Properties

Label 3856.2.a.j.1.4
Level $3856$
Weight $2$
Character 3856.1
Self dual yes
Analytic conductor $30.790$
Analytic rank $0$
Dimension $7$
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] = [3856,2,Mod(1,3856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3856, 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("3856.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.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: \(30.7903150194\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.27758\) of defining polynomial
Character \(\chi\) \(=\) 3856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.494846 q^{3} -1.23324 q^{5} -1.36627 q^{7} -2.75513 q^{9} +O(q^{10})\) \(q+0.494846 q^{3} -1.23324 q^{5} -1.36627 q^{7} -2.75513 q^{9} +4.69806 q^{11} -0.0431968 q^{13} -0.610264 q^{15} -7.31430 q^{17} +0.697489 q^{19} -0.676090 q^{21} -1.41195 q^{23} -3.47911 q^{25} -2.84790 q^{27} +8.30334 q^{29} -3.39655 q^{31} +2.32481 q^{33} +1.68494 q^{35} +7.15948 q^{37} -0.0213757 q^{39} +5.45541 q^{41} +11.7568 q^{43} +3.39774 q^{45} +5.24836 q^{47} -5.13332 q^{49} -3.61945 q^{51} -8.57769 q^{53} -5.79385 q^{55} +0.345149 q^{57} +12.9925 q^{59} +10.1636 q^{61} +3.76423 q^{63} +0.0532721 q^{65} -10.1259 q^{67} -0.698697 q^{69} -1.86703 q^{71} +6.47826 q^{73} -1.72162 q^{75} -6.41880 q^{77} +12.9436 q^{79} +6.85611 q^{81} +2.32915 q^{83} +9.02030 q^{85} +4.10887 q^{87} -14.5180 q^{89} +0.0590183 q^{91} -1.68077 q^{93} -0.860173 q^{95} -2.23725 q^{97} -12.9438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9} + 18 q^{11} - q^{13} + 11 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 22 q^{23} + 5 q^{25} - 3 q^{27} - 16 q^{29} + 18 q^{31} + 4 q^{33} - 7 q^{35} + 8 q^{37} + 9 q^{39} - 15 q^{41} - 14 q^{43} + 3 q^{45} + 10 q^{47} + 6 q^{49} - 13 q^{51} + 15 q^{53} - 29 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{61} + 16 q^{63} - 7 q^{65} - 18 q^{67} + 26 q^{69} + 50 q^{71} - 16 q^{75} + 17 q^{77} + 15 q^{79} - 9 q^{81} + 24 q^{83} - 2 q^{85} - 12 q^{87} - 13 q^{89} + 12 q^{91} + 14 q^{93} + 41 q^{95} + q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.494846 0.285699 0.142850 0.989744i \(-0.454373\pi\)
0.142850 + 0.989744i \(0.454373\pi\)
\(4\) 0 0
\(5\) −1.23324 −0.551523 −0.275761 0.961226i \(-0.588930\pi\)
−0.275761 + 0.961226i \(0.588930\pi\)
\(6\) 0 0
\(7\) −1.36627 −0.516400 −0.258200 0.966092i \(-0.583129\pi\)
−0.258200 + 0.966092i \(0.583129\pi\)
\(8\) 0 0
\(9\) −2.75513 −0.918376
\(10\) 0 0
\(11\) 4.69806 1.41652 0.708259 0.705952i \(-0.249481\pi\)
0.708259 + 0.705952i \(0.249481\pi\)
\(12\) 0 0
\(13\) −0.0431968 −0.0119806 −0.00599032 0.999982i \(-0.501907\pi\)
−0.00599032 + 0.999982i \(0.501907\pi\)
\(14\) 0 0
\(15\) −0.610264 −0.157570
\(16\) 0 0
\(17\) −7.31430 −1.77398 −0.886989 0.461790i \(-0.847207\pi\)
−0.886989 + 0.461790i \(0.847207\pi\)
\(18\) 0 0
\(19\) 0.697489 0.160015 0.0800075 0.996794i \(-0.474506\pi\)
0.0800075 + 0.996794i \(0.474506\pi\)
\(20\) 0 0
\(21\) −0.676090 −0.147535
\(22\) 0 0
\(23\) −1.41195 −0.294412 −0.147206 0.989106i \(-0.547028\pi\)
−0.147206 + 0.989106i \(0.547028\pi\)
\(24\) 0 0
\(25\) −3.47911 −0.695823
\(26\) 0 0
\(27\) −2.84790 −0.548078
\(28\) 0 0
\(29\) 8.30334 1.54189 0.770946 0.636901i \(-0.219784\pi\)
0.770946 + 0.636901i \(0.219784\pi\)
\(30\) 0 0
\(31\) −3.39655 −0.610038 −0.305019 0.952346i \(-0.598663\pi\)
−0.305019 + 0.952346i \(0.598663\pi\)
\(32\) 0 0
\(33\) 2.32481 0.404698
\(34\) 0 0
\(35\) 1.68494 0.284806
\(36\) 0 0
\(37\) 7.15948 1.17701 0.588506 0.808493i \(-0.299716\pi\)
0.588506 + 0.808493i \(0.299716\pi\)
\(38\) 0 0
\(39\) −0.0213757 −0.00342286
\(40\) 0 0
\(41\) 5.45541 0.851992 0.425996 0.904725i \(-0.359924\pi\)
0.425996 + 0.904725i \(0.359924\pi\)
\(42\) 0 0
\(43\) 11.7568 1.79290 0.896448 0.443149i \(-0.146139\pi\)
0.896448 + 0.443149i \(0.146139\pi\)
\(44\) 0 0
\(45\) 3.39774 0.506505
\(46\) 0 0
\(47\) 5.24836 0.765551 0.382776 0.923841i \(-0.374968\pi\)
0.382776 + 0.923841i \(0.374968\pi\)
\(48\) 0 0
\(49\) −5.13332 −0.733331
\(50\) 0 0
\(51\) −3.61945 −0.506824
\(52\) 0 0
\(53\) −8.57769 −1.17824 −0.589118 0.808047i \(-0.700525\pi\)
−0.589118 + 0.808047i \(0.700525\pi\)
\(54\) 0 0
\(55\) −5.79385 −0.781242
\(56\) 0 0
