Properties

Label 308.2.a.c.1.2
Level $308$
Weight $2$
Character 308.1
Self dual yes
Analytic conductor $2.459$
Analytic rank $0$
Dimension $3$
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] = [308,2,Mod(1,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, 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("308.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 308.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: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.321637\) of defining polynomial
Character \(\chi\) \(=\) 308.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.321637 q^{3} +3.89655 q^{5} +1.00000 q^{7} -2.89655 q^{9} +O(q^{10})\) \(q-0.321637 q^{3} +3.89655 q^{5} +1.00000 q^{7} -2.89655 q^{9} +1.00000 q^{11} -0.218187 q^{13} -1.25328 q^{15} +4.86146 q^{17} -0.643274 q^{19} -0.321637 q^{21} -6.53982 q^{23} +10.1831 q^{25} +1.89655 q^{27} +10.4364 q^{29} -3.03509 q^{31} -0.321637 q^{33} +3.89655 q^{35} -1.89655 q^{37} +0.0701770 q^{39} -4.86146 q^{41} -1.35673 q^{43} -11.2865 q^{45} -5.57491 q^{47} +1.00000 q^{49} -1.56363 q^{51} -7.79310 q^{53} +3.89655 q^{55} +0.206901 q^{57} -7.67836 q^{59} -8.01129 q^{61} -2.89655 q^{63} -0.850175 q^{65} +2.53982 q^{67} +2.10345 q^{69} -14.3329 q^{71} -8.86146 q^{73} -3.27526 q^{75} +1.00000 q^{77} +13.1498 q^{79} +8.07965 q^{81} +9.72292 q^{83} +18.9429 q^{85} -3.35673 q^{87} -10.6100 q^{89} -0.218187 q^{91} +0.976197 q^{93} -2.50655 q^{95} +0.746725 q^{97} -2.89655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{11} + 12 q^{13} + 9 q^{15} + 2 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 18 q^{25} - 7 q^{27} + 6 q^{29} - 9 q^{31} - q^{33} - q^{35} + 7 q^{37} - 2 q^{41} - 4 q^{43} - 34 q^{45} - 4 q^{47} + 3 q^{49} - 30 q^{51} + 2 q^{53} - q^{55} + 26 q^{57} - 23 q^{59} + 14 q^{61} + 4 q^{63} - 28 q^{65} - 5 q^{67} + 19 q^{69} - 5 q^{71} - 14 q^{73} - 48 q^{75} + 3 q^{77} + 14 q^{79} - q^{81} + 4 q^{83} + 6 q^{85} - 10 q^{87} - 19 q^{89} + 12 q^{91} - 35 q^{93} + 18 q^{95} + 15 q^{97} + 4 q^{99}+O(q^{100}) \) 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 0
\(3\) −0.321637 −0.185697 −0.0928487 0.995680i \(-0.529597\pi\)
−0.0928487 + 0.995680i \(0.529597\pi\)
\(4\) 0 0
\(5\) 3.89655 1.74259 0.871295 0.490760i \(-0.163281\pi\)
0.871295 + 0.490760i \(0.163281\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.89655 −0.965517
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −0.218187 −0.0605141 −0.0302571 0.999542i \(-0.509633\pi\)
−0.0302571 + 0.999542i \(0.509633\pi\)
\(14\) 0 0
\(15\) −1.25328 −0.323594
\(16\) 0 0
\(17\) 4.86146 1.17908 0.589539 0.807740i \(-0.299310\pi\)
0.589539 + 0.807740i \(0.299310\pi\)
\(18\) 0 0
\(19\) −0.643274 −0.147577 −0.0737886 0.997274i \(-0.523509\pi\)
−0.0737886 + 0.997274i \(0.523509\pi\)
\(20\) 0 0
\(21\) −0.321637 −0.0701870
\(22\) 0 0
\(23\) −6.53982 −1.36365 −0.681824 0.731516i \(-0.738813\pi\)
−0.681824 + 0.731516i \(0.738813\pi\)
\(24\) 0 0
\(25\) 10.1831 2.03662
\(26\) 0 0
\(27\) 1.89655 0.364991
\(28\) 0 0
\(29\) 10.4364 1.93799 0.968993 0.247088i \(-0.0794738\pi\)
0.968993 + 0.247088i \(0.0794738\pi\)
\(30\) 0 0
\(31\) −3.03509 −0.545118 −0.272559 0.962139i \(-0.587870\pi\)
−0.272559 + 0.962139i \(0.587870\pi\)
\(32\) 0 0
\(33\) −0.321637 −0.0559898
\(34\) 0 0
\(35\) 3.89655 0.658637
\(36\) 0 0
\(37\) −1.89655 −0.311791 −0.155895 0.987774i \(-0.549826\pi\)
−0.155895 + 0.987774i \(0.549826\pi\)
\(38\) 0 0
\(39\) 0.0701770 0.0112373
\(40\) 0 0
\(41\) −4.86146 −0.759233 −0.379616 0.925144i \(-0.623944\pi\)
−0.379616 + 0.925144i \(0.623944\pi\)
\(42\) 0 0
\(43\) −1.35673 −0.206899 −0.103449 0.994635i \(-0.532988\pi\)
−0.103449 + 0.994635i \(0.532988\pi\)
\(44\) 0 0
\(45\) −11.2865 −1.68250
\(46\) 0 0
\(47\) −5.57491 −0.813185 −0.406592 0.913610i \(-0.633283\pi\)
−0.406592 + 0.913610i \(0.633283\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.56363 −0.218952
\(52\) 0 0
\(53\) −7.79310 −1.07046 −0.535232 0.844705i \(-0.679776\pi\)
−0.535232 + 0.844705i \(0.679776\pi\)
\(54\) 0 0
\(55\) 3.89655 0.525411
\(56\) 0 0
\(57\) 0.206901 0.0274047
\(58\) 0 0
\(59\) −7.67836 −0.999638 −0.499819 0.866130i \(-0.666600\pi\)
−0.499819 + 0.866130i \(0.666600\pi\)
\(60\) 0 0
\(61\) −8.01129 −1.02574 −0.512870 0.858466i \(-0.671418\pi\)
−0.512870 + 0.858466i \(0.671418\pi\)
