) 0 0
\(301\) −36.5155 −2.10472
\(302\) −0.856799 −0.0493033
\(303\) −1.61730 −0.0929115
\(304\) 24.6354 1.41294
\(305\) 0 0
\(306\) 0.705817 0.0403489
\(307\) 3.07724 0.175627 0.0878135 0.996137i \(-0.472012\pi\)
0.0878135 + 0.996137i \(0.472012\pi\)
\(308\) −32.5561 −1.85505
\(309\) −2.47404 −0.140743
\(310\) 0 0
\(311\) 12.6191 0.715562 0.357781 0.933806i \(-0.383533\pi\)
0.357781 + 0.933806i \(0.383533\pi\)
\(312\) −3.80877 −0.215629
\(313\) 24.2861 1.37273 0.686364 0.727258i \(-0.259206\pi\)
0.686364 + 0.727258i \(0.259206\pi\)
\(314\) 0.0936093 0.00528268
\(315\) 0 0
\(316\) −6.52679 −0.367161
\(317\) −9.52720 −0.535101 −0.267550 0.963544i \(-0.586214\pi\)
−0.267550 + 0.963544i \(0.586214\pi\)
\(318\) −1.94122 −0.108858
\(319\) 14.0792 0.788284
\(320\) 0 0
\(321\) 1.00000 0.0558146
\(322\) −0.946156 −0.0527272
\(323\) 28.9672 1.61178
\(324\) −1.97576 −0.109764
\(325\) 0 0
\(326\) 3.37395 0.186866
\(327\) −16.1850 −0.895033
\(328\) −4.44683 −0.245535
\(329\) 46.9307 2.58737
\(330\) 0 0
\(331\) −0.655437 −0.0360261 −0.0180130 0.999838i \(-0.505734\pi\)
−0.0180130 + 0.999838i \(0.505734\pi\)
\(332\) 7.65828 0.420303
\(333\) −7.16219 −0.392485
\(334\) −1.07330 −0.0587283
\(335\) 0 0
\(336\) 15.1040 0.823991
\(337\) 17.5012 0.953352 0.476676 0.879079i \(-0.341841\pi\)
0.476676 + 0.879079i \(0.341841\pi\)
\(338\) 3.86998 0.210499
\(339\) 16.5476 0.898742
\(340\) 0 0
\(341\) 17.3619 0.940202
\(342\) 0.995015 0.0538042
\(343\) −5.28919 −0.285589
\(344\) −5.76967 −0.311080
\(345\) 0 0
\(346\) −2.10169 −0.112988
\(347\) 9.99897 0.536773 0.268386 0.963311i \(-0.413510\pi\)
0.268386 + 0.963311i \(0.413510\pi\)
\(348\) −6.61404 −0.354550
\(349\) 3.77143 0.201880 0.100940 0.994893i \(-0.467815\pi\)
0.100940 + 0.994893i \(0.467815\pi\)
\(350\) 0 0
\(351\) 6.15258 0.328400
\(352\) −7.73177 −0.412104
\(353\) −17.2527 −0.918267 −0.459133 0.888367i \(-0.651840\pi\)
−0.459133 + 0.888367i \(0.651840\pi\)
\(354\) −0.938908 −0.0499024
\(355\) 0 0
\(356\) 13.2102 0.700137
\(357\) 17.7598 0.939950
\(358\) −3.59308 −0.189900
\(359\) 20.3780 1.07551 0.537755 0.843101i \(-0.319272\pi\)
0.537755 + 0.843101i \(0.319272\pi\)
\(360\) 0 0
\(361\) 21.8360 1.14926
\(362\) 0.503275 0.0264515
\(363\) −6.68844 −0.351052
\(364\) −47.6261 −2.49628
\(365\) 0 0
\(366\) −1.42423 −0.0744456
\(367\) −17.4162 −0.909119 −0.454560 0.890716i \(-0.650203\pi\)
−0.454560 + 0.890716i \(0.650203\pi\)
\(368\) 5.97913 0.311684
\(369\) 7.18328 0.373947
\(370\) 0 0
\(371\) −48.8451 −2.53591
\(372\) −8.15618 −0.422878
\(373\) 0.0544363 0.00281861 0.00140930 0.999999i \(-0.499551\pi\)
0.00140930 + 0.999999i \(0.499551\pi\)
\(374\) −2.96850 −0.153498
\(375\) 0 0
\(376\) 7.41533 0.382417
\(377\) 20.5964 1.06077
\(378\) 0.610045 0.0313773
\(379\) 5.64875 0.290157 0.145079 0.989420i \(-0.453657\pi\)
0.145079 + 0.989420i \(0.453657\pi\)
\(380\) 0 0
\(381\) 21.3131 1.09190
\(382\) 1.16550 0.0596323
\(383\) 33.3118 1.70216 0.851078 0.525040i \(-0.175950\pi\)
0.851078 + 0.525040i \(0.175950\pi\)
\(384\) 4.83272 0.246618
\(385\) 0 0
\(386\) 3.68393 0.187507