\(57\) 0.345149 0.0457162
\(58\) 0 0
\(59\) 12.9925 1.69148 0.845738 0.533598i \(-0.179160\pi\)
0.845738 + 0.533598i \(0.179160\pi\)
\(60\) 0 0
\(61\) 10.1636 1.30132 0.650658 0.759371i \(-0.274493\pi\)
0.650658 + 0.759371i \(0.274493\pi\)
\(62\) 0 0
\(63\) 3.76423 0.474249
\(64\) 0 0
\(65\) 0.0532721 0.00660759
\(66\) 0 0
\(67\) −10.1259 −1.23707 −0.618536 0.785757i \(-0.712274\pi\)
−0.618536 + 0.785757i \(0.712274\pi\)
\(68\) 0 0
\(69\) −0.698697 −0.0841132
\(70\) 0 0
\(71\) −1.86703 −0.221576 −0.110788 0.993844i \(-0.535337\pi\)
−0.110788 + 0.993844i \(0.535337\pi\)
\(72\) 0 0
\(73\) 6.47826 0.758223 0.379111 0.925351i \(-0.376230\pi\)
0.379111 + 0.925351i \(0.376230\pi\)
\(74\) 0 0
\(75\) −1.72162 −0.198796
\(76\) 0 0
\(77\) −6.41880 −0.731490
\(78\) 0 0
\(79\) 12.9436 1.45627 0.728136 0.685432i \(-0.240387\pi\)
0.728136 + 0.685432i \(0.240387\pi\)
\(80\) 0 0
\(81\) 6.85611 0.761790
\(82\) 0 0
\(83\) 2.32915 0.255657 0.127829 0.991796i \(-0.459199\pi\)
0.127829 + 0.991796i \(0.459199\pi\)
\(84\) 0 0
\(85\) 9.02030 0.978389
\(86\) 0 0
\(87\) 4.10887 0.440517
\(88\) 0 0
\(89\) −14.5180 −1.53890 −0.769450 0.638707i \(-0.779470\pi\)
−0.769450 + 0.638707i \(0.779470\pi\)
\(90\) 0 0
\(91\) 0.0590183 0.00618680
\(92\) 0 0
\(93\) −1.68077 −0.174287
\(94\) 0 0
\(95\) −0.860173 −0.0882519
\(96\) 0 0
\(97\) −2.23725 −0.227158 −0.113579 0.993529i \(-0.536231\pi\)
−0.113579 + 0.993529i \(0.536231\pi\)
\(98\) 0 0
\(99\) −12.9438 −1.30090
\(100\) 0 0
\(101\) −4.01607 −0.399614 −0.199807 0.979835i \(-0.564031\pi\)
−0.199807 + 0.979835i \(0.564031\pi\)
\(102\) 0 0
\(103\) 5.25187 0.517482 0.258741 0.965947i \(-0.416692\pi\)
0.258741 + 0.965947i \(0.416692\pi\)
\(104\) 0 0
\(105\) 0.833783 0.0813689
\(106\) 0 0
\(107\) −5.60385 −0.541745 −0.270873 0.962615i \(-0.587312\pi\)
−0.270873 + 0.962615i \(0.587312\pi\)
\(108\) 0 0
\(109\) −4.77557 −0.457416 −0.228708 0.973495i \(-0.573450\pi\)
−0.228708 + 0.973495i \(0.573450\pi\)
\(110\) 0 0
\(111\) 3.54284 0.336271
\(112\) 0 0
\(113\) −12.6186 −1.18706 −0.593531 0.804811i \(-0.702267\pi\)
−0.593531 + 0.804811i \(0.702267\pi\)
\(114\) 0 0
\(115\) 1.74127 0.162375
\(116\) 0 0
\(117\) 0.119013 0.0110027
\(118\) 0 0
\(119\) 9.99327 0.916082
\(120\) 0 0
\(121\) 11.0718 1.00653
\(122\) 0 0
\(123\) 2.69959 0.243413
\(124\) 0 0
\(125\) 10.4568 0.935285
\(126\) 0 0
\(127\) −0.427651 −0.0379479 −0.0189740 0.999820i \(-0.506040\pi\)
−0.0189740 + 0.999820i \(0.506040\pi\)
\(128\) 0 0
\(129\) 5.81780 0.512229
\(130\) 0 0
\(131\) 13.3633 1.16756 0.583780 0.811912i \(-0.301573\pi\)
0.583780 + 0.811912i \(0.301573\pi\)
\(132\) 0 0
\(133\) −0.952955 −0.0826317
\(134\) 0 0
\(135\) 3.51215 0.302278
\(136\) 0 0
\(137\) 11.2702 0.962874 0.481437 0.876481i \(-0.340115\pi\)
0.481437 + 0.876481i \(0.340115\pi\)
\(138\) 0 0
\(139\) 0.927184 0.0786427 0.0393214 0.999227i \(-0.487480\pi\)
0.0393214 + 0.999227i \(0.487480\pi\)
\(140\) 0 0
\(141\) 2.59713 0.218717
\(142\) 0 0
\(143\) −0.202941 −0.0169708
\(144\) 0 0
\(145\) −10.2400 −0.850388
\(146\) 0 0
\(147\) −2.54020 −0.209512
\(148\) 0 0
\(149\) 12.8757 1.05481 0.527407 0.849612i \(-0.323164\pi\)
0.527407 + 0.849612i \(0.323164\pi\)
\(150\) 0 0
\(151\) 21.2436 1.72878 0.864390 0.502823i \(-0.167705\pi\)
0.864390 + 0.502823i \(0.167705\pi\)
\(152\) 0 0
\(153\) 20.1518 1.62918
\(154\) 0 0
\(155\) 4.18876 0.336450
\(156\) 0 0
\(157\) 17.1798 1.37110 0.685550 0.728026i \(-0.259562\pi\)
0.685550 + 0.728026i \(0.259562\pi\)
\(158\) 0 0
\(159\) −4.24463 −0.336621
\(160\) 0 0
\(161\) 1.92910 0.152034
\(162\) 0 0
\(163\) −11.1081 −0.870055 −0.435028 0.900417i \(-0.643261\pi\)
−0.435028 + 0.900417i \(0.643261\pi\)
\(164\) 0 0
\(165\) −2.86706 −0.223200
\(166\) 0 0
\(167\) 22.3791 1.73174 0.865872 0.500266i \(-0.166764\pi\)
0.865872 + 0.500266i \(0.166764\pi\)
\(168\) 0 0
\(169\) −12.9981 −0.999856
\(170\) 0 0
\(171\) −1.92167 −0.146954
\(172\) 0 0
\(173\) 4.16100 0.316355 0.158178 0.987411i \(-0.449438\pi\)
0.158178 + 0.987411i \(0.449438\pi\)
\(174\) 0 0
\(175\) 4.75339 0.359323
\(176\) 0 0
\(177\) 6.42927 0.483254
\(178\) 0 0
\(179\) 5.79009 0.432772 0.216386 0.976308i \(-0.430573\pi\)
0.216386 + 0.976308i \(0.430573\pi\)
\(180\) 0 0
\(181\) 17.4917 1.30015 0.650076 0.759869i \(-0.274737\pi\)