\(62\) 0 0
\(63\) −2.89655 −0.364931
\(64\) 0 0
\(65\) −0.850175 −0.105451
\(66\) 0 0
\(67\) 2.53982 0.310289 0.155144 0.987892i \(-0.450416\pi\)
0.155144 + 0.987892i \(0.450416\pi\)
\(68\) 0 0
\(69\) 2.10345 0.253226
\(70\) 0 0
\(71\) −14.3329 −1.70101 −0.850503 0.525971i \(-0.823702\pi\)
−0.850503 + 0.525971i \(0.823702\pi\)
\(72\) 0 0
\(73\) −8.86146 −1.03716 −0.518578 0.855030i \(-0.673538\pi\)
−0.518578 + 0.855030i \(0.673538\pi\)
\(74\) 0 0
\(75\) −3.27526 −0.378195
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 13.1498 1.47947 0.739735 0.672898i \(-0.234951\pi\)
0.739735 + 0.672898i \(0.234951\pi\)
\(80\) 0 0
\(81\) 8.07965 0.897739
\(82\) 0 0
\(83\) 9.72292 1.06723 0.533615 0.845728i \(-0.320833\pi\)
0.533615 + 0.845728i \(0.320833\pi\)
\(84\) 0 0
\(85\) 18.9429 2.05465
\(86\) 0 0
\(87\) −3.35673 −0.359879
\(88\) 0 0
\(89\) −10.6100 −1.12466 −0.562329 0.826914i \(-0.690095\pi\)
−0.562329 + 0.826914i \(0.690095\pi\)
\(90\) 0 0
\(91\) −0.218187 −0.0228722
\(92\) 0 0
\(93\) 0.976197 0.101227
\(94\) 0 0
\(95\) −2.50655 −0.257167
\(96\) 0 0
\(97\) 0.746725 0.0758184 0.0379092 0.999281i \(-0.487930\pi\)
0.0379092 + 0.999281i \(0.487930\pi\)
\(98\) 0 0
\(99\) −2.89655 −0.291114
\(100\) 0 0
\(101\) −8.21819 −0.817740 −0.408870 0.912593i \(-0.634077\pi\)
−0.408870 + 0.912593i \(0.634077\pi\)
\(102\) 0 0
\(103\) 6.86146 0.676080 0.338040 0.941132i \(-0.390236\pi\)
0.338040 + 0.941132i \(0.390236\pi\)
\(104\) 0 0
\(105\) −1.25328 −0.122307
\(106\) 0 0
\(107\) −7.79310 −0.753387 −0.376694 0.926338i \(-0.622939\pi\)
−0.376694 + 0.926338i \(0.622939\pi\)
\(108\) 0 0
\(109\) 0.0701770 0.00672173 0.00336087 0.999994i \(-0.498930\pi\)
0.00336087 + 0.999994i \(0.498930\pi\)
\(110\) 0 0
\(111\) 0.610001 0.0578987
\(112\) 0 0
\(113\) 18.4031 1.73122 0.865609 0.500721i \(-0.166932\pi\)
0.865609 + 0.500721i \(0.166932\pi\)
\(114\) 0 0
\(115\) −25.4827 −2.37628
\(116\) 0 0
\(117\) 0.631989 0.0584274
\(118\) 0 0
\(119\) 4.86146 0.445649
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 1.56363 0.140987
\(124\) 0 0
\(125\) 20.1962 1.80640
\(126\) 0 0
\(127\) 21.0796 1.87052 0.935258 0.353966i \(-0.115167\pi\)
0.935258 + 0.353966i \(0.115167\pi\)
\(128\) 0 0
\(129\) 0.436373 0.0384205
\(130\) 0 0
\(131\) −19.7931 −1.72933 −0.864666 0.502348i \(-0.832470\pi\)
−0.864666 + 0.502348i \(0.832470\pi\)
\(132\) 0 0
\(133\) −0.643274 −0.0557790
\(134\) 0 0
\(135\) 7.39000 0.636030
\(136\) 0 0
\(137\) −10.9762 −0.937760 −0.468880 0.883262i \(-0.655342\pi\)
−0.468880 + 0.883262i \(0.655342\pi\)
\(138\) 0 0
\(139\) −16.2295 −1.37657 −0.688283 0.725442i \(-0.741635\pi\)
−0.688283 + 0.725442i \(0.741635\pi\)
\(140\) 0 0
\(141\) 1.79310 0.151006
\(142\) 0 0
\(143\) −0.218187 −0.0182457
\(144\) 0 0
\(145\) 40.6658 3.37711
\(146\) 0 0
\(147\) −0.321637 −0.0265282
\(148\) 0 0
\(149\) −14.8727 −1.21842 −0.609211 0.793008i \(-0.708514\pi\)
−0.609211 + 0.793008i \(0.708514\pi\)
\(150\) 0 0
\(151\) 13.1498 1.07012 0.535059 0.844815i \(-0.320289\pi\)
0.535059 + 0.844815i \(0.320289\pi\)
\(152\) 0 0
\(153\) −14.0815 −1.13842
\(154\) 0 0
\(155\) −11.8264 −0.949917
\(156\) 0 0
\(157\) 11.8965 0.949448 0.474724 0.880135i \(-0.342548\pi\)
0.474724 + 0.880135i \(0.342548\pi\)
\(158\) 0 0
\(159\) 2.50655 0.198782
\(160\) 0 0
\(161\) −6.53982 −0.515410
\(162\) 0 0
\(163\) 7.14982 0.560017 0.280009 0.959997i \(-0.409663\pi\)
0.280009 + 0.959997i \(0.409663\pi\)
\(164\) 0 0
\(165\) −1.25328 −0.0975673
\(166\) 0 0
\(167\) 25.0796 1.94072 0.970361 0.241661i \(-0.0776922\pi\)
0.970361 + 0.241661i \(0.0776922\pi\)
\(168\) 0 0
\(169\) −12.9524 −0.996338
\(170\) 0 0
\(171\) 1.86328 0.142488
\(172\) 0 0
\(173\) −7.36801 −0.560180 −0.280090 0.959974i \(-0.590364\pi\)
−0.280090 + 0.959974i \(0.590364\pi\)
\(174\) 0 0
\(175\) 10.1831 0.769770
\(176\) 0 0
\(177\) 2.46965 0.185630
\(178\) 0 0
\(179\) 8.81690 0.659006 0.329503 0.944154i \(-0.393119\pi\)
0.329503 + 0.944154i \(0.393119\pi\)
\(180\) 0 0
\(181\) 17.4126 1.29427 0.647133 0.762377i \(-0.275968\pi\)
0.647133 + 0.762377i \(0.275968\pi\)
\(182\) 0 0
\(183\) 2.57673 0.190477
\(184\) 0 0
\(185\) −7.39000 −0.543324
\(186\) 0 0
\(187\) 4.86146 0.355505
\(188\) 0 0
\(189\) 1.89655 0.137954
\(190\) 0 0
\(191\) −9.89655 −0.716089 −0.358045 0.933704i \(-0.616556\pi\)