\(387\) 9.32016 0.473771
\(388\) −12.9657 −0.658235
\(389\) −8.32421 −0.422054 −0.211027 0.977480i \(-0.567681\pi\)
−0.211027 + 0.977480i \(0.567681\pi\)
\(390\) 0 0
\(391\) 7.03048 0.355547
\(392\) −5.16909 −0.261079
\(393\) −7.55127 −0.380911
\(394\) 0.0782325 0.00394130
\(395\) 0 0
\(396\) 8.30956 0.417571
\(397\) −7.56190 −0.379521 −0.189760 0.981830i \(-0.560771\pi\)
−0.189760 + 0.981830i \(0.560771\pi\)
\(398\) 1.87123 0.0937961
\(399\) 25.0366 1.25340
\(400\) 0 0
\(401\) −13.2500 −0.661676 −0.330838 0.943688i \(-0.607331\pi\)
−0.330838 + 0.943688i \(0.607331\pi\)
\(402\) 0.888640 0.0443213
\(403\) 25.3987 1.26520
\(404\) −3.19539 −0.158977
\(405\) 0 0
\(406\) 2.04219 0.101352
\(407\) 30.1225 1.49311
\(408\) 2.80616 0.138925
\(409\) −26.6915 −1.31981 −0.659906 0.751348i \(-0.729404\pi\)
−0.659906 + 0.751348i \(0.729404\pi\)
\(410\) 0 0
\(411\) −13.2009 −0.651153
\(412\) −4.88810 −0.240820
\(413\) −23.6249 −1.16250
\(414\) 0.241495 0.0118688
\(415\) 0 0
\(416\) −11.3107 −0.554555
\(417\) 13.2352 0.648131
\(418\) −4.18480 −0.204685
\(419\) −26.2407 −1.28194 −0.640971 0.767565i \(-0.721468\pi\)
−0.640971 + 0.767565i \(0.721468\pi\)
\(420\) 0 0
\(421\) 23.0876 1.12522 0.562609 0.826723i \(-0.309798\pi\)
0.562609 + 0.826723i \(0.309798\pi\)
\(422\) −3.51042 −0.170884
\(423\) −11.9785 −0.582415
\(424\) −7.71782 −0.374810
\(425\) 0 0
\(426\) −1.22328 −0.0592679
\(427\) −35.8365 −1.73425
\(428\) 1.97576 0.0955017
\(429\) −25.8763 −1.24932
\(430\) 0 0
\(431\) −22.4525 −1.08150 −0.540748 0.841184i \(-0.681859\pi\)
−0.540748 + 0.841184i \(0.681859\pi\)
\(432\) −3.85512 −0.185480
\(433\) 10.3947 0.499538 0.249769 0.968305i \(-0.419645\pi\)
0.249769 + 0.968305i \(0.419645\pi\)
\(434\) 2.51835 0.120885
\(435\) 0 0
\(436\) −31.9776 −1.53145
\(437\) 9.91111 0.474112
\(438\) −1.42555 −0.0681155
\(439\) −31.2659 −1.49224 −0.746120 0.665812i \(-0.768085\pi\)
−0.746120 + 0.665812i \(0.768085\pi\)
\(440\) 0 0
\(441\) 8.35000 0.397619
\(442\) −4.34260 −0.206556
\(443\) 3.36078 0.159676 0.0798378 0.996808i \(-0.474560\pi\)
0.0798378 + 0.996808i \(0.474560\pi\)
\(444\) −14.1507 −0.671564
\(445\) 0 0
\(446\) 1.74715 0.0827301
\(447\) 0.813693 0.0384864
\(448\) 29.0865 1.37421
\(449\) 5.41796 0.255689 0.127845 0.991794i \(-0.459194\pi\)
0.127845 + 0.991794i \(0.459194\pi\)
\(450\) 0 0
\(451\) −30.2112 −1.42259
\(452\) 32.6940 1.53780
\(453\) 5.50264 0.258537
\(454\) −0.590768 −0.0277261
\(455\) 0 0
\(456\) 3.95594 0.185254
\(457\) −14.4160 −0.674354 −0.337177 0.941441i \(-0.609472\pi\)
−0.337177 + 0.941441i \(0.609472\pi\)
\(458\) −3.39747 −0.158753
\(459\) −4.53299 −0.211582
\(460\) 0 0
\(461\) 28.3177 1.31889 0.659444 0.751754i \(-0.270792\pi\)
0.659444 + 0.751754i \(0.270792\pi\)
\(462\) −2.56571 −0.119367
\(463\) −17.7568 −0.825227 −0.412614 0.910906i \(-0.635384\pi\)
−0.412614 + 0.910906i \(0.635384\pi\)
\(464\) −12.9054 −0.599118
\(465\) 0 0
\(466\) −0.393824 −0.0182435
\(467\) 32.4812 1.50305 0.751525 0.659705i \(-0.229319\pi\)
0.751525 + 0.659705i \(0.229319\pi\)
\(468\) 12.1560 0.561911
\(469\) 22.3600 1.03249