0.650076 + 0.759869i \(0.274737\pi\)
\(182\) 0 0
\(183\) 5.02941 0.371785
\(184\) 0 0
\(185\) −8.82937 −0.649148
\(186\) 0 0
\(187\) −34.3630 −2.51287
\(188\) 0 0
\(189\) 3.89099 0.283028
\(190\) 0 0
\(191\) 21.6074 1.56346 0.781728 0.623620i \(-0.214339\pi\)
0.781728 + 0.623620i \(0.214339\pi\)
\(192\) 0 0
\(193\) 2.30886 0.166195 0.0830977 0.996541i \(-0.473519\pi\)
0.0830977 + 0.996541i \(0.473519\pi\)
\(194\) 0 0
\(195\) 0.0263615 0.00188778
\(196\) 0 0
\(197\) 1.91876 0.136706 0.0683531 0.997661i \(-0.478226\pi\)
0.0683531 + 0.997661i \(0.478226\pi\)
\(198\) 0 0
\(199\) −21.2430 −1.50588 −0.752938 0.658091i \(-0.771364\pi\)
−0.752938 + 0.658091i \(0.771364\pi\)
\(200\) 0 0
\(201\) −5.01074 −0.353430
\(202\) 0 0
\(203\) −11.3446 −0.796232
\(204\) 0 0
\(205\) −6.72784 −0.469893
\(206\) 0 0
\(207\) 3.89010 0.270381
\(208\) 0 0
\(209\) 3.27685 0.226664
\(210\) 0 0
\(211\) 19.0961 1.31463 0.657314 0.753616i \(-0.271692\pi\)
0.657314 + 0.753616i \(0.271692\pi\)
\(212\) 0 0
\(213\) −0.923892 −0.0633040
\(214\) 0 0
\(215\) −14.4990 −0.988823
\(216\) 0 0
\(217\) 4.64058 0.315023
\(218\) 0 0
\(219\) 3.20574 0.216624
\(220\) 0 0
\(221\) 0.315954 0.0212534
\(222\) 0 0
\(223\) −8.41781 −0.563698 −0.281849 0.959459i \(-0.590948\pi\)
−0.281849 + 0.959459i \(0.590948\pi\)
\(224\) 0 0
\(225\) 9.58540 0.639027
\(226\) 0 0
\(227\) −1.82854 −0.121364 −0.0606821 0.998157i \(-0.519328\pi\)
−0.0606821 + 0.998157i \(0.519328\pi\)
\(228\) 0 0
\(229\) 2.52038 0.166551 0.0832756 0.996527i \(-0.473462\pi\)
0.0832756 + 0.996527i \(0.473462\pi\)
\(230\) 0 0
\(231\) −3.17631 −0.208986
\(232\) 0 0
\(233\) 16.3812 1.07317 0.536585 0.843847i \(-0.319714\pi\)
0.536585 + 0.843847i \(0.319714\pi\)
\(234\) 0 0
\(235\) −6.47249 −0.422219
\(236\) 0 0
\(237\) 6.40510 0.416056
\(238\) 0 0
\(239\) 10.7430 0.694909 0.347455 0.937697i \(-0.387046\pi\)
0.347455 + 0.937697i \(0.387046\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 0 0
\(243\) 11.9364 0.765721
\(244\) 0 0
\(245\) 6.33063 0.404449
\(246\) 0 0
\(247\) −0.0301293 −0.00191708
\(248\) 0 0
\(249\) 1.15257 0.0730411
\(250\) 0 0
\(251\) −23.8819 −1.50741 −0.753706 0.657212i \(-0.771736\pi\)
−0.753706 + 0.657212i \(0.771736\pi\)
\(252\) 0 0
\(253\) −6.63342 −0.417040
\(254\) 0 0
\(255\) 4.46366 0.279525
\(256\) 0 0
\(257\) 5.25407 0.327740 0.163870 0.986482i \(-0.447602\pi\)
0.163870 + 0.986482i \(0.447602\pi\)
\(258\) 0 0
\(259\) −9.78175 −0.607808
\(260\) 0 0
\(261\) −22.8768 −1.41604
\(262\) 0 0
\(263\) 1.10964 0.0684235 0.0342118 0.999415i \(-0.489108\pi\)
0.0342118 + 0.999415i \(0.489108\pi\)
\(264\) 0 0
\(265\) 10.5784 0.649824
\(266\) 0 0
\(267\) −7.18414 −0.439662
\(268\) 0 0
\(269\) −2.84634 −0.173544 −0.0867722 0.996228i \(-0.527655\pi\)
−0.0867722 + 0.996228i \(0.527655\pi\)
\(270\) 0 0
\(271\) −13.1029 −0.795946 −0.397973 0.917397i \(-0.630286\pi\)
−0.397973 + 0.917397i \(0.630286\pi\)
\(272\) 0 0
\(273\) 0.0292049 0.00176756
\(274\) 0 0
\(275\) −16.3451 −0.985646
\(276\) 0 0
\(277\) −23.4634 −1.40978 −0.704888 0.709318i \(-0.749003\pi\)
−0.704888 + 0.709318i \(0.749003\pi\)
\(278\) 0 0
\(279\) 9.35792 0.560244
\(280\) 0 0
\(281\) 8.91165 0.531625 0.265812 0.964025i \(-0.414360\pi\)
0.265812 + 0.964025i \(0.414360\pi\)
\(282\) 0 0
\(283\) 15.8040 0.939453 0.469726 0.882812i \(-0.344353\pi\)
0.469726 + 0.882812i \(0.344353\pi\)
\(284\) 0 0
\(285\) −0.425653 −0.0252135
\(286\) 0 0
\(287\) −7.45354 −0.439968
\(288\) 0 0
\(289\) 36.4990 2.14700
\(290\) 0 0
\(291\) −1.10709 −0.0648988
\(292\) 0 0
\(293\) 32.9425 1.92452 0.962261 0.272127i \(-0.0877272\pi\)
0.962261 + 0.272127i \(0.0877272\pi\)
\(294\) 0 0
\(295\) −16.0229 −0.932888
\(296\) 0 0
\(297\) −13.3796 −0.776363
\(298\) 0 0
\(299\) 0.0609917 0.00352724
\(300\) 0 0
\(301\) −16.0629 −0.925851
\(302\) 0 0
\(303\) −1.98733 −0.114169
\(304\) 0 0
\(305\) −12.5342 −0.717705
\(306\) 0 0
\(307\) −20.2343 −1.15483 −0.577416 0.816450i \(-0.695939\pi\)
−0.577416 + 0.816450i \(0.695939\pi\)
\(308\) 0 0
\(309\) 2.59886 0.147844
\(310\) 0 0
\(311\) −2.27289 −0.128884 −0.0644420 0.997921i \(-0.520527\pi\)
−0.0644420 + 0.997921i \(0.520527\pi\)
\(312\) 0 0
\(313\) 13.1142 0.741257 0.370629 0.928781i \(-0.379142\pi\)
0.370629 + 0.928781i \(0.379142\pi\)
\(314\) 0 0