−0.358045 + 0.933704i \(0.616556\pi\)
\(192\) 0 0
\(193\) 17.5862 1.26588 0.632941 0.774200i \(-0.281848\pi\)
0.632941 + 0.774200i \(0.281848\pi\)
\(194\) 0 0
\(195\) 0.273448 0.0195820
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 22.4477 1.59127 0.795636 0.605775i \(-0.207137\pi\)
0.795636 + 0.605775i \(0.207137\pi\)
\(200\) 0 0
\(201\) −0.816902 −0.0576198
\(202\) 0 0
\(203\) 10.4364 0.732490
\(204\) 0 0
\(205\) −18.9429 −1.32303
\(206\) 0 0
\(207\) 18.9429 1.31662
\(208\) 0 0
\(209\) −0.643274 −0.0444962
\(210\) 0 0
\(211\) 0.206901 0.0142436 0.00712182 0.999975i \(-0.497733\pi\)
0.00712182 + 0.999975i \(0.497733\pi\)
\(212\) 0 0
\(213\) 4.61000 0.315872
\(214\) 0 0
\(215\) −5.28655 −0.360540
\(216\) 0 0
\(217\) −3.03509 −0.206035
\(218\) 0 0
\(219\) 2.85018 0.192597
\(220\) 0 0
\(221\) −1.06071 −0.0713508
\(222\) 0 0
\(223\) −22.4809 −1.50543 −0.752717 0.658344i \(-0.771257\pi\)
−0.752717 + 0.658344i \(0.771257\pi\)
\(224\) 0 0
\(225\) −29.4958 −1.96639
\(226\) 0 0
\(227\) 15.5862 1.03449 0.517246 0.855837i \(-0.326957\pi\)
0.517246 + 0.855837i \(0.326957\pi\)
\(228\) 0 0
\(229\) −7.04637 −0.465638 −0.232819 0.972520i \(-0.574795\pi\)
−0.232819 + 0.972520i \(0.574795\pi\)
\(230\) 0 0
\(231\) −0.321637 −0.0211622
\(232\) 0 0
\(233\) −1.56363 −0.102437 −0.0512183 0.998687i \(-0.516310\pi\)
−0.0512183 + 0.998687i \(0.516310\pi\)
\(234\) 0 0
\(235\) −21.7229 −1.41705
\(236\) 0 0
\(237\) −4.22947 −0.274734
\(238\) 0 0
\(239\) 26.1593 1.69210 0.846052 0.533100i \(-0.178973\pi\)
0.846052 + 0.533100i \(0.178973\pi\)
\(240\) 0 0
\(241\) −16.6546 −1.07281 −0.536407 0.843959i \(-0.680219\pi\)
−0.536407 + 0.843959i \(0.680219\pi\)
\(242\) 0 0
\(243\) −8.28836 −0.531699
\(244\) 0 0
\(245\) 3.89655 0.248941
\(246\) 0 0
\(247\) 0.140354 0.00893051
\(248\) 0 0
\(249\) −3.12725 −0.198182
\(250\) 0 0
\(251\) −29.4013 −1.85579 −0.927896 0.372838i \(-0.878385\pi\)
−0.927896 + 0.372838i \(0.878385\pi\)
\(252\) 0 0
\(253\) −6.53982 −0.411155
\(254\) 0 0
\(255\) −6.09275 −0.381543
\(256\) 0 0
\(257\) 12.7135 0.793043 0.396522 0.918025i \(-0.370217\pi\)
0.396522 + 0.918025i \(0.370217\pi\)
\(258\) 0 0
\(259\) −1.89655 −0.117846
\(260\) 0 0
\(261\) −30.2295 −1.87116
\(262\) 0 0
\(263\) 4.20690 0.259409 0.129704 0.991553i \(-0.458597\pi\)
0.129704 + 0.991553i \(0.458597\pi\)
\(264\) 0 0
\(265\) −30.3662 −1.86538
\(266\) 0 0
\(267\) 3.41257 0.208846
\(268\) 0 0
\(269\) 3.72292 0.226991 0.113495 0.993539i \(-0.463795\pi\)
0.113495 + 0.993539i \(0.463795\pi\)
\(270\) 0 0
\(271\) 5.72292 0.347643 0.173821 0.984777i \(-0.444388\pi\)
0.173821 + 0.984777i \(0.444388\pi\)
\(272\) 0 0
\(273\) 0.0701770 0.00424730
\(274\) 0 0
\(275\) 10.1831 0.614064
\(276\) 0 0
\(277\) −12.5066 −0.751446 −0.375723 0.926732i \(-0.622606\pi\)
−0.375723 + 0.926732i \(0.622606\pi\)
\(278\) 0 0
\(279\) 8.79128 0.526320
\(280\) 0 0
\(281\) 6.22947 0.371619 0.185810 0.982586i \(-0.440509\pi\)
0.185810 + 0.982586i \(0.440509\pi\)
\(282\) 0 0
\(283\) 16.8727 1.00298 0.501490 0.865163i \(-0.332785\pi\)
0.501490 + 0.865163i \(0.332785\pi\)
\(284\) 0 0
\(285\) 0.806200 0.0477552
\(286\) 0 0
\(287\) −4.86146 −0.286963
\(288\) 0 0
\(289\) 6.63380 0.390224
\(290\) 0 0
\(291\) −0.240174 −0.0140793
\(292\) 0 0
\(293\) 13.0018 0.759574 0.379787 0.925074i \(-0.375997\pi\)
0.379787 + 0.925074i \(0.375997\pi\)
\(294\) 0 0
\(295\) −29.9191 −1.74196
\(296\) 0 0
\(297\) 1.89655 0.110049
\(298\) 0 0
\(299\) 1.42690 0.0825199
\(300\) 0 0
\(301\) −1.35673 −0.0782004
\(302\) 0 0
\(303\) 2.64327 0.151852
\(304\) 0 0
\(305\) −31.2164 −1.78744
\(306\) 0 0
\(307\) −0.413802 −0.0236169 −0.0118085 0.999930i \(-0.503759\pi\)
−0.0118085 + 0.999930i \(0.503759\pi\)
\(308\) 0 0
\(309\) −2.20690 −0.125546
\(310\) 0 0
\(311\) 27.8746 1.58062 0.790311 0.612706i \(-0.209919\pi\)
0.790311 + 0.612706i \(0.209919\pi\)
\(312\) 0 0
\(313\) 14.4031 0.814111 0.407056 0.913403i \(-0.366555\pi\)
0.407056 + 0.913403i \(0.366555\pi\)
\(314\) 0 0
\(315\) −11.2865 −0.635925
\(316\) 0 0
\(317\) 4.97620 0.279491 0.139746 0.990187i \(-0.455372\pi\)
0.139746 + 0.990187i \(0.455372\pi\)
\(318\) 0 0
\(319\) 10.4364 0.584325
\(320\) 0 0
\(321\) 2.50655 0.139902
\(322\) 0 0
\(323\) −3.12725 −0.174005
\(324\) 0 0
\(325\) −2.22182 −0.123244
\(326\) 0 0
\(327\) −0.0225715 −0.00124821
