Properties

Label 9196.2.a.n.1.5
Level $9196$
Weight $2$
Character 9196.1
Self dual yes
Analytic conductor $73.430$
Analytic rank $0$
Dimension $6$
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] = [9196,2,Mod(1,9196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9196, 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("9196.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9196.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: \(73.4304296988\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.744786576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 836)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.82559\) of defining polynomial
Character \(\chi\) \(=\) 9196.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} +O(q^{10})\) \(q+2.82559 q^{3} -0.343563 q^{5} -4.77460 q^{7} +4.98394 q^{9} -1.86340 q^{13} -0.970768 q^{15} +5.69469 q^{17} +1.00000 q^{19} -13.4910 q^{21} -1.06385 q^{23} -4.88196 q^{25} +5.60580 q^{27} +9.31093 q^{29} +2.97077 q^{31} +1.64038 q^{35} -3.54764 q^{37} -5.26521 q^{39} -2.82372 q^{41} -8.09454 q^{43} -1.71230 q^{45} +3.00756 q^{47} +15.7968 q^{49} +16.0908 q^{51} -4.33830 q^{53} +2.82559 q^{57} +14.6566 q^{59} -1.03595 q^{61} -23.7963 q^{63} +0.640197 q^{65} +7.93109 q^{67} -3.00601 q^{69} -11.4107 q^{71} +16.3062 q^{73} -13.7944 q^{75} +13.3459 q^{79} +0.887845 q^{81} +3.40742 q^{83} -1.95649 q^{85} +26.3088 q^{87} +0.0562918 q^{89} +8.89700 q^{91} +8.39416 q^{93} -0.343563 q^{95} +11.4778 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} - 8 q^{13} + 9 q^{15} + 2 q^{17} + 6 q^{19} - 18 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} - 10 q^{29} + 3 q^{31} + 4 q^{35} + 7 q^{37} + 10 q^{39} - 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + q^{57} + 15 q^{59} - 24 q^{61} + 20 q^{63} + 28 q^{65} + 25 q^{67} - 33 q^{69} - 9 q^{71} + 26 q^{73} + 28 q^{75} + 16 q^{79} + 58 q^{81} + 2 q^{83} - 12 q^{85} + 36 q^{87} + 7 q^{89} + 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97}+O(q^{100}) \) 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
\(3\) 2.82559 1.63135 0.815677 0.578508i \(-0.196365\pi\)
0.815677 + 0.578508i \(0.196365\pi\)
\(4\) 0 0
\(5\) −0.343563 −0.153646 −0.0768231 0.997045i \(-0.524478\pi\)
−0.0768231 + 0.997045i \(0.524478\pi\)
\(6\) 0 0
\(7\) −4.77460 −1.80463 −0.902314 0.431079i \(-0.858133\pi\)
−0.902314 + 0.431079i \(0.858133\pi\)
\(8\) 0 0
\(9\) 4.98394 1.66131
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.86340 −0.516815 −0.258407 0.966036i \(-0.583198\pi\)
−0.258407 + 0.966036i \(0.583198\pi\)
\(14\) 0 0
\(15\) −0.970768 −0.250651
\(16\) 0 0
\(17\) 5.69469 1.38116 0.690582 0.723254i \(-0.257354\pi\)
0.690582 + 0.723254i \(0.257354\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −13.4910 −2.94399
\(22\) 0 0
\(23\) −1.06385 −0.221829 −0.110914 0.993830i \(-0.535378\pi\)
−0.110914 + 0.993830i \(0.535378\pi\)
\(24\) 0 0
\(25\) −4.88196 −0.976393
\(26\) 0 0
\(27\) 5.60580 1.07884
\(28\) 0 0
\(29\) 9.31093 1.72900 0.864498 0.502636i \(-0.167636\pi\)
0.864498 + 0.502636i \(0.167636\pi\)
\(30\) 0 0
\(31\) 2.97077 0.533566 0.266783 0.963757i \(-0.414039\pi\)
0.266783 + 0.963757i \(0.414039\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.64038 0.277274
\(36\) 0 0
\(37\) −3.54764 −0.583229 −0.291614 0.956536i \(-0.594192\pi\)
−0.291614 + 0.956536i \(0.594192\pi\)
\(38\) 0 0
\(39\) −5.26521 −0.843108
\(40\) 0 0
\(41\) −2.82372 −0.440992 −0.220496 0.975388i \(-0.570768\pi\)
−0.220496 + 0.975388i \(0.570768\pi\)
\(42\) 0 0
\(43\) −8.09454 −1.23441 −0.617203 0.786804i \(-0.711734\pi\)
−0.617203 + 0.786804i \(0.711734\pi\)
\(44\) 0 0
\(45\) −1.71230 −0.255255
\(46\) 0 0
\(47\) 3.00756 0.438698 0.219349 0.975646i \(-0.429607\pi\)
0.219349 + 0.975646i \(0.429607\pi\)
\(48\) 0 0
\(49\) 15.7968 2.25668
\(50\) 0 0
\(51\) 16.0908 2.25317