\(315\) −4.64221 −0.261559
\(316\) 0 0
\(317\) −7.86525 −0.441757 −0.220878 0.975301i \(-0.570892\pi\)
−0.220878 + 0.975301i \(0.570892\pi\)
\(318\) 0 0
\(319\) 39.0096 2.18412
\(320\) 0 0
\(321\) −2.77304 −0.154776
\(322\) 0 0
\(323\) −5.10164 −0.283863
\(324\) 0 0
\(325\) 0.150287 0.00833640
\(326\) 0 0
\(327\) −2.36317 −0.130683
\(328\) 0 0
\(329\) −7.17065 −0.395331
\(330\) 0 0
\(331\) 19.3752 1.06496 0.532480 0.846443i \(-0.321260\pi\)
0.532480 + 0.846443i \(0.321260\pi\)
\(332\) 0 0
\(333\) −19.7253 −1.08094
\(334\) 0 0
\(335\) 12.4876 0.682273
\(336\) 0 0
\(337\) −26.2459 −1.42970 −0.714852 0.699276i \(-0.753506\pi\)
−0.714852 + 0.699276i \(0.753506\pi\)
\(338\) 0 0
\(339\) −6.24428 −0.339143
\(340\) 0 0
\(341\) −15.9572 −0.864130
\(342\) 0 0
\(343\) 16.5773 0.895092
\(344\) 0 0
\(345\) 0.861662 0.0463903
\(346\) 0 0
\(347\) 3.65189 0.196044 0.0980219 0.995184i \(-0.468748\pi\)
0.0980219 + 0.995184i \(0.468748\pi\)
\(348\) 0 0
\(349\) −0.818652 −0.0438214 −0.0219107 0.999760i \(-0.506975\pi\)
−0.0219107 + 0.999760i \(0.506975\pi\)
\(350\) 0 0
\(351\) 0.123020 0.00656633
\(352\) 0 0
\(353\) −28.5367 −1.51886 −0.759428 0.650591i \(-0.774521\pi\)
−0.759428 + 0.650591i \(0.774521\pi\)
\(354\) 0 0
\(355\) 2.30250 0.122204
\(356\) 0 0
\(357\) 4.94513 0.261724
\(358\) 0 0
\(359\) 15.8623 0.837180 0.418590 0.908175i \(-0.362524\pi\)
0.418590 + 0.908175i \(0.362524\pi\)
\(360\) 0 0
\(361\) −18.5135 −0.974395
\(362\) 0 0
\(363\) 5.47882 0.287563
\(364\) 0 0
\(365\) −7.98926 −0.418177
\(366\) 0 0
\(367\) −13.9937 −0.730463 −0.365232 0.930917i \(-0.619010\pi\)
−0.365232 + 0.930917i \(0.619010\pi\)
\(368\) 0 0
\(369\) −15.0304 −0.782449
\(370\) 0 0
\(371\) 11.7194 0.608441
\(372\) 0 0
\(373\) 12.2336 0.633434 0.316717 0.948520i \(-0.397419\pi\)
0.316717 + 0.948520i \(0.397419\pi\)
\(374\) 0 0
\(375\) 5.17450 0.267210
\(376\) 0 0
\(377\) −0.358678 −0.0184728
\(378\) 0 0
\(379\) −0.284829 −0.0146307 −0.00731535 0.999973i \(-0.502329\pi\)
−0.00731535 + 0.999973i \(0.502329\pi\)
\(380\) 0 0
\(381\) −0.211621 −0.0108417
\(382\) 0 0
\(383\) −29.8268 −1.52408 −0.762038 0.647532i \(-0.775801\pi\)
−0.762038 + 0.647532i \(0.775801\pi\)
\(384\) 0 0
\(385\) 7.91593 0.403433
\(386\) 0 0
\(387\) −32.3915 −1.64655
\(388\) 0 0
\(389\) −8.88544 −0.450510 −0.225255 0.974300i \(-0.572321\pi\)
−0.225255 + 0.974300i \(0.572321\pi\)
\(390\) 0 0
\(391\) 10.3274 0.522280
\(392\) 0 0
\(393\) 6.61279 0.333571
\(394\) 0 0
\(395\) −15.9626 −0.803167
\(396\) 0 0
\(397\) −29.5831 −1.48473 −0.742367 0.669993i \(-0.766297\pi\)
−0.742367 + 0.669993i \(0.766297\pi\)
\(398\) 0 0
\(399\) −0.471566 −0.0236078
\(400\) 0 0
\(401\) 6.30306 0.314760 0.157380 0.987538i \(-0.449695\pi\)
0.157380 + 0.987538i \(0.449695\pi\)
\(402\) 0 0
\(403\) 0.146720 0.00730864
\(404\) 0 0
\(405\) −8.45525 −0.420145
\(406\) 0 0
\(407\) 33.6357 1.66726
\(408\) 0 0
\(409\) −7.63163 −0.377360 −0.188680 0.982039i \(-0.560421\pi\)
−0.188680 + 0.982039i \(0.560421\pi\)
\(410\) 0 0
\(411\) 5.57699 0.275092
\(412\) 0 0
\(413\) −17.7512 −0.873478
\(414\) 0 0
\(415\) −2.87240 −0.141001
\(416\) 0 0
\(417\) 0.458813 0.0224682
\(418\) 0 0
\(419\) −2.34760 −0.114688 −0.0573439 0.998354i \(-0.518263\pi\)
−0.0573439 + 0.998354i \(0.518263\pi\)
\(420\) 0 0
\(421\) 8.61693 0.419963 0.209982 0.977705i \(-0.432660\pi\)
0.209982 + 0.977705i \(0.432660\pi\)
\(422\) 0 0
\(423\) −14.4599 −0.703064
\(424\) 0 0
\(425\) 25.4473 1.23437
\(426\) 0 0
\(427\) −13.8862 −0.671999
\(428\) 0 0
\(429\) −0.100425 −0.00484854
\(430\) 0 0
\(431\) 2.73295 0.131641 0.0658207 0.997831i \(-0.479033\pi\)
0.0658207 + 0.997831i \(0.479033\pi\)
\(432\) 0 0
\(433\) 14.8768 0.714934 0.357467 0.933926i \(-0.383640\pi\)
0.357467 + 0.933926i \(0.383640\pi\)
\(434\) 0 0
\(435\) −5.06723 −0.242955
\(436\) 0 0
\(437\) −0.984819 −0.0471103
\(438\) 0 0
\(439\) 19.8115 0.945552 0.472776 0.881183i \(-0.343252\pi\)
0.472776 + 0.881183i \(0.343252\pi\)
\(440\) 0 0
\(441\) 14.1430 0.673474
\(442\) 0 0
\(443\) 39.5818 1.88059 0.940293 0.340366i \(-0.110551\pi\)
0.940293 + 0.340366i \(0.110551\pi\)
\(444\) 0 0
\(445\) 17.9041 0.848738
\(446\) 0 0
\(447\) 6.37146 0.301360
\(448\) 0 0
\(449\) −22.9205 −1.08169 −0.540843 0.841124i \(-0.681895\pi\)