\(328\) 0 0
\(329\) −5.57491 −0.307355
\(330\) 0 0
\(331\) −4.40310 −0.242016 −0.121008 0.992652i \(-0.538613\pi\)
−0.121008 + 0.992652i \(0.538613\pi\)
\(332\) 0 0
\(333\) 5.49345 0.301039
\(334\) 0 0
\(335\) 9.89655 0.540706
\(336\) 0 0
\(337\) 25.1498 1.37000 0.684999 0.728544i \(-0.259803\pi\)
0.684999 + 0.728544i \(0.259803\pi\)
\(338\) 0 0
\(339\) −5.91912 −0.321483
\(340\) 0 0
\(341\) −3.03509 −0.164359
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 8.19620 0.441269
\(346\) 0 0
\(347\) −3.72292 −0.199857 −0.0999285 0.994995i \(-0.531861\pi\)
−0.0999285 + 0.994995i \(0.531861\pi\)
\(348\) 0 0
\(349\) 9.73421 0.521060 0.260530 0.965466i \(-0.416103\pi\)
0.260530 + 0.965466i \(0.416103\pi\)
\(350\) 0 0
\(351\) −0.413802 −0.0220871
\(352\) 0 0
\(353\) −28.3555 −1.50921 −0.754605 0.656179i \(-0.772172\pi\)
−0.754605 + 0.656179i \(0.772172\pi\)
\(354\) 0 0
\(355\) −55.8489 −2.96415
\(356\) 0 0
\(357\) −1.56363 −0.0827559
\(358\) 0 0
\(359\) 4.07018 0.214816 0.107408 0.994215i \(-0.465745\pi\)
0.107408 + 0.994215i \(0.465745\pi\)
\(360\) 0 0
\(361\) −18.5862 −0.978221
\(362\) 0 0
\(363\) −0.321637 −0.0168816
\(364\) 0 0
\(365\) −34.5291 −1.80734
\(366\) 0 0
\(367\) −16.3442 −0.853161 −0.426580 0.904450i \(-0.640282\pi\)
−0.426580 + 0.904450i \(0.640282\pi\)
\(368\) 0 0
\(369\) 14.0815 0.733052
\(370\) 0 0
\(371\) −7.79310 −0.404598
\(372\) 0 0
\(373\) 27.0796 1.40213 0.701066 0.713097i \(-0.252708\pi\)
0.701066 + 0.713097i \(0.252708\pi\)
\(374\) 0 0
\(375\) −6.49585 −0.335444
\(376\) 0 0
\(377\) −2.27708 −0.117275
\(378\) 0 0
\(379\) 8.90602 0.457472 0.228736 0.973489i \(-0.426541\pi\)
0.228736 + 0.973489i \(0.426541\pi\)
\(380\) 0 0
\(381\) −6.78000 −0.347350
\(382\) 0 0
\(383\) −11.6118 −0.593336 −0.296668 0.954981i \(-0.595875\pi\)
−0.296668 + 0.954981i \(0.595875\pi\)
\(384\) 0 0
\(385\) 3.89655 0.198587
\(386\) 0 0
\(387\) 3.92982 0.199764
\(388\) 0 0
\(389\) −12.7693 −0.647429 −0.323715 0.946155i \(-0.604932\pi\)
−0.323715 + 0.946155i \(0.604932\pi\)
\(390\) 0 0
\(391\) −31.7931 −1.60785
\(392\) 0 0
\(393\) 6.36620 0.321132
\(394\) 0 0
\(395\) 51.2389 2.57811
\(396\) 0 0
\(397\) 24.2996 1.21956 0.609782 0.792569i \(-0.291257\pi\)
0.609782 + 0.792569i \(0.291257\pi\)
\(398\) 0 0
\(399\) 0.206901 0.0103580
\(400\) 0 0
\(401\) 12.7135 0.634879 0.317440 0.948278i \(-0.397177\pi\)
0.317440 + 0.948278i \(0.397177\pi\)
\(402\) 0 0
\(403\) 0.662216 0.0329873
\(404\) 0 0
\(405\) 31.4827 1.56439
\(406\) 0 0
\(407\) −1.89655 −0.0940085
\(408\) 0 0
\(409\) 6.72474 0.332517 0.166258 0.986082i \(-0.446831\pi\)
0.166258 + 0.986082i \(0.446831\pi\)
\(410\) 0 0
\(411\) 3.53035 0.174139
\(412\) 0 0
\(413\) −7.67836 −0.377828
\(414\) 0 0
\(415\) 37.8858 1.85974
\(416\) 0 0
\(417\) 5.22000 0.255625
\(418\) 0 0
\(419\) 24.2884 1.18656 0.593282 0.804995i \(-0.297832\pi\)
0.593282 + 0.804995i \(0.297832\pi\)
\(420\) 0 0
\(421\) −11.7931 −0.574760 −0.287380 0.957817i \(-0.592784\pi\)
−0.287380 + 0.957817i \(0.592784\pi\)
\(422\) 0 0
\(423\) 16.1480 0.785143
\(424\) 0 0
\(425\) 49.5047 2.40133
\(426\) 0 0
\(427\) −8.01129 −0.387693
\(428\) 0 0
\(429\) 0.0701770 0.00338818
\(430\) 0 0
\(431\) −10.7836 −0.519429 −0.259715 0.965685i \(-0.583628\pi\)
−0.259715 + 0.965685i \(0.583628\pi\)
\(432\) 0 0
\(433\) −25.4126 −1.22125 −0.610625 0.791920i \(-0.709082\pi\)
−0.610625 + 0.791920i \(0.709082\pi\)
\(434\) 0 0
\(435\) −13.0796 −0.627121
\(436\) 0 0
\(437\) 4.20690 0.201243
\(438\) 0 0
\(439\) −19.3567 −0.923846 −0.461923 0.886920i \(-0.652840\pi\)
−0.461923 + 0.886920i \(0.652840\pi\)
\(440\) 0 0
\(441\) −2.89655 −0.137931
\(442\) 0 0
\(443\) 22.9762 1.09163 0.545816 0.837905i \(-0.316220\pi\)
0.545816 + 0.837905i \(0.316220\pi\)
\(444\) 0 0
\(445\) −41.3424 −1.95982
\(446\) 0 0
\(447\) 4.78363 0.226258
\(448\) 0 0
\(449\) 4.26275 0.201171 0.100586 0.994928i \(-0.467928\pi\)
0.100586 + 0.994928i \(0.467928\pi\)
\(450\) 0 0
\(451\) −4.86146 −0.228917
\(452\) 0 0
\(453\) −4.22947 −0.198718
\(454\) 0 0
\(455\) −0.850175 −0.0398568
\(456\) 0 0
\(457\) 32.0927 1.50124 0.750618 0.660737i \(-0.229756\pi\)
0.750618 + 0.660737i \(0.229756\pi\)
\(458\) 0 0
\(459\) 9.22000 0.430353
\(460\) 0 0
\(461\) 17.7116 0.824913 0.412457 0.910977i \(-0.364671\pi\)
0.412457 + 0.910977i \(0.364671\pi\)
\(462\) 0 0