\(470\) 0 0
\(471\) −0.601189 −0.0277013
\(472\) −3.73287 −0.171819
\(473\) −39.1984 −1.80234
\(474\) −0.514369 −0.0236257
\(475\) 0 0
\(476\) 35.0891 1.60830
\(477\) 12.4671 0.570831
\(478\) −0.0148316 −0.000678383 0
\(479\) 32.5182 1.48579 0.742897 0.669405i \(-0.233451\pi\)
0.742897 + 0.669405i \(0.233451\pi\)
\(480\) 0 0
\(481\) 44.0659 2.00923
\(482\) −3.08430 −0.140486
\(483\) 6.07652 0.276491
\(484\) −13.2147 −0.600669
\(485\) 0 0
\(486\) −0.155707 −0.00706301
\(487\) 34.8343 1.57849 0.789245 0.614078i \(-0.210472\pi\)
0.789245 + 0.614078i \(0.210472\pi\)
\(488\) −5.66238 −0.256324
\(489\) −21.6686 −0.979886
\(490\) 0 0
\(491\) −9.42858 −0.425506 −0.212753 0.977106i \(-0.568243\pi\)
−0.212753 + 0.977106i \(0.568243\pi\)
\(492\) 14.1924 0.639843
\(493\) −15.1746 −0.683431
\(494\) −6.12191 −0.275438
\(495\) 0 0
\(496\) −15.9144 −0.714580
\(497\) −30.7802 −1.38068
\(498\) 0.603540 0.0270453
\(499\) 11.1288 0.498192 0.249096 0.968479i \(-0.419866\pi\)
0.249096 + 0.968479i \(0.419866\pi\)
\(500\) 0 0
\(501\) 6.89307 0.307960
\(502\) −0.958418 −0.0427763
\(503\) 5.12488 0.228507 0.114253 0.993452i \(-0.463552\pi\)
0.114253 + 0.993452i \(0.463552\pi\)
\(504\) 2.42539 0.108036
\(505\) 0 0
\(506\) −1.01567 −0.0451521
\(507\) −24.8543 −1.10382
\(508\) 42.1094 1.86830
\(509\) −32.5419 −1.44240 −0.721198 0.692729i \(-0.756408\pi\)
−0.721198 + 0.692729i \(0.756408\pi\)
\(510\) 0 0
\(511\) −35.8698 −1.58679
\(512\) 11.8602 0.524152
\(513\) −6.39031 −0.282139
\(514\) −1.72082 −0.0759021
\(515\) 0 0
\(516\) 18.4144 0.810647
\(517\) 50.3788 2.21566
\(518\) 4.36926 0.191974
\(519\) 13.4977 0.592485
\(520\) 0 0
\(521\) −29.6282 −1.29803 −0.649017 0.760774i \(-0.724820\pi\)
−0.649017 + 0.760774i \(0.724820\pi\)
\(522\) −0.521244 −0.0228143
\(523\) −18.7648 −0.820528 −0.410264 0.911967i \(-0.634563\pi\)
−0.410264 + 0.911967i \(0.634563\pi\)
\(524\) −14.9195 −0.651760
\(525\) 0 0
\(526\) −2.66837 −0.116347
\(527\) −18.7128 −0.815141
\(528\) 16.2137 0.705612
\(529\) −20.5945 −0.895414
\(530\) 0 0
\(531\) 6.02997 0.261678
\(532\) 49.4663 2.14463
\(533\) −44.1957 −1.91433
\(534\) 1.04108 0.0450518
\(535\) 0 0
\(536\) 3.53302 0.152603
\(537\) 23.0759 0.995801
\(538\) −0.422551 −0.0182175
\(539\) −35.1181 −1.51264
\(540\) 0 0
\(541\) −33.0478 −1.42084 −0.710419 0.703779i \(-0.751494\pi\)
−0.710419 + 0.703779i \(0.751494\pi\)
\(542\) −1.29496 −0.0556235
\(543\) −3.23219 −0.138707
\(544\) 8.33333 0.357288
\(545\) 0 0
\(546\) −3.75335 −0.160629
\(547\) −20.6974 −0.884958 −0.442479 0.896779i \(-0.645901\pi\)
−0.442479 + 0.896779i \(0.645901\pi\)
\(548\) −26.0818 −1.11416
\(549\) 9.14686 0.390378
\(550\) 0 0
\(551\) −21.3922 −0.911338
\(552\) 0.960125 0.0408657
\(553\) −12.9426 −0.550375
\(554\) −0.0264105 −0.00112208
\(555\) 0 0
\(556\) 26.1495 1.10899
\(557\) −14.0895 −0.596991 −0.298496 0.954411i \(-0.596485\pi\)
−0.298496 + 0.954411i \(0.596485\pi\)
\(558\) −0.642779 −0.0272110
\(559\) −57.3431 −2.42535
\(560\) 0 0
\(561\) 19.0647 0.804911
\(562\) −4.62339 −0.195026
\(563\) −39.1610 −1.65044 −0.825221 0.564811i \(-0.808949\pi\)