−0.540843 + 0.841124i \(0.681895\pi\)
\(450\) 0 0
\(451\) 25.6299 1.20686
\(452\) 0 0
\(453\) 10.5123 0.493911
\(454\) 0 0
\(455\) −0.0727838 −0.00341216
\(456\) 0 0
\(457\) −9.66494 −0.452107 −0.226053 0.974115i \(-0.572582\pi\)
−0.226053 + 0.974115i \(0.572582\pi\)
\(458\) 0 0
\(459\) 20.8304 0.972279
\(460\) 0 0
\(461\) 6.28046 0.292510 0.146255 0.989247i \(-0.453278\pi\)
0.146255 + 0.989247i \(0.453278\pi\)
\(462\) 0 0
\(463\) 7.64337 0.355218 0.177609 0.984101i \(-0.443164\pi\)
0.177609 + 0.984101i \(0.443164\pi\)
\(464\) 0 0
\(465\) 2.07279 0.0961234
\(466\) 0 0
\(467\) −25.8943 −1.19824 −0.599122 0.800658i \(-0.704484\pi\)
−0.599122 + 0.800658i \(0.704484\pi\)
\(468\) 0 0
\(469\) 13.8346 0.638823
\(470\) 0 0
\(471\) 8.50136 0.391722
\(472\) 0 0
\(473\) 55.2342 2.53967
\(474\) 0 0
\(475\) −2.42664 −0.111342
\(476\) 0 0
\(477\) 23.6326 1.08206
\(478\) 0 0
\(479\) −12.6131 −0.576309 −0.288155 0.957584i \(-0.593042\pi\)
−0.288155 + 0.957584i \(0.593042\pi\)
\(480\) 0 0
\(481\) −0.309267 −0.0141013
\(482\) 0 0
\(483\) 0.954605 0.0434360
\(484\) 0 0
\(485\) 2.75907 0.125283
\(486\) 0 0
\(487\) −31.7807 −1.44012 −0.720061 0.693910i \(-0.755886\pi\)
−0.720061 + 0.693910i \(0.755886\pi\)
\(488\) 0 0
\(489\) −5.49680 −0.248574
\(490\) 0 0
\(491\) 10.2151 0.461001 0.230500 0.973072i \(-0.425964\pi\)
0.230500 + 0.973072i \(0.425964\pi\)
\(492\) 0 0
\(493\) −60.7331 −2.73528
\(494\) 0 0
\(495\) 15.9628 0.717474
\(496\) 0 0
\(497\) 2.55086 0.114422
\(498\) 0 0
\(499\) 9.35390 0.418738 0.209369 0.977837i \(-0.432859\pi\)
0.209369 + 0.977837i \(0.432859\pi\)
\(500\) 0 0
\(501\) 11.0742 0.494758
\(502\) 0 0
\(503\) 13.7659 0.613792 0.306896 0.951743i \(-0.400710\pi\)
0.306896 + 0.951743i \(0.400710\pi\)
\(504\) 0 0
\(505\) 4.95278 0.220396
\(506\) 0 0
\(507\) −6.43207 −0.285658
\(508\) 0 0
\(509\) 22.6367 1.00335 0.501677 0.865055i \(-0.332717\pi\)
0.501677 + 0.865055i \(0.332717\pi\)
\(510\) 0 0
\(511\) −8.85102 −0.391546
\(512\) 0 0
\(513\) −1.98638 −0.0877008
\(514\) 0 0
\(515\) −6.47682 −0.285403
\(516\) 0 0
\(517\) 24.6571 1.08442
\(518\) 0 0
\(519\) 2.05905 0.0903824
\(520\) 0 0
\(521\) −10.8738 −0.476389 −0.238194 0.971218i \(-0.576555\pi\)
−0.238194 + 0.971218i \(0.576555\pi\)
\(522\) 0 0
\(523\) 3.43232 0.150085 0.0750424 0.997180i \(-0.476091\pi\)
0.0750424 + 0.997180i \(0.476091\pi\)
\(524\) 0 0
\(525\) 2.35219 0.102658
\(526\) 0 0
\(527\) 24.8434 1.08219
\(528\) 0 0
\(529\) −21.0064 −0.913322
\(530\) 0 0
\(531\) −35.7959 −1.55341
\(532\) 0 0
\(533\) −0.235656 −0.0102074
\(534\) 0 0
\(535\) 6.91091 0.298785
\(536\) 0 0
\(537\) 2.86520 0.123643
\(538\) 0 0
\(539\) −24.1167 −1.03878
\(540\) 0 0
\(541\) −1.00758 −0.0433193 −0.0216597 0.999765i \(-0.506895\pi\)
−0.0216597 + 0.999765i \(0.506895\pi\)
\(542\) 0 0
\(543\) 8.65571 0.371452
\(544\) 0 0
\(545\) 5.88943 0.252275
\(546\) 0 0
\(547\) −10.6391 −0.454894 −0.227447 0.973791i \(-0.573038\pi\)
−0.227447 + 0.973791i \(0.573038\pi\)
\(548\) 0 0
\(549\) −28.0020 −1.19510
\(550\) 0 0
\(551\) 5.79149 0.246726
\(552\) 0 0
\(553\) −17.6844 −0.752019
\(554\) 0 0
\(555\) −4.36917 −0.185461
\(556\) 0 0
\(557\) 18.9092 0.801207 0.400603 0.916252i \(-0.368801\pi\)
0.400603 + 0.916252i \(0.368801\pi\)
\(558\) 0 0
\(559\) −0.507856 −0.0214800
\(560\) 0 0
\(561\) −17.0044 −0.717926
\(562\) 0 0
\(563\) −28.1768 −1.18751 −0.593756 0.804645i \(-0.702356\pi\)
−0.593756 + 0.804645i \(0.702356\pi\)
\(564\) 0 0
\(565\) 15.5618 0.654692
\(566\) 0 0
\(567\) −9.36727 −0.393388
\(568\) 0 0
\(569\) −0.499818 −0.0209534 −0.0104767 0.999945i \(-0.503335\pi\)
−0.0104767 + 0.999945i \(0.503335\pi\)
\(570\) 0 0
\(571\) 28.2303 1.18140 0.590702 0.806890i \(-0.298851\pi\)
0.590702 + 0.806890i \(0.298851\pi\)
\(572\) 0 0
\(573\) 10.6923 0.446678
\(574\) 0 0
\(575\) 4.91233 0.204858
\(576\) 0 0
\(577\) −36.3728 −1.51422 −0.757111 0.653287i \(-0.773390\pi\)
−0.757111 + 0.653287i \(0.773390\pi\)
\(578\) 0 0
\(579\) 1.14253 0.0474819
\(580\) 0 0
\(581\) −3.18224 −0.132021
\(582\) 0 0
\(583\) −40.2985 −1.66899
\(584\) 0 0
\(585\) −0.146771 −0.00606825
\(586\) 0 0
\(587\) −16.1223 −0.665437 −0.332718 0.943026i \(-0.607966\pi\)
−0.332718 + 0.943026i \(0.607966\pi\)
\(588\) 0 0
\(589\) −2.36905 −0.0976152
\(590\) 0 0
\(591\) 0.949491 0.0390569