\(463\) 23.4126 1.08807 0.544037 0.839061i \(-0.316895\pi\)
0.544037 + 0.839061i \(0.316895\pi\)
\(464\) 0 0
\(465\) 3.80380 0.176397
\(466\) 0 0
\(467\) 25.1944 1.16586 0.582929 0.812523i \(-0.301907\pi\)
0.582929 + 0.812523i \(0.301907\pi\)
\(468\) 0 0
\(469\) 2.53982 0.117278
\(470\) 0 0
\(471\) −3.82637 −0.176310
\(472\) 0 0
\(473\) −1.35673 −0.0623823
\(474\) 0 0
\(475\) −6.55053 −0.300559
\(476\) 0 0
\(477\) 22.5731 1.03355
\(478\) 0 0
\(479\) 33.7229 1.54084 0.770420 0.637537i \(-0.220047\pi\)
0.770420 + 0.637537i \(0.220047\pi\)
\(480\) 0 0
\(481\) 0.413802 0.0188677
\(482\) 0 0
\(483\) 2.10345 0.0957103
\(484\) 0 0
\(485\) 2.90965 0.132120
\(486\) 0 0
\(487\) 24.4922 1.10985 0.554924 0.831901i \(-0.312747\pi\)
0.554924 + 0.831901i \(0.312747\pi\)
\(488\) 0 0
\(489\) −2.29965 −0.103994
\(490\) 0 0
\(491\) 0.136724 0.00617027 0.00308513 0.999995i \(-0.499018\pi\)
0.00308513 + 0.999995i \(0.499018\pi\)
\(492\) 0 0
\(493\) 50.7360 2.28504
\(494\) 0 0
\(495\) −11.2865 −0.507293
\(496\) 0 0
\(497\) −14.3329 −0.642919
\(498\) 0 0
\(499\) 4.29602 0.192316 0.0961581 0.995366i \(-0.469345\pi\)
0.0961581 + 0.995366i \(0.469345\pi\)
\(500\) 0 0
\(501\) −8.06655 −0.360387
\(502\) 0 0
\(503\) −39.4684 −1.75981 −0.879905 0.475150i \(-0.842394\pi\)
−0.879905 + 0.475150i \(0.842394\pi\)
\(504\) 0 0
\(505\) −32.0226 −1.42499
\(506\) 0 0
\(507\) 4.16597 0.185017
\(508\) 0 0
\(509\) −39.0464 −1.73070 −0.865350 0.501168i \(-0.832904\pi\)
−0.865350 + 0.501168i \(0.832904\pi\)
\(510\) 0 0
\(511\) −8.86146 −0.392008
\(512\) 0 0
\(513\) −1.22000 −0.0538644
\(514\) 0 0
\(515\) 26.7360 1.17813
\(516\) 0 0
\(517\) −5.57491 −0.245184
\(518\) 0 0
\(519\) 2.36983 0.104024
\(520\) 0 0
\(521\) −3.39363 −0.148678 −0.0743388 0.997233i \(-0.523685\pi\)
−0.0743388 + 0.997233i \(0.523685\pi\)
\(522\) 0 0
\(523\) −30.1593 −1.31877 −0.659387 0.751804i \(-0.729184\pi\)
−0.659387 + 0.751804i \(0.729184\pi\)
\(524\) 0 0
\(525\) −3.27526 −0.142944
\(526\) 0 0
\(527\) −14.7550 −0.642736
\(528\) 0 0
\(529\) 19.7693 0.859535
\(530\) 0 0
\(531\) 22.2408 0.965167
\(532\) 0 0
\(533\) 1.06071 0.0459443
\(534\) 0 0
\(535\) −30.3662 −1.31285
\(536\) 0 0
\(537\) −2.83584 −0.122376
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −17.7931 −0.764985 −0.382493 0.923959i \(-0.624934\pi\)
−0.382493 + 0.923959i \(0.624934\pi\)
\(542\) 0 0
\(543\) −5.60053 −0.240342
\(544\) 0 0
\(545\) 0.273448 0.0117132
\(546\) 0 0
\(547\) −16.7324 −0.715425 −0.357713 0.933832i \(-0.616443\pi\)
−0.357713 + 0.933832i \(0.616443\pi\)
\(548\) 0 0
\(549\) 23.2051 0.990369
\(550\) 0 0
\(551\) −6.71345 −0.286003
\(552\) 0 0
\(553\) 13.1498 0.559187
\(554\) 0 0
\(555\) 2.37690 0.100894
\(556\) 0 0
\(557\) −16.2996 −0.690638 −0.345319 0.938485i \(-0.612229\pi\)
−0.345319 + 0.938485i \(0.612229\pi\)
\(558\) 0 0
\(559\) 0.296019 0.0125203
\(560\) 0 0
\(561\) −1.56363 −0.0660164
\(562\) 0 0
\(563\) −6.13672 −0.258632 −0.129316 0.991603i \(-0.541278\pi\)
−0.129316 + 0.991603i \(0.541278\pi\)
\(564\) 0 0
\(565\) 71.7086 3.01680
\(566\) 0 0
\(567\) 8.07965 0.339313
\(568\) 0 0
\(569\) −29.1498 −1.22202 −0.611012 0.791621i \(-0.709237\pi\)
−0.611012 + 0.791621i \(0.709237\pi\)
\(570\) 0 0
\(571\) −12.8727 −0.538708 −0.269354 0.963041i \(-0.586810\pi\)
−0.269354 + 0.963041i \(0.586810\pi\)
\(572\) 0 0
\(573\) 3.18310 0.132976
\(574\) 0 0
\(575\) −66.5957 −2.77723
\(576\) 0 0
\(577\) 16.3329 0.679948 0.339974 0.940435i \(-0.389582\pi\)
0.339974 + 0.940435i \(0.389582\pi\)
\(578\) 0 0
\(579\) −5.65638 −0.235071
\(580\) 0 0
\(581\) 9.72292 0.403375
\(582\) 0 0
\(583\) −7.79310 −0.322757
\(584\) 0 0
\(585\) 2.46257 0.101815
\(586\) 0 0
\(587\) −26.4477 −1.09161 −0.545806 0.837912i \(-0.683776\pi\)
−0.545806 + 0.837912i \(0.683776\pi\)
\(588\) 0 0
\(589\) 1.95239 0.0804470
\(590\) 0 0
\(591\) 1.92982 0.0793823
\(592\) 0 0
\(593\) −26.5844 −1.09169 −0.545845 0.837886i \(-0.683791\pi\)
−0.545845 + 0.837886i \(0.683791\pi\)
\(594\) 0 0
\(595\) 18.9429 0.776584
\(596\) 0 0
\(597\) −7.22000 −0.295495
\(598\) 0 0
\(599\) 38.7360 1.58271 0.791355 0.611356i \(-0.209376\pi\)
0.791355 + 0.611356i \(0.209376\pi\)
\(600\) 0 0
\(601\) −14.4953 −0.591274 −0.295637 0.955300i \(-0.595532\pi\)
−0.295637 + 0.955300i \(0.595532\pi\)
\(602\) 0 0
\(603\) −7.35673 −0.299589
\(604\) 0 0
\(605\) 3.89655 0.158417