−0.825221 + 0.564811i \(0.808949\pi\)
\(564\) −23.6666 −0.996544
\(565\) 0 0
\(566\) 1.76284 0.0740977
\(567\) −3.91791 −0.164537
\(568\) −4.86345 −0.204066
\(569\) 16.7962 0.704134 0.352067 0.935975i \(-0.385479\pi\)
0.352067 + 0.935975i \(0.385479\pi\)
\(570\) 0 0
\(571\) −7.40957 −0.310081 −0.155040 0.987908i \(-0.549551\pi\)
−0.155040 + 0.987908i \(0.549551\pi\)
\(572\) −51.1252 −2.13765
\(573\) −7.48523 −0.312700
\(574\) −4.38213 −0.182906
\(575\) 0 0
\(576\) −7.42399 −0.309333
\(577\) 4.43326 0.184559 0.0922795 0.995733i \(-0.470585\pi\)
0.0922795 + 0.995733i \(0.470585\pi\)
\(578\) 0.552444 0.0229786
\(579\) −23.6594 −0.983250
\(580\) 0 0
\(581\) 15.1863 0.630034
\(582\) −1.02181 −0.0423555
\(583\) −52.4338 −2.17159
\(584\) −5.66764 −0.234529
\(585\) 0 0
\(586\) 4.99011 0.206139
\(587\) 33.3266 1.37554 0.687769 0.725930i \(-0.258590\pi\)
0.687769 + 0.725930i \(0.258590\pi\)
\(588\) 16.4976 0.680348
\(589\) −26.3800 −1.08697
\(590\) 0 0
\(591\) −0.502435 −0.0206674
\(592\) −27.6111 −1.13481
\(593\) 23.9282 0.982612 0.491306 0.870987i \(-0.336520\pi\)
0.491306 + 0.870987i \(0.336520\pi\)
\(594\) 0.654866 0.0268695
\(595\) 0 0
\(596\) 1.60766 0.0658522
\(597\) −12.0176 −0.491848
\(598\) −1.48582 −0.0607596
\(599\) 11.5448 0.471707 0.235854 0.971789i \(-0.424211\pi\)
0.235854 + 0.971789i \(0.424211\pi\)
\(600\) 0 0
\(601\) 16.2357 0.662268 0.331134 0.943584i \(-0.392569\pi\)
0.331134 + 0.943584i \(0.392569\pi\)
\(602\) −5.68572 −0.231733
\(603\) −5.70713 −0.232412
\(604\) 10.8719 0.442370
\(605\) 0 0
\(606\) −0.251825 −0.0102297
\(607\) 10.3573 0.420391 0.210196 0.977659i \(-0.432590\pi\)
0.210196 + 0.977659i \(0.432590\pi\)
\(608\) 11.7478 0.476435
\(609\) −13.1156 −0.531470
\(610\) 0 0
\(611\) 73.6988 2.98153
\(612\) −8.95607 −0.362028
\(613\) −5.74382 −0.231991 −0.115995 0.993250i \(-0.537006\pi\)
−0.115995 + 0.993250i \(0.537006\pi\)
\(614\) 0.479147 0.0193368
\(615\) 0 0
\(616\) −10.2006 −0.410995
\(617\) −11.0518 −0.444929 −0.222465 0.974941i \(-0.571410\pi\)
−0.222465 + 0.974941i \(0.571410\pi\)
\(618\) −0.385226 −0.0154960
\(619\) 11.5390 0.463792 0.231896 0.972741i \(-0.425507\pi\)
0.231896 + 0.972741i \(0.425507\pi\)
\(620\) 0 0
\(621\) −1.55096 −0.0622379
\(622\) 1.96488 0.0787844
\(623\) 26.1956 1.04951
\(624\) 23.7189 0.949517
\(625\) 0 0
\(626\) 3.78151 0.151139
\(627\) 26.8761 1.07333
\(628\) −1.18780 −0.0473985
\(629\) −32.4661 −1.29451
\(630\) 0 0
\(631\) 17.9927 0.716278 0.358139 0.933668i \(-0.383411\pi\)
0.358139 + 0.933668i \(0.383411\pi\)
\(632\) −2.04500 −0.0813459
\(633\) 22.5450 0.896084
\(634\) −1.48345 −0.0589154
\(635\) 0 0
\(636\) 24.6320 0.976723
\(637\) −51.3741 −2.03551
\(638\) 2.19223 0.0867912
\(639\) 7.85627 0.310789
\(640\) 0 0
\(641\) 12.1989 0.481826 0.240913 0.970547i \(-0.422553\pi\)
0.240913 + 0.970547i \(0.422553\pi\)
\(642\) 0.155707 0.00614526
\(643\) −32.4279 −1.27883 −0.639416 0.768861i \(-0.720824\pi\)
−0.639416 + 0.768861i \(0.720824\pi\)
\(644\) 12.0057 0.473091
\(645\) 0 0
\(646\) 4.51039 0.177459
\(647\) −24.5804 −0.966354 −0.483177 0.875523i \(-0.660517\pi\)