\(592\) 0 0
\(593\) 24.8487 1.02041 0.510207 0.860052i \(-0.329569\pi\)
0.510207 + 0.860052i \(0.329569\pi\)
\(594\) 0 0
\(595\) −12.3241 −0.505240
\(596\) 0 0
\(597\) −10.5120 −0.430228
\(598\) 0 0
\(599\) 1.50155 0.0613517 0.0306759 0.999529i \(-0.490234\pi\)
0.0306759 + 0.999529i \(0.490234\pi\)
\(600\) 0 0
\(601\) −24.0828 −0.982356 −0.491178 0.871059i \(-0.663434\pi\)
−0.491178 + 0.871059i \(0.663434\pi\)
\(602\) 0 0
\(603\) 27.8981 1.13610
\(604\) 0 0
\(605\) −13.6542 −0.555121
\(606\) 0 0
\(607\) −16.0760 −0.652506 −0.326253 0.945283i \(-0.605786\pi\)
−0.326253 + 0.945283i \(0.605786\pi\)
\(608\) 0 0
\(609\) −5.61381 −0.227483
\(610\) 0 0
\(611\) −0.226712 −0.00917179
\(612\) 0 0
\(613\) 43.5233 1.75789 0.878945 0.476923i \(-0.158248\pi\)
0.878945 + 0.476923i \(0.158248\pi\)
\(614\) 0 0
\(615\) −3.32924 −0.134248
\(616\) 0 0
\(617\) −3.82544 −0.154006 −0.0770032 0.997031i \(-0.524535\pi\)
−0.0770032 + 0.997031i \(0.524535\pi\)
\(618\) 0 0
\(619\) −8.69881 −0.349635 −0.174817 0.984601i \(-0.555934\pi\)
−0.174817 + 0.984601i \(0.555934\pi\)
\(620\) 0 0
\(621\) 4.02109 0.161361
\(622\) 0 0
\(623\) 19.8354 0.794687
\(624\) 0 0
\(625\) 4.49981 0.179992
\(626\) 0 0
\(627\) 1.62153 0.0647578
\(628\) 0 0
\(629\) −52.3666 −2.08799
\(630\) 0 0
\(631\) 28.0264 1.11571 0.557856 0.829938i \(-0.311624\pi\)
0.557856 + 0.829938i \(0.311624\pi\)
\(632\) 0 0
\(633\) 9.44961 0.375588
\(634\) 0 0
\(635\) 0.527398 0.0209291
\(636\) 0 0
\(637\) 0.221743 0.00878578
\(638\) 0 0
\(639\) 5.14391 0.203490
\(640\) 0 0
\(641\) 22.7621 0.899048 0.449524 0.893268i \(-0.351594\pi\)
0.449524 + 0.893268i \(0.351594\pi\)
\(642\) 0 0
\(643\) −35.9559 −1.41796 −0.708981 0.705228i \(-0.750845\pi\)
−0.708981 + 0.705228i \(0.750845\pi\)
\(644\) 0 0
\(645\) −7.17476 −0.282506
\(646\) 0 0
\(647\) 1.94619 0.0765127 0.0382564 0.999268i \(-0.487820\pi\)
0.0382564 + 0.999268i \(0.487820\pi\)
\(648\) 0 0
\(649\) 61.0395 2.39601
\(650\) 0 0
\(651\) 2.29637 0.0900019
\(652\) 0 0
\(653\) 24.3392 0.952467 0.476233 0.879319i \(-0.342002\pi\)
0.476233 + 0.879319i \(0.342002\pi\)
\(654\) 0 0
\(655\) −16.4802 −0.643936
\(656\) 0 0
\(657\) −17.8484 −0.696334
\(658\) 0 0
\(659\) 6.16907 0.240313 0.120156 0.992755i \(-0.461660\pi\)
0.120156 + 0.992755i \(0.461660\pi\)
\(660\) 0 0
\(661\) −48.1704 −1.87361 −0.936807 0.349848i \(-0.886233\pi\)
−0.936807 + 0.349848i \(0.886233\pi\)
\(662\) 0 0
\(663\) 0.156349 0.00607208
\(664\) 0 0
\(665\) 1.17522 0.0455732
\(666\) 0 0
\(667\) −11.7239 −0.453951
\(668\) 0 0
\(669\) −4.16551 −0.161048
\(670\) 0 0
\(671\) 47.7492 1.84334
\(672\) 0 0
\(673\) −19.2071 −0.740381 −0.370190 0.928956i \(-0.620708\pi\)
−0.370190 + 0.928956i \(0.620708\pi\)
\(674\) 0 0
\(675\) 9.90817 0.381366
\(676\) 0 0
\(677\) −14.9094 −0.573014 −0.286507 0.958078i \(-0.592494\pi\)
−0.286507 + 0.958078i \(0.592494\pi\)
\(678\) 0 0
\(679\) 3.05667 0.117304
\(680\) 0 0
\(681\) −0.904843 −0.0346737
\(682\) 0 0
\(683\) 22.8682 0.875029 0.437515 0.899211i \(-0.355859\pi\)
0.437515 + 0.899211i \(0.355859\pi\)
\(684\) 0 0
\(685\) −13.8988 −0.531047
\(686\) 0 0
\(687\) 1.24720 0.0475835
\(688\) 0 0
\(689\) 0.370529 0.0141160
\(690\) 0 0
\(691\) 35.5210 1.35128 0.675641 0.737231i \(-0.263867\pi\)
0.675641 + 0.737231i \(0.263867\pi\)
\(692\) 0 0
\(693\) 17.6846 0.671783
\(694\) 0 0
\(695\) −1.14344 −0.0433732
\(696\) 0 0
\(697\) −39.9025 −1.51142
\(698\) 0 0
\(699\) 8.10617 0.306604
\(700\) 0 0
\(701\) 9.22303 0.348349 0.174175 0.984715i \(-0.444274\pi\)
0.174175 + 0.984715i \(0.444274\pi\)
\(702\) 0 0
\(703\) 4.99366 0.188339
\(704\) 0 0
\(705\) −3.20288 −0.120628
\(706\) 0 0
\(707\) 5.48701 0.206360
\(708\) 0 0
\(709\) 28.7763 1.08072 0.540358 0.841435i \(-0.318289\pi\)
0.540358 + 0.841435i \(0.318289\pi\)
\(710\) 0 0
\(711\) −35.6614 −1.33741
\(712\) 0 0
\(713\) 4.79575 0.179602
\(714\) 0 0
\(715\) 0.250276 0.00935978
\(716\) 0 0
\(717\) 5.31614 0.198535
\(718\) 0 0
\(719\) 24.1878 0.902055 0.451027 0.892510i \(-0.351058\pi\)
0.451027 + 0.892510i \(0.351058\pi\)
\(720\) 0 0
\(721\) −7.17544 −0.267227
\(722\) 0 0
\(723\) −0.494846 −0.0184035
\(724\) 0 0
\(725\) −28.8883 −1.07288
\(726\) 0 0
\(727\) −9.73994 −0.361235 −0.180617 0.983553i \(-0.557810\pi\)
−0.180617 + 0.983553i \(0.557810\pi\)