\(606\) 0 0
\(607\) 10.7800 0.437547 0.218773 0.975776i \(-0.429794\pi\)
0.218773 + 0.975776i \(0.429794\pi\)
\(608\) 0 0
\(609\) −3.35673 −0.136021
\(610\) 0 0
\(611\) 1.21637 0.0492091
\(612\) 0 0
\(613\) −43.1022 −1.74088 −0.870441 0.492273i \(-0.836166\pi\)
−0.870441 + 0.492273i \(0.836166\pi\)
\(614\) 0 0
\(615\) 6.09275 0.245683
\(616\) 0 0
\(617\) −4.93929 −0.198848 −0.0994242 0.995045i \(-0.531700\pi\)
−0.0994242 + 0.995045i \(0.531700\pi\)
\(618\) 0 0
\(619\) −3.88526 −0.156162 −0.0780810 0.996947i \(-0.524879\pi\)
−0.0780810 + 0.996947i \(0.524879\pi\)
\(620\) 0 0
\(621\) −12.4031 −0.497719
\(622\) 0 0
\(623\) −10.6100 −0.425081
\(624\) 0 0
\(625\) 27.7800 1.11120
\(626\) 0 0
\(627\) 0.206901 0.00826283
\(628\) 0 0
\(629\) −9.22000 −0.367626
\(630\) 0 0
\(631\) 10.9097 0.434306 0.217153 0.976138i \(-0.430323\pi\)
0.217153 + 0.976138i \(0.430323\pi\)
\(632\) 0 0
\(633\) −0.0665470 −0.00264501
\(634\) 0 0
\(635\) 82.1379 3.25954
\(636\) 0 0
\(637\) −0.218187 −0.00864487
\(638\) 0 0
\(639\) 41.5160 1.64235
\(640\) 0 0
\(641\) 14.1035 0.557053 0.278526 0.960429i \(-0.410154\pi\)
0.278526 + 0.960429i \(0.410154\pi\)
\(642\) 0 0
\(643\) −43.7009 −1.72340 −0.861698 0.507421i \(-0.830599\pi\)
−0.861698 + 0.507421i \(0.830599\pi\)
\(644\) 0 0
\(645\) 1.70035 0.0669512
\(646\) 0 0
\(647\) 2.04456 0.0803799 0.0401900 0.999192i \(-0.487204\pi\)
0.0401900 + 0.999192i \(0.487204\pi\)
\(648\) 0 0
\(649\) −7.67836 −0.301402
\(650\) 0 0
\(651\) 0.976197 0.0382602
\(652\) 0 0
\(653\) −6.67655 −0.261274 −0.130637 0.991430i \(-0.541702\pi\)
−0.130637 + 0.991430i \(0.541702\pi\)
\(654\) 0 0
\(655\) −77.1248 −3.01351
\(656\) 0 0
\(657\) 25.6677 1.00139
\(658\) 0 0
\(659\) −15.4495 −0.601826 −0.300913 0.953652i \(-0.597291\pi\)
−0.300913 + 0.953652i \(0.597291\pi\)
\(660\) 0 0
\(661\) 31.0689 1.20844 0.604221 0.796817i \(-0.293484\pi\)
0.604221 + 0.796817i \(0.293484\pi\)
\(662\) 0 0
\(663\) 0.341163 0.0132497
\(664\) 0 0
\(665\) −2.50655 −0.0971999
\(666\) 0 0
\(667\) −68.2520 −2.64273
\(668\) 0 0
\(669\) 7.23070 0.279555
\(670\) 0 0
\(671\) −8.01129 −0.309272
\(672\) 0 0
\(673\) 24.8026 0.956069 0.478034 0.878341i \(-0.341349\pi\)
0.478034 + 0.878341i \(0.341349\pi\)
\(674\) 0 0
\(675\) 19.3128 0.743348
\(676\) 0 0
\(677\) 31.0208 1.19222 0.596112 0.802901i \(-0.296711\pi\)
0.596112 + 0.802901i \(0.296711\pi\)
\(678\) 0 0
\(679\) 0.746725 0.0286567
\(680\) 0 0
\(681\) −5.01310 −0.192102
\(682\) 0 0
\(683\) 20.8727 0.798673 0.399337 0.916804i \(-0.369240\pi\)
0.399337 + 0.916804i \(0.369240\pi\)
\(684\) 0 0
\(685\) −42.7693 −1.63413
\(686\) 0 0
\(687\) 2.26638 0.0864676
\(688\) 0 0
\(689\) 1.70035 0.0647782
\(690\) 0 0
\(691\) 3.05766 0.116319 0.0581594 0.998307i \(-0.481477\pi\)
0.0581594 + 0.998307i \(0.481477\pi\)
\(692\) 0 0
\(693\) −2.89655 −0.110031
\(694\) 0 0
\(695\) −63.2389 −2.39879
\(696\) 0 0
\(697\) −23.6338 −0.895194
\(698\) 0 0
\(699\) 0.502920 0.0190222
\(700\) 0 0
\(701\) −23.5160 −0.888188 −0.444094 0.895980i \(-0.646474\pi\)
−0.444094 + 0.895980i \(0.646474\pi\)
\(702\) 0 0
\(703\) 1.22000 0.0460132
\(704\) 0 0
\(705\) 6.98690 0.263142
\(706\) 0 0
\(707\) −8.21819 −0.309077
\(708\) 0 0
\(709\) −12.1035 −0.454555 −0.227277 0.973830i \(-0.572982\pi\)
−0.227277 + 0.973830i \(0.572982\pi\)
\(710\) 0 0
\(711\) −38.0891 −1.42845
\(712\) 0 0
\(713\) 19.8489 0.743349
\(714\) 0 0
\(715\) −0.850175 −0.0317948
\(716\) 0 0
\(717\) −8.41380 −0.314219
\(718\) 0 0
\(719\) −26.0446 −0.971298 −0.485649 0.874154i \(-0.661417\pi\)
−0.485649 + 0.874154i \(0.661417\pi\)
\(720\) 0 0
\(721\) 6.86146 0.255534
\(722\) 0 0
\(723\) 5.35673 0.199219
\(724\) 0 0
\(725\) 106.275 3.94694
\(726\) 0 0
\(727\) −5.19438 −0.192649 −0.0963245 0.995350i \(-0.530709\pi\)
−0.0963245 + 0.995350i \(0.530709\pi\)
\(728\) 0 0
\(729\) −21.5731 −0.799004
\(730\) 0 0
\(731\) −6.59567 −0.243950
\(732\) 0 0
\(733\) 40.7949 1.50679 0.753397 0.657566i \(-0.228414\pi\)
0.753397 + 0.657566i \(0.228414\pi\)
\(734\) 0 0
\(735\) −1.25328 −0.0462278
\(736\) 0 0
\(737\) 2.53982 0.0935556
\(738\) 0 0
\(739\) −43.2389 −1.59057 −0.795285 0.606236i \(-0.792679\pi\)
−0.795285 + 0.606236i \(0.792679\pi\)
\(740\) 0 0
\(741\) −0.0451430 −0.00165837
\(742\) 0 0
\(743\) −31.1724 −1.14360 −0.571802 0.820392i \(-0.693756\pi\)