−0.483177 + 0.875523i \(0.660517\pi\)
\(648\) −0.619053 −0.0243187
\(649\) −25.3606 −0.995492
\(650\) 0 0
\(651\) −16.1736 −0.633895
\(652\) −42.8118 −1.67664
\(653\) 48.6767 1.90487 0.952433 0.304749i \(-0.0985727\pi\)
0.952433 + 0.304749i \(0.0985727\pi\)
\(654\) −2.52012 −0.0985444
\(655\) 0 0
\(656\) 27.6924 1.08121
\(657\) 9.15535 0.357184
\(658\) 7.30744 0.284874
\(659\) 41.9601 1.63453 0.817267 0.576259i \(-0.195488\pi\)
0.817267 + 0.576259i \(0.195488\pi\)
\(660\) 0 0
\(661\) −39.9051 −1.55213 −0.776064 0.630654i \(-0.782787\pi\)
−0.776064 + 0.630654i \(0.782787\pi\)
\(662\) −0.102056 −0.00396652
\(663\) 27.8896 1.08314
\(664\) 2.39953 0.0931197
\(665\) 0 0
\(666\) −1.11520 −0.0432132
\(667\) −5.19199 −0.201035
\(668\) 13.6190 0.526936
\(669\) −11.2208 −0.433821
\(670\) 0 0
\(671\) −38.4695 −1.48510
\(672\) 7.20258 0.277846
\(673\) −29.9856 −1.15586 −0.577930 0.816086i \(-0.696139\pi\)
−0.577930 + 0.816086i \(0.696139\pi\)
\(674\) 2.72506 0.104965
\(675\) 0 0
\(676\) −49.1059 −1.88869
\(677\) −27.3960 −1.05291 −0.526456 0.850202i \(-0.676480\pi\)
−0.526456 + 0.850202i \(0.676480\pi\)
\(678\) 2.57658 0.0989528
\(679\) −25.7109 −0.986695
\(680\) 0 0
\(681\) 3.79410 0.145390
\(682\) 2.70338 0.103518
\(683\) −3.71237 −0.142050 −0.0710249 0.997475i \(-0.522627\pi\)
−0.0710249 + 0.997475i \(0.522627\pi\)
\(684\) −12.6257 −0.482755
\(685\) 0 0
\(686\) −0.823563 −0.0314438
\(687\) 21.8196 0.832471
\(688\) 35.9303 1.36983
\(689\) −76.7051 −2.92223
\(690\) 0 0
\(691\) −29.2686 −1.11343 −0.556714 0.830704i \(-0.687938\pi\)
−0.556714 + 0.830704i \(0.687938\pi\)
\(692\) 26.6682 1.01377
\(693\) 16.4778 0.625939
\(694\) 1.55691 0.0590994
\(695\) 0 0
\(696\) −2.07234 −0.0785519
\(697\) 32.5617 1.23336
\(698\) 0.587237 0.0222273
\(699\) 2.52926 0.0956655
\(700\) 0 0
\(701\) 39.3801 1.48736 0.743682 0.668534i \(-0.233078\pi\)
0.743682 + 0.668534i \(0.233078\pi\)
\(702\) 0.957999 0.0361574
\(703\) −45.7686 −1.72619
\(704\) 31.2235 1.17678
\(705\) 0 0
\(706\) −2.68636 −0.101102
\(707\) −6.33644 −0.238306
\(708\) 11.9137 0.447746
\(709\) 24.6721 0.926582 0.463291 0.886206i \(-0.346669\pi\)
0.463291 + 0.886206i \(0.346669\pi\)
\(710\) 0 0
\(711\) 3.30344 0.123889
\(712\) 4.13907 0.155118
\(713\) −6.40257 −0.239778
\(714\) 2.76533 0.103490
\(715\) 0 0
\(716\) 45.5924 1.70387
\(717\) 0.0952535 0.00355731
\(718\) 3.17300 0.118415
\(719\) −23.8360 −0.888933 −0.444467 0.895795i \(-0.646607\pi\)
−0.444467 + 0.895795i \(0.646607\pi\)
\(720\) 0 0
\(721\) −9.69307 −0.360989
\(722\) 3.40002 0.126536
\(723\) 19.8084 0.736681
\(724\) −6.38602 −0.237335
\(725\) 0 0
\(726\) −1.04144 −0.0386513
\(727\) −47.3237 −1.75514 −0.877570 0.479449i \(-0.840837\pi\)
−0.877570 + 0.479449i \(0.840837\pi\)
\(728\) −14.9224 −0.553062
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.2482 1.56261
\(732\) 18.0719 0.667958
\(733\) −39.1381 −1.44560 −0.722799 0.691058i \(-0.757145\pi\)
−0.722799 + 0.691058i \(0.757145\pi\)
\(734\) −2.71183 −0.100095
\(735\) 0 0
\(736\) 2.85124 0.105098
\(737\) 24.0028 0.884156
\(738\) 1.11849 0.0411720