\(728\) 0 0
\(729\) −14.6617 −0.543024
\(730\) 0 0
\(731\) −85.9928 −3.18056
\(732\) 0 0
\(733\) −4.38060 −0.161801 −0.0809006 0.996722i \(-0.525780\pi\)
−0.0809006 + 0.996722i \(0.525780\pi\)
\(734\) 0 0
\(735\) 3.13268 0.115551
\(736\) 0 0
\(737\) −47.5719 −1.75234
\(738\) 0 0
\(739\) 6.21520 0.228630 0.114315 0.993445i \(-0.463533\pi\)
0.114315 + 0.993445i \(0.463533\pi\)
\(740\) 0 0
\(741\) −0.0149094 −0.000547709 0
\(742\) 0 0
\(743\) −26.7097 −0.979885 −0.489942 0.871755i \(-0.662982\pi\)
−0.489942 + 0.871755i \(0.662982\pi\)
\(744\) 0 0
\(745\) −15.8788 −0.581754
\(746\) 0 0
\(747\) −6.41710 −0.234790
\(748\) 0 0
\(749\) 7.65635 0.279757
\(750\) 0 0
\(751\) −46.9924 −1.71478 −0.857388 0.514670i \(-0.827915\pi\)
−0.857388 + 0.514670i \(0.827915\pi\)
\(752\) 0 0
\(753\) −11.8178 −0.430666
\(754\) 0 0
\(755\) −26.1985 −0.953461
\(756\) 0 0
\(757\) −35.2425 −1.28091 −0.640456 0.767995i \(-0.721255\pi\)
−0.640456 + 0.767995i \(0.721255\pi\)
\(758\) 0 0
\(759\) −3.28252 −0.119148
\(760\) 0 0
\(761\) −47.1109 −1.70777 −0.853885 0.520461i \(-0.825760\pi\)
−0.853885 + 0.520461i \(0.825760\pi\)
\(762\) 0 0
\(763\) 6.52469 0.236210
\(764\) 0 0
\(765\) −24.8521 −0.898529
\(766\) 0 0
\(767\) −0.561234 −0.0202650
\(768\) 0 0
\(769\) −53.8827 −1.94306 −0.971530 0.236915i \(-0.923864\pi\)
−0.971530 + 0.236915i \(0.923864\pi\)
\(770\) 0 0
\(771\) 2.59995 0.0936350
\(772\) 0 0
\(773\) −7.33241 −0.263728 −0.131864 0.991268i \(-0.542096\pi\)
−0.131864 + 0.991268i \(0.542096\pi\)
\(774\) 0 0
\(775\) 11.8170 0.424478
\(776\) 0 0
\(777\) −4.84045 −0.173650
\(778\) 0 0
\(779\) 3.80509 0.136332
\(780\) 0 0
\(781\) −8.77143 −0.313866
\(782\) 0 0
\(783\) −23.6471 −0.845078
\(784\) 0 0
\(785\) −21.1869 −0.756193
\(786\) 0 0
\(787\) 0.274880 0.00979841 0.00489921 0.999988i \(-0.498441\pi\)
0.00489921 + 0.999988i \(0.498441\pi\)
\(788\) 0 0
\(789\) 0.549102 0.0195485
\(790\) 0 0
\(791\) 17.2404 0.612998
\(792\) 0 0
\(793\) −0.439035 −0.0155906
\(794\) 0 0
\(795\) 5.23466 0.185654
\(796\) 0 0
\(797\) −7.42858 −0.263134 −0.131567 0.991307i \(-0.542001\pi\)
−0.131567 + 0.991307i \(0.542001\pi\)
\(798\) 0 0
\(799\) −38.3881 −1.35807
\(800\) 0 0
\(801\) 39.9988 1.41329
\(802\) 0 0
\(803\) 30.4353 1.07404
\(804\) 0 0
\(805\) −2.37904 −0.0838502
\(806\) 0 0
\(807\) −1.40850 −0.0495815
\(808\) 0 0
\(809\) 42.6491 1.49946 0.749730 0.661743i \(-0.230183\pi\)
0.749730 + 0.661743i \(0.230183\pi\)
\(810\) 0 0
\(811\) −25.6139 −0.899425 −0.449713 0.893173i \(-0.648474\pi\)
−0.449713 + 0.893173i \(0.648474\pi\)
\(812\) 0 0
\(813\) −6.48392 −0.227401
\(814\) 0 0
\(815\) 13.6990 0.479855
\(816\) 0 0
\(817\) 8.20024 0.286890
\(818\) 0 0
\(819\) −0.162603 −0.00568181
\(820\) 0 0
\(821\) 21.8573 0.762826 0.381413 0.924405i \(-0.375438\pi\)
0.381413 + 0.924405i \(0.375438\pi\)
\(822\) 0 0
\(823\) −1.58270 −0.0551693 −0.0275847 0.999619i \(-0.508782\pi\)
−0.0275847 + 0.999619i \(0.508782\pi\)
\(824\) 0 0
\(825\) −8.08830 −0.281598
\(826\) 0 0
\(827\) −36.2198 −1.25949 −0.629743 0.776803i \(-0.716840\pi\)
−0.629743 + 0.776803i \(0.716840\pi\)
\(828\) 0 0
\(829\) 30.5957 1.06263 0.531316 0.847174i \(-0.321698\pi\)
0.531316 + 0.847174i \(0.321698\pi\)
\(830\) 0 0
\(831\) −11.6107 −0.402772
\(832\) 0 0
\(833\) 37.5466 1.30091
\(834\) 0 0
\(835\) −27.5988 −0.955096
\(836\) 0 0
\(837\) 9.67303 0.334349
\(838\) 0 0
\(839\) 9.85906 0.340373 0.170186 0.985412i \(-0.445563\pi\)
0.170186 + 0.985412i \(0.445563\pi\)
\(840\) 0 0
\(841\) 39.9455 1.37743
\(842\) 0 0
\(843\) 4.40989 0.151885
\(844\) 0 0
\(845\) 16.0298 0.551443
\(846\) 0 0
\(847\) −15.1270 −0.519769
\(848\) 0 0
\(849\) 7.82056 0.268401
\(850\) 0 0
\(851\) −10.1088 −0.346526
\(852\) 0 0
\(853\) 20.6701 0.707730 0.353865 0.935296i \(-0.384867\pi\)
0.353865 + 0.935296i \(0.384867\pi\)
\(854\) 0 0
\(855\) 2.36989 0.0810484
\(856\) 0 0
\(857\) 36.0858 1.23267 0.616334 0.787485i \(-0.288617\pi\)
0.616334 + 0.787485i \(0.288617\pi\)
\(858\) 0 0
\(859\) −40.7566 −1.39060 −0.695299 0.718720i \(-0.744728\pi\)
−0.695299 + 0.718720i \(0.744728\pi\)
\(860\) 0 0
\(861\) −3.68835 −0.125699
\(862\) 0 0
\(863\) −21.5125 −0.732295 −0.366148 0.930557i \(-0.619323\pi\)
−0.366148 + 0.930557i \(0.619323\pi\)
\(864\) 0 0
\(865\) −5.13152 −0.174477