−0.571802 + 0.820392i \(0.693756\pi\)
\(744\) 0 0
\(745\) −57.9524 −2.12321
\(746\) 0 0
\(747\) −28.1629 −1.03043
\(748\) 0 0
\(749\) −7.79310 −0.284754
\(750\) 0 0
\(751\) −6.33292 −0.231092 −0.115546 0.993302i \(-0.536862\pi\)
−0.115546 + 0.993302i \(0.536862\pi\)
\(752\) 0 0
\(753\) 9.45655 0.344616
\(754\) 0 0
\(755\) 51.2389 1.86478
\(756\) 0 0
\(757\) 31.0131 1.12719 0.563595 0.826051i \(-0.309418\pi\)
0.563595 + 0.826051i \(0.309418\pi\)
\(758\) 0 0
\(759\) 2.10345 0.0763504
\(760\) 0 0
\(761\) −14.3549 −0.520365 −0.260183 0.965559i \(-0.583783\pi\)
−0.260183 + 0.965559i \(0.583783\pi\)
\(762\) 0 0
\(763\) 0.0701770 0.00254058
\(764\) 0 0
\(765\) −54.8691 −1.98380
\(766\) 0 0
\(767\) 1.67532 0.0604922
\(768\) 0 0
\(769\) −25.6451 −0.924786 −0.462393 0.886675i \(-0.653009\pi\)
−0.462393 + 0.886675i \(0.653009\pi\)
\(770\) 0 0
\(771\) −4.08912 −0.147266
\(772\) 0 0
\(773\) −23.1688 −0.833323 −0.416661 0.909062i \(-0.636800\pi\)
−0.416661 + 0.909062i \(0.636800\pi\)
\(774\) 0 0
\(775\) −30.9066 −1.11020
\(776\) 0 0
\(777\) 0.610001 0.0218837
\(778\) 0 0
\(779\) 3.12725 0.112045
\(780\) 0 0
\(781\) −14.3329 −0.512872
\(782\) 0 0
\(783\) 19.7931 0.707348
\(784\) 0 0
\(785\) 46.3555 1.65450
\(786\) 0 0
\(787\) −41.6564 −1.48489 −0.742445 0.669907i \(-0.766334\pi\)
−0.742445 + 0.669907i \(0.766334\pi\)
\(788\) 0 0
\(789\) −1.35310 −0.0481715
\(790\) 0 0
\(791\) 18.4031 0.654339
\(792\) 0 0
\(793\) 1.74796 0.0620717
\(794\) 0 0
\(795\) 9.76690 0.346396
\(796\) 0 0
\(797\) 19.0238 0.673858 0.336929 0.941530i \(-0.390612\pi\)
0.336929 + 0.941530i \(0.390612\pi\)
\(798\) 0 0
\(799\) −27.1022 −0.958808
\(800\) 0 0
\(801\) 30.7324 1.08588
\(802\) 0 0
\(803\) −8.86146 −0.312714
\(804\) 0 0
\(805\) −25.4827 −0.898149
\(806\) 0 0
\(807\) −1.19743 −0.0421515
\(808\) 0 0
\(809\) 23.0131 0.809098 0.404549 0.914516i \(-0.367429\pi\)
0.404549 + 0.914516i \(0.367429\pi\)
\(810\) 0 0
\(811\) 24.6658 0.866135 0.433067 0.901361i \(-0.357431\pi\)
0.433067 + 0.901361i \(0.357431\pi\)
\(812\) 0 0
\(813\) −1.84070 −0.0645563
\(814\) 0 0
\(815\) 27.8596 0.975881
\(816\) 0 0
\(817\) 0.872747 0.0305335
\(818\) 0 0
\(819\) 0.631989 0.0220835
\(820\) 0 0
\(821\) 44.6182 1.55719 0.778594 0.627528i \(-0.215933\pi\)
0.778594 + 0.627528i \(0.215933\pi\)
\(822\) 0 0
\(823\) 38.7693 1.35141 0.675706 0.737171i \(-0.263839\pi\)
0.675706 + 0.737171i \(0.263839\pi\)
\(824\) 0 0
\(825\) −3.27526 −0.114030
\(826\) 0 0
\(827\) 39.1022 1.35972 0.679859 0.733343i \(-0.262041\pi\)
0.679859 + 0.733343i \(0.262041\pi\)
\(828\) 0 0
\(829\) −3.68965 −0.128147 −0.0640734 0.997945i \(-0.520409\pi\)
−0.0640734 + 0.997945i \(0.520409\pi\)
\(830\) 0 0
\(831\) 4.02257 0.139541
\(832\) 0 0
\(833\) 4.86146 0.168440
\(834\) 0 0
\(835\) 97.7241 3.38188
\(836\) 0 0
\(837\) −5.75620 −0.198963
\(838\) 0 0
\(839\) −26.0446 −0.899158 −0.449579 0.893241i \(-0.648426\pi\)
−0.449579 + 0.893241i \(0.648426\pi\)
\(840\) 0 0
\(841\) 79.9179 2.75579
\(842\) 0 0
\(843\) −2.00363 −0.0690087
\(844\) 0 0
\(845\) −50.4696 −1.73621
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −5.42690 −0.186251
\(850\) 0 0
\(851\) 12.4031 0.425173
\(852\) 0 0
\(853\) 40.1742 1.37554 0.687769 0.725929i \(-0.258590\pi\)
0.687769 + 0.725929i \(0.258590\pi\)
\(854\) 0 0
\(855\) 7.26035 0.248299
\(856\) 0 0
\(857\) −7.64146 −0.261027 −0.130514 0.991447i \(-0.541663\pi\)
−0.130514 + 0.991447i \(0.541663\pi\)
\(858\) 0 0
\(859\) −4.11474 −0.140393 −0.0701965 0.997533i \(-0.522363\pi\)
−0.0701965 + 0.997533i \(0.522363\pi\)
\(860\) 0 0
\(861\) 1.56363 0.0532883
\(862\) 0 0
\(863\) 2.15930 0.0735033 0.0367516 0.999324i \(-0.488299\pi\)
0.0367516 + 0.999324i \(0.488299\pi\)
\(864\) 0 0
\(865\) −28.7098 −0.976163
\(866\) 0 0
\(867\) −2.13368 −0.0724635
\(868\) 0 0
\(869\) 13.1498 0.446077
\(870\) 0 0
\(871\) −0.554156 −0.0187769
\(872\) 0 0
\(873\) −2.16293 −0.0732039
\(874\) 0 0
\(875\) 20.1962 0.682756
\(876\) 0 0
\(877\) −27.8633 −0.940876 −0.470438 0.882433i \(-0.655904\pi\)
−0.470438 + 0.882433i \(0.655904\pi\)
\(878\) 0 0
\(879\) −4.18187 −0.141051
\(880\) 0 0
\(881\) −26.6991 −0.899516 −0.449758 0.893150i \(-0.648490\pi\)
−0.449758 + 0.893150i \(0.648490\pi\)
\(882\) 0 0
\(883\) −26.2996 −0.885054 −0.442527 0.896755i \(-0.645918\pi\)
−0.442527 + 0.896755i \(0.645918\pi\)
\(884\) 0 0