\(739\) 20.0503 0.737564 0.368782 0.929516i \(-0.379775\pi\)
0.368782 + 0.929516i \(0.379775\pi\)
\(740\) 0 0
\(741\) 39.3169 1.44434
\(742\) −7.60552 −0.279208
\(743\) −52.0682 −1.91020 −0.955099 0.296287i \(-0.904251\pi\)
−0.955099 + 0.296287i \(0.904251\pi\)
\(744\) −2.55553 −0.0936903
\(745\) 0 0
\(746\) 0.00847611 0.000310332 0
\(747\) −3.87613 −0.141820
\(748\) 37.6671 1.37725
\(749\) 3.91791 0.143157
\(750\) 0 0
\(751\) −18.5212 −0.675848 −0.337924 0.941173i \(-0.609725\pi\)
−0.337924 + 0.941173i \(0.609725\pi\)
\(752\) −46.1786 −1.68396
\(753\) 6.15527 0.224310
\(754\) 3.20700 0.116792
\(755\) 0 0
\(756\) −7.74083 −0.281531
\(757\) 12.1790 0.442655 0.221327 0.975200i \(-0.428961\pi\)
0.221327 + 0.975200i \(0.428961\pi\)
\(758\) 0.879550 0.0319467
\(759\) 6.52297 0.236769
\(760\) 0 0
\(761\) −38.4420 −1.39352 −0.696760 0.717304i \(-0.745376\pi\)
−0.696760 + 0.717304i \(0.745376\pi\)
\(762\) 3.31859 0.120220
\(763\) −63.4114 −2.29565
\(764\) −14.7890 −0.535047
\(765\) 0 0
\(766\) 5.18688 0.187410
\(767\) −37.0999 −1.33960
\(768\) −14.0955 −0.508627
\(769\) 16.9516 0.611290 0.305645 0.952145i \(-0.401128\pi\)
0.305645 + 0.952145i \(0.401128\pi\)
\(770\) 0 0
\(771\) 11.0517 0.398016
\(772\) −46.7451 −1.68239
\(773\) 43.7652 1.57412 0.787062 0.616874i \(-0.211601\pi\)
0.787062 + 0.616874i \(0.211601\pi\)
\(774\) 1.45121 0.0521628
\(775\) 0 0
\(776\) −4.06248 −0.145835
\(777\) −28.0608 −1.00668
\(778\) −1.29614 −0.0464688
\(779\) 45.9033 1.64466
\(780\) 0 0
\(781\) −33.0416 −1.18232
\(782\) 1.09469 0.0391462
\(783\) 3.34760 0.119633
\(784\) 32.1903 1.14965
\(785\) 0 0
\(786\) −1.17579 −0.0419389
\(787\) 37.0068 1.31915 0.659574 0.751639i \(-0.270737\pi\)
0.659574 + 0.751639i \(0.270737\pi\)
\(788\) −0.992688 −0.0353630
\(789\) 17.1372 0.610099
\(790\) 0 0
\(791\) 64.8320 2.30516
\(792\) 2.60359 0.0925145
\(793\) −56.2768 −1.99845
\(794\) −1.17744 −0.0417858
\(795\) 0 0
\(796\) −23.7439 −0.841579
\(797\) −1.30055 −0.0460680 −0.0230340 0.999735i \(-0.507333\pi\)
−0.0230340 + 0.999735i \(0.507333\pi\)
\(798\) 3.89838 0.138001
\(799\) −54.2985 −1.92094
\(800\) 0 0
\(801\) −6.68613 −0.236243
\(802\) −2.06312 −0.0728515
\(803\) −38.5052 −1.35882
\(804\) −11.2759 −0.397670
\(805\) 0 0
\(806\) 3.95475 0.139300
\(807\) 2.71376 0.0955287
\(808\) −1.00119 −0.0352219
\(809\) 18.8423 0.662459 0.331230 0.943550i \(-0.392537\pi\)
0.331230 + 0.943550i \(0.392537\pi\)
\(810\) 0 0
\(811\) −10.6066 −0.372448 −0.186224 0.982507i \(-0.559625\pi\)
−0.186224 + 0.982507i \(0.559625\pi\)
\(812\) −25.9132 −0.909375
\(813\) 8.31668 0.291679
\(814\) 4.69027 0.164394
\(815\) 0 0
\(816\) −17.4752 −0.611755
\(817\) 59.5587 2.08370
\(818\) −4.15606 −0.145313
\(819\) 24.1052 0.842305
\(820\) 0 0
\(821\) 13.1312 0.458281 0.229141 0.973393i \(-0.426408\pi\)
0.229141 + 0.973393i \(0.426408\pi\)
\(822\) −2.05547 −0.0716929
\(823\) 27.6426 0.963561 0.481781 0.876292i \(-0.339990\pi\)
0.481781 + 0.876292i \(0.339990\pi\)
\(824\) −1.53156 −0.0533545
\(825\) 0 0
\(826\) −3.67856 −0.127993
\(827\) −33.2253 −1.15536 −0.577679 0.816264i \(-0.696041\pi\)