\(866\) 0 0
\(867\) 18.0614 0.613396
\(868\) 0 0
\(869\) 60.8100 2.06284
\(870\) 0 0
\(871\) 0.437405 0.0148209
\(872\) 0 0
\(873\) 6.16390 0.208616
\(874\) 0 0
\(875\) −14.2868 −0.482981
\(876\) 0 0
\(877\) −47.8912 −1.61717 −0.808586 0.588378i \(-0.799767\pi\)
−0.808586 + 0.588378i \(0.799767\pi\)
\(878\) 0 0
\(879\) 16.3015 0.549835
\(880\) 0 0
\(881\) −18.2689 −0.615496 −0.307748 0.951468i \(-0.599575\pi\)
−0.307748 + 0.951468i \(0.599575\pi\)
\(882\) 0 0
\(883\) 37.0354 1.24634 0.623170 0.782087i \(-0.285845\pi\)
0.623170 + 0.782087i \(0.285845\pi\)
\(884\) 0 0
\(885\) −7.92885 −0.266525
\(886\) 0 0
\(887\) 13.7619 0.462080 0.231040 0.972944i \(-0.425787\pi\)
0.231040 + 0.972944i \(0.425787\pi\)
\(888\) 0 0
\(889\) 0.584285 0.0195963
\(890\) 0 0
\(891\) 32.2104 1.07909
\(892\) 0 0
\(893\) 3.66067 0.122500
\(894\) 0 0
\(895\) −7.14059 −0.238684
\(896\) 0 0
\(897\) 0.0301815 0.00100773
\(898\) 0 0
\(899\) −28.2027 −0.940613
\(900\) 0 0
\(901\) 62.7398 2.09017
\(902\) 0 0
\(903\) −7.94866 −0.264515
\(904\) 0 0
\(905\) −21.5716 −0.717063
\(906\) 0 0
\(907\) 27.1842 0.902637 0.451319 0.892363i \(-0.350954\pi\)
0.451319 + 0.892363i \(0.350954\pi\)
\(908\) 0 0
\(909\) 11.0648 0.366996
\(910\) 0 0
\(911\) −29.4894 −0.977027 −0.488513 0.872556i \(-0.662461\pi\)
−0.488513 + 0.872556i \(0.662461\pi\)
\(912\) 0 0
\(913\) 10.9425 0.362143
\(914\) 0 0
\(915\) −6.20248 −0.205048
\(916\) 0 0
\(917\) −18.2579 −0.602928
\(918\) 0 0
\(919\) 44.2354 1.45919 0.729596 0.683878i \(-0.239708\pi\)
0.729596 + 0.683878i \(0.239708\pi\)
\(920\) 0 0
\(921\) −10.0128 −0.329935
\(922\) 0 0
\(923\) 0.0806498 0.00265462
\(924\) 0 0
\(925\) −24.9086 −0.818991
\(926\) 0 0
\(927\) −14.4696 −0.475243
\(928\) 0 0
\(929\) 17.7425 0.582112 0.291056 0.956706i \(-0.405993\pi\)
0.291056 + 0.956706i \(0.405993\pi\)
\(930\) 0 0
\(931\) −3.58043 −0.117344
\(932\) 0 0
\(933\) −1.12473 −0.0368220
\(934\) 0 0
\(935\) 42.3779 1.38591
\(936\) 0 0
\(937\) 46.1680 1.50824 0.754121 0.656735i \(-0.228063\pi\)
0.754121 + 0.656735i \(0.228063\pi\)
\(938\) 0 0
\(939\) 6.48949 0.211777
\(940\) 0 0
\(941\) −22.0383 −0.718428 −0.359214 0.933255i \(-0.616955\pi\)
−0.359214 + 0.933255i \(0.616955\pi\)
\(942\) 0 0
\(943\) −7.70276 −0.250836
\(944\) 0 0
\(945\) −4.79853 −0.156096
\(946\) 0 0
\(947\) 34.5940 1.12416 0.562078 0.827085i \(-0.310002\pi\)
0.562078 + 0.827085i \(0.310002\pi\)
\(948\) 0 0
\(949\) −0.279840 −0.00908399
\(950\) 0 0
\(951\) −3.89209 −0.126210
\(952\) 0 0
\(953\) −20.9799 −0.679607 −0.339803 0.940496i \(-0.610361\pi\)
−0.339803 + 0.940496i \(0.610361\pi\)
\(954\) 0 0
\(955\) −26.6471 −0.862281
\(956\) 0 0
\(957\) 19.3037 0.624001
\(958\) 0 0
\(959\) −15.3980 −0.497228
\(960\) 0 0
\(961\) −19.4635 −0.627854
\(962\) 0 0
\(963\) 15.4393 0.497526
\(964\) 0 0
\(965\) −2.84738 −0.0916605
\(966\) 0 0
\(967\) 0.0716187 0.00230310 0.00115155 0.999999i \(-0.499633\pi\)
0.00115155 + 0.999999i \(0.499633\pi\)
\(968\) 0 0
\(969\) −2.52453 −0.0810995
\(970\) 0 0
\(971\) 8.18205 0.262575 0.131287 0.991344i \(-0.458089\pi\)
0.131287 + 0.991344i \(0.458089\pi\)
\(972\) 0 0
\(973\) −1.26678 −0.0406111
\(974\) 0 0
\(975\) 0.0743687 0.00238170
\(976\) 0 0
\(977\) 22.6886 0.725872 0.362936 0.931814i \(-0.381774\pi\)
0.362936 + 0.931814i \(0.381774\pi\)
\(978\) 0 0
\(979\) −68.2062 −2.17988
\(980\) 0 0
\(981\) 13.1573 0.420080
\(982\) 0 0
\(983\) 18.8463 0.601103 0.300551 0.953766i \(-0.402829\pi\)
0.300551 + 0.953766i \(0.402829\pi\)
\(984\) 0 0
\(985\) −2.36630 −0.0753966
\(986\) 0 0
\(987\) −3.54836 −0.112946
\(988\) 0 0
\(989\) −16.6000 −0.527850
\(990\) 0 0
\(991\) 16.8310 0.534655 0.267328 0.963606i \(-0.413859\pi\)
0.267328 + 0.963606i \(0.413859\pi\)
\(992\) 0 0
\(993\) 9.58775 0.304258
\(994\) 0 0
\(995\) 26.1978 0.830525
\(996\) 0 0
\(997\) 60.1343 1.90447 0.952236 0.305363i \(-0.0987779\pi\)
0.952236 + 0.305363i \(0.0987779\pi\)
\(998\) 0 0
\(999\) −20.3895 −0.645095
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 3856.2.a.j.1.4 7
4.3 odd 2 241.2.a.a.1.5 7
12.11 even 2 2169.2.a.e.1.3 7
20.19 odd 2 6025.2.a.f.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.5 7 4.3 odd 2
2169.2.a.e.1.3 7 12.11 even 2
3856.2.a.j.1.4 7 1.1 even 1 trivial
6025.2.a.f.1.3 7 20.19 odd 2