\(885\) 9.62310 0.323477
\(886\) 0 0
\(887\) 51.3567 1.72439 0.862195 0.506576i \(-0.169089\pi\)
0.862195 + 0.506576i \(0.169089\pi\)
\(888\) 0 0
\(889\) 21.0796 0.706989
\(890\) 0 0
\(891\) 8.07965 0.270678
\(892\) 0 0
\(893\) 3.58620 0.120008
\(894\) 0 0
\(895\) 34.3555 1.14838
\(896\) 0 0
\(897\) −0.458945 −0.0153237
\(898\) 0 0
\(899\) −31.6753 −1.05643
\(900\) 0 0
\(901\) −37.8858 −1.26216
\(902\) 0 0
\(903\) 0.436373 0.0145216
\(904\) 0 0
\(905\) 67.8489 2.25538
\(906\) 0 0
\(907\) 14.5957 0.484641 0.242321 0.970196i \(-0.422091\pi\)
0.242321 + 0.970196i \(0.422091\pi\)
\(908\) 0 0
\(909\) 23.8044 0.789542
\(910\) 0 0
\(911\) −27.0095 −0.894864 −0.447432 0.894318i \(-0.647661\pi\)
−0.447432 + 0.894318i \(0.647661\pi\)
\(912\) 0 0
\(913\) 9.72292 0.321782
\(914\) 0 0
\(915\) 10.0403 0.331924
\(916\) 0 0
\(917\) −19.7931 −0.653626
\(918\) 0 0
\(919\) 49.7455 1.64095 0.820476 0.571681i \(-0.193708\pi\)
0.820476 + 0.571681i \(0.193708\pi\)
\(920\) 0 0
\(921\) 0.133094 0.00438560
\(922\) 0 0
\(923\) 3.12725 0.102935
\(924\) 0 0
\(925\) −19.3128 −0.634999
\(926\) 0 0
\(927\) −19.8746 −0.652766
\(928\) 0 0
\(929\) −7.30912 −0.239804 −0.119902 0.992786i \(-0.538258\pi\)
−0.119902 + 0.992786i \(0.538258\pi\)
\(930\) 0 0
\(931\) −0.643274 −0.0210825
\(932\) 0 0
\(933\) −8.96550 −0.293517
\(934\) 0 0
\(935\) 18.9429 0.619500
\(936\) 0 0
\(937\) −39.5309 −1.29142 −0.645710 0.763583i \(-0.723438\pi\)
−0.645710 + 0.763583i \(0.723438\pi\)
\(938\) 0 0
\(939\) −4.63257 −0.151178
\(940\) 0 0
\(941\) −53.8971 −1.75700 −0.878498 0.477746i \(-0.841454\pi\)
−0.878498 + 0.477746i \(0.841454\pi\)
\(942\) 0 0
\(943\) 31.7931 1.03533
\(944\) 0 0
\(945\) 7.39000 0.240397
\(946\) 0 0
\(947\) 28.7217 0.933330 0.466665 0.884434i \(-0.345455\pi\)
0.466665 + 0.884434i \(0.345455\pi\)
\(948\) 0 0
\(949\) 1.93345 0.0627625
\(950\) 0 0
\(951\) −1.60053 −0.0519007
\(952\) 0 0
\(953\) 21.9335 0.710494 0.355247 0.934772i \(-0.384397\pi\)
0.355247 + 0.934772i \(0.384397\pi\)
\(954\) 0 0
\(955\) −38.5624 −1.24785
\(956\) 0 0
\(957\) −3.35673 −0.108508
\(958\) 0 0
\(959\) −10.9762 −0.354440
\(960\) 0 0
\(961\) −21.7882 −0.702846
\(962\) 0 0
\(963\) 22.5731 0.727408
\(964\) 0 0
\(965\) 68.5255 2.20591
\(966\) 0 0
\(967\) 40.7360 1.30998 0.654991 0.755637i \(-0.272672\pi\)
0.654991 + 0.755637i \(0.272672\pi\)
\(968\) 0 0
\(969\) 1.00584 0.0323123
\(970\) 0 0
\(971\) −38.0446 −1.22091 −0.610454 0.792052i \(-0.709013\pi\)
−0.610454 + 0.792052i \(0.709013\pi\)
\(972\) 0 0
\(973\) −16.2295 −0.520293
\(974\) 0 0
\(975\) 0.714619 0.0228861
\(976\) 0 0
\(977\) −37.2282 −1.19104 −0.595518 0.803342i \(-0.703053\pi\)
−0.595518 + 0.803342i \(0.703053\pi\)
\(978\) 0 0
\(979\) −10.6100 −0.339097
\(980\) 0 0
\(981\) −0.203271 −0.00648994
\(982\) 0 0
\(983\) −50.0446 −1.59617 −0.798087 0.602543i \(-0.794154\pi\)
−0.798087 + 0.602543i \(0.794154\pi\)
\(984\) 0 0
\(985\) −23.3793 −0.744926
\(986\) 0 0
\(987\) 1.79310 0.0570750
\(988\) 0 0
\(989\) 8.87275 0.282137
\(990\) 0 0
\(991\) 11.9774 0.380476 0.190238 0.981738i \(-0.439074\pi\)
0.190238 + 0.981738i \(0.439074\pi\)
\(992\) 0 0
\(993\) 1.41620 0.0449418
\(994\) 0 0
\(995\) 87.4684 2.77294
\(996\) 0 0
\(997\) 23.3906 0.740787 0.370394 0.928875i \(-0.379223\pi\)
0.370394 + 0.928875i \(0.379223\pi\)
\(998\) 0 0
\(999\) −3.59690 −0.113801
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 308.2.a.c.1.2 3
3.2 odd 2 2772.2.a.s.1.1 3
4.3 odd 2 1232.2.a.r.1.2 3
5.2 odd 4 7700.2.e.p.1849.4 6
5.3 odd 4 7700.2.e.p.1849.3 6
5.4 even 2 7700.2.a.y.1.2 3
7.2 even 3 2156.2.i.m.1145.2 6
7.3 odd 6 2156.2.i.k.177.2 6
7.4 even 3 2156.2.i.m.177.2 6
7.5 odd 6 2156.2.i.k.1145.2 6
7.6 odd 2 2156.2.a.j.1.2 3
8.3 odd 2 4928.2.a.bx.1.2 3
8.5 even 2 4928.2.a.ca.1.2 3
11.10 odd 2 3388.2.a.o.1.2 3
28.27 even 2 8624.2.a.cj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
308.2.a.c.1.2 3 1.1 even 1 trivial
1232.2.a.r.1.2 3 4.3 odd 2
2156.2.a.j.1.2 3 7.6 odd 2
2156.2.i.k.177.2 6 7.3 odd 6
2156.2.i.k.1145.2 6 7.5 odd 6
2156.2.i.m.177.2 6 7.4 even 3
2156.2.i.m.1145.2 6 7.2 even 3
2772.2.a.s.1.1 3 3.2 odd 2
3388.2.a.o.1.2 3 11.10 odd 2
4928.2.a.bx.1.2 3 8.3 odd 2
4928.2.a.ca.1.2 3 8.5 even 2
7700.2.a.y.1.2 3 5.4 even 2
7700.2.e.p.1849.3 6 5.3 odd 4
7700.2.e.p.1849.4 6 5.2 odd 4
8624.2.a.cj.1.2 3 28.27 even 2