−0.577679 + 0.816264i \(0.696041\pi\)
\(828\) −3.06432 −0.106492
\(829\) 23.1934 0.805540 0.402770 0.915301i \(-0.368047\pi\)
0.402770 + 0.915301i \(0.368047\pi\)
\(830\) 0 0
\(831\) 0.169617 0.00588395
\(832\) 45.6767 1.58356
\(833\) 37.8505 1.31144
\(834\) 2.06081 0.0713601
\(835\) 0 0
\(836\) 53.1006 1.83652
\(837\) 4.12813 0.142689
\(838\) −4.08586 −0.141144
\(839\) 22.1976 0.766347 0.383174 0.923676i \(-0.374831\pi\)
0.383174 + 0.923676i \(0.374831\pi\)
\(840\) 0 0
\(841\) −17.7936 −0.613572
\(842\) 3.59489 0.123888
\(843\) 29.6929 1.02268
\(844\) 44.5434 1.53325
\(845\) 0 0
\(846\) −1.86514 −0.0641248
\(847\) −26.2047 −0.900404
\(848\) 48.0623 1.65047
\(849\) −11.3215 −0.388554
\(850\) 0 0
\(851\) −11.1083 −0.380786
\(852\) 15.5221 0.531778
\(853\) 33.7654 1.15611 0.578053 0.815999i \(-0.303813\pi\)
0.578053 + 0.815999i \(0.303813\pi\)
\(854\) −5.58000 −0.190944
\(855\) 0 0
\(856\) 0.619053 0.0211588
\(857\) 8.43718 0.288209 0.144104 0.989562i \(-0.453970\pi\)
0.144104 + 0.989562i \(0.453970\pi\)
\(858\) −4.02912 −0.137552
\(859\) 10.2973 0.351338 0.175669 0.984449i \(-0.443791\pi\)
0.175669 + 0.984449i \(0.443791\pi\)
\(860\) 0 0
\(861\) 28.1434 0.959125
\(862\) −3.49600 −0.119074
\(863\) 42.9481 1.46197 0.730985 0.682394i \(-0.239061\pi\)
0.730985 + 0.682394i \(0.239061\pi\)
\(864\) −1.83837 −0.0625428
\(865\) 0 0
\(866\) 1.61853 0.0549999
\(867\) −3.54797 −0.120495
\(868\) −31.9552 −1.08463
\(869\) −13.8935 −0.471304
\(870\) 0 0
\(871\) 35.1136 1.18978
\(872\) −10.0194 −0.339299
\(873\) 6.56241 0.222104
\(874\) 1.54323 0.0522005
\(875\) 0 0
\(876\) 18.0887 0.611162
\(877\) −39.5482 −1.33545 −0.667724 0.744409i \(-0.732731\pi\)
−0.667724 + 0.744409i \(0.732731\pi\)
\(878\) −4.86832 −0.164298
\(879\) −32.0481 −1.08095
\(880\) 0 0
\(881\) 0.331700 0.0111752 0.00558762 0.999984i \(-0.498221\pi\)
0.00558762 + 0.999984i \(0.498221\pi\)
\(882\) 1.30015 0.0437784
\(883\) 3.97236 0.133680 0.0668402 0.997764i \(-0.478708\pi\)
0.0668402 + 0.997764i \(0.478708\pi\)
\(884\) 55.1030 1.85331
\(885\) 0 0
\(886\) 0.523297 0.0175805
\(887\) −41.9397 −1.40820 −0.704099 0.710102i \(-0.748649\pi\)
−0.704099 + 0.710102i \(0.748649\pi\)
\(888\) −4.43377 −0.148788
\(889\) 83.5027 2.80059
\(890\) 0 0
\(891\) −4.20576 −0.140898
\(892\) −22.1695 −0.742291
\(893\) −76.5464 −2.56153
\(894\) 0.126698 0.00423740
\(895\) 0 0
\(896\) 18.9341 0.632545
\(897\) 9.54240 0.318612
\(898\) 0.843613 0.0281517
\(899\) 13.8193 0.460901
\(900\) 0 0
\(901\) 56.5134 1.88273
\(902\) −4.70409 −0.156629
\(903\) 36.5155 1.21516
\(904\) 10.2438 0.340705
\(905\) 0 0
\(906\) 0.856799 0.0284653
\(907\) −19.3926 −0.643922 −0.321961 0.946753i \(-0.604342\pi\)
−0.321961 + 0.946753i \(0.604342\pi\)
\(908\) 7.49622 0.248771
\(909\) 1.61730 0.0536425
\(910\) 0 0
\(911\) −14.3443 −0.475248 −0.237624 0.971357i \(-0.576369\pi\)
−0.237624 + 0.971357i \(0.576369\pi\)
\(912\) −24.6354 −0.815760
\(913\) 16.3021 0.539520
\(914\) −2.24468 −0.0742473
\(915\) 0 0
\(916\) 43.1103 1.42440
\(917\) −29.5852 −0.976989
\(918\) −0.705817 −0.0232954
\(919\) −44.0493 −1.45305 −0.726526 0.687138i \(-0.758867\pi\)
−0.726526 + 0.687138i \(0.758867\pi\)
\(920\) 0 0
\(921\) −3.07724 −0.101398
\(922\) 4.40926 0.145211
\(923\) −48.3364 −1.59101
\(924\) 32.5561 1.07102
\(925\) 0 0
\(926\) −2.76485 −0.0908587
\(927\) 2.47404 0.0812582
\(928\) −6.15414 −0.202020
\(929\) −11.7163 −0.384400 −0.192200 0.981356i \(-0.561562\pi\)
−0.192200 + 0.981356i \(0.561562\pi\)
\(930\) 0 0
\(931\) 53.3591 1.74877
\(932\) 4.99721 0.163689
\(933\) −12.6191 −0.413130
\(934\) 5.05755 0.165488
\(935\) 0 0
\(936\) 3.80877 0.124494
\(937\) 14.2892 0.466808 0.233404 0.972380i \(-0.425013\pi\)
0.233404 + 0.972380i \(0.425013\pi\)
\(938\) 3.48161 0.113679
\(939\) −24.2861 −0.792545
\(940\) 0 0
\(941\) −11.1338 −0.362950 −0.181475 0.983396i \(-0.558087\pi\)
−0.181475 + 0.983396i \(0.558087\pi\)
\(942\) −0.0936093 −0.00304995
\(943\) 11.1410 0.362800
\(944\) 23.2463 0.756601
\(945\) 0 0
\(946\) −6.10346 −0.198441
\(947\) 41.4663 1.34747 0.673737 0.738971i \(-0.264688\pi\)
0.673737 + 0.738971i \(0.264688\pi\)
\(948\) 6.52679 0.211980
\(949\) −56.3290 −1.82852
\(950\) 0 0
\(951\) 9.52720 0.308941
\(952\) 10.9943 0.356326
\(953\) 33.3251 1.07951 0.539753 0.841824i \(-0.318518\pi\)
0.539753 + 0.841824i \(0.318518\pi\)
\(954\) 1.94122 0.0628493
\(955\) 0 0
\(956\) 0.188198 0.00608674
\(957\) −14.0792 −0.455116
\(958\) 5.06331 0.163588
\(959\) −51.7200 −1.67013
\(960\) 0 0
\(961\) −13.9585 −0.450275
\(962\) 6.86137 0.221219
\(963\) −1.00000 −0.0322245
\(964\) 39.1365 1.26050
\(965\) 0 0
\(966\) 0.946156 0.0304421
\(967\) −50.1545 −1.61286 −0.806431 0.591329i \(-0.798604\pi\)
−0.806431 + 0.591329i \(0.798604\pi\)
\(968\) −4.14050 −0.133081
\(969\) −28.9672 −0.930560
\(970\) 0 0
\(971\) 1.04282 0.0334657 0.0167329 0.999860i \(-0.494674\pi\)
0.0167329 + 0.999860i \(0.494674\pi\)
\(972\) 1.97576 0.0633724
\(973\) 51.8543 1.66237
\(974\) 5.42393 0.173794
\(975\) 0 0
\(976\) 35.2622 1.12872
\(977\) 59.7517 1.91163 0.955813 0.293976i \(-0.0949785\pi\)
0.955813 + 0.293976i \(0.0949785\pi\)
\(978\) −3.37395 −0.107887
\(979\) 28.1203 0.898728
\(980\) 0 0
\(981\) 16.1850 0.516748
\(982\) −1.46809 −0.0468488
\(983\) −19.2009 −0.612413 −0.306207 0.951965i \(-0.599060\pi\)
−0.306207 + 0.951965i \(0.599060\pi\)
\(984\) 4.44683 0.141760
\(985\) 0 0
\(986\) −2.36279 −0.0752467
\(987\) −46.9307 −1.49382
\(988\) 77.6805 2.47135
\(989\) 14.4552 0.459648
\(990\) 0 0
\(991\) −26.6856 −0.847695 −0.423848 0.905734i \(-0.639321\pi\)
−0.423848 + 0.905734i \(0.639321\pi\)
\(992\) −7.58905 −0.240953
\(993\) 0.655437 0.0207997
\(994\) −4.79268 −0.152015
\(995\) 0 0
\(996\) −7.65828 −0.242662
\(997\) −10.7703 −0.341099 −0.170550 0.985349i \(-0.554554\pi\)
−0.170550 + 0.985349i \(0.554554\pi\)
\(998\) 1.73283 0.0548517
\(999\) 7.16219 0.226602
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 8025.2.a.bf.1.8 12
5.4 even 2 1605.2.a.n.1.5 12
15.14 odd 2 4815.2.a.u.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1605.2.a.n.1.5 12 5.4 even 2
4815.2.a.u.1.8 12 15.14 odd 2
8025.2.a.bf.1.8 12 1.1 even 1 trivial