\(52\) 0 0
\(53\) −4.33830 −0.595911 −0.297956 0.954580i \(-0.596305\pi\)
−0.297956 + 0.954580i \(0.596305\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.82559 0.374258
\(58\) 0 0
\(59\) 14.6566 1.90812 0.954061 0.299613i \(-0.0968574\pi\)
0.954061 + 0.299613i \(0.0968574\pi\)
\(60\) 0 0
\(61\) −1.03595 −0.132640 −0.0663201 0.997798i \(-0.521126\pi\)
−0.0663201 + 0.997798i \(0.521126\pi\)
\(62\) 0 0
\(63\) −23.7963 −2.99805
\(64\) 0 0
\(65\) 0.640197 0.0794067
\(66\) 0 0
\(67\) 7.93109 0.968937 0.484468 0.874809i \(-0.339013\pi\)
0.484468 + 0.874809i \(0.339013\pi\)
\(68\) 0 0
\(69\) −3.00601 −0.361881
\(70\) 0 0
\(71\) −11.4107 −1.35421 −0.677103 0.735889i \(-0.736765\pi\)
−0.677103 + 0.735889i \(0.736765\pi\)
\(72\) 0 0
\(73\) 16.3062 1.90849 0.954247 0.299019i \(-0.0966594\pi\)
0.954247 + 0.299019i \(0.0966594\pi\)
\(74\) 0 0
\(75\) −13.7944 −1.59284
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.3459 1.50153 0.750763 0.660571i \(-0.229686\pi\)
0.750763 + 0.660571i \(0.229686\pi\)
\(80\) 0 0
\(81\) 0.887845 0.0986494
\(82\) 0 0
\(83\) 3.40742 0.374012 0.187006 0.982359i \(-0.440122\pi\)
0.187006 + 0.982359i \(0.440122\pi\)
\(84\) 0 0
\(85\) −1.95649 −0.212211
\(86\) 0 0
\(87\) 26.3088 2.82060
\(88\) 0 0
\(89\) 0.0562918 0.00596692 0.00298346 0.999996i \(-0.499050\pi\)
0.00298346 + 0.999996i \(0.499050\pi\)
\(90\) 0 0
\(91\) 8.89700 0.932659
\(92\) 0 0
\(93\) 8.39416 0.870434
\(94\) 0 0
\(95\) −0.343563 −0.0352489
\(96\) 0 0
\(97\) 11.4778 1.16539 0.582696 0.812690i \(-0.301998\pi\)
0.582696 + 0.812690i \(0.301998\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.64361 −0.462057 −0.231028 0.972947i \(-0.574209\pi\)
−0.231028 + 0.972947i \(0.574209\pi\)
\(102\) 0 0
\(103\) 9.73334 0.959055 0.479527 0.877527i \(-0.340808\pi\)
0.479527 + 0.877527i \(0.340808\pi\)
\(104\) 0 0
\(105\) 4.63503 0.452333
\(106\) 0 0
\(107\) 9.06541 0.876386 0.438193 0.898881i \(-0.355619\pi\)
0.438193 + 0.898881i \(0.355619\pi\)
\(108\) 0 0
\(109\) −5.74943 −0.550695 −0.275348 0.961345i \(-0.588793\pi\)
−0.275348 + 0.961345i \(0.588793\pi\)
\(110\) 0 0
\(111\) −10.0242 −0.951452
\(112\) 0 0
\(113\) 2.68930 0.252988 0.126494 0.991967i \(-0.459628\pi\)
0.126494 + 0.991967i \(0.459628\pi\)
\(114\) 0 0
\(115\) 0.365501 0.0340831
\(116\) 0 0
\(117\) −9.28709 −0.858592
\(118\) 0 0
\(119\) −27.1898 −2.49249
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −7.97868 −0.719413
\(124\) 0 0
\(125\) 3.39508 0.303665
\(126\) 0 0
\(127\) 13.1746 1.16906 0.584530 0.811372i \(-0.301279\pi\)
0.584530 + 0.811372i \(0.301279\pi\)
\(128\) 0 0
\(129\) −22.8718 −2.01375
\(130\) 0 0
\(131\) −19.6659 −1.71822 −0.859110 0.511791i \(-0.828982\pi\)
−0.859110 + 0.511791i \(0.828982\pi\)
\(132\) 0 0
\(133\) −4.77460 −0.414010
\(134\) 0 0
\(135\) −1.92595 −0.165759
\(136\) 0 0
\(137\) 9.00937 0.769722 0.384861 0.922974i \(-0.374249\pi\)
0.384861 + 0.922974i \(0.374249\pi\)
\(138\) 0 0
\(139\) 19.8943 1.68741 0.843707 0.536804i \(-0.180369\pi\)
0.843707 + 0.536804i \(0.180369\pi\)
\(140\) 0 0
\(141\) 8.49812 0.715671
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3.19890 −0.265654
\(146\) 0 0
\(147\) 44.6352 3.68145
\(148\) 0 0
\(149\) 13.0292 1.06739 0.533695 0.845677i \(-0.320803\pi\)
0.533695 + 0.845677i \(0.320803\pi\)
\(150\) 0 0
\(151\) 1.73820 0.141453 0.0707264 0.997496i \(-0.477468\pi\)
0.0707264 + 0.997496i \(0.477468\pi\)
\(152\) 0 0
\(153\) 28.3820 2.29455
\(154\) 0 0
\(155\) −1.02065 −0.0819804
\(156\) 0 0
\(157\) −6.95506 −0.555074 −0.277537 0.960715i \(-0.589518\pi\)
−0.277537 + 0.960715i \(0.589518\pi\)
\(158\) 0 0
\(159\) −12.2582 −0.972142
\(160\) 0 0
\(161\) 5.07947 0.400318
\(162\) 0 0
\(163\) 14.7201 1.15297 0.576484 0.817108i \(-0.304424\pi\)
0.576484 + 0.817108i \(0.304424\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.95276 0.460638 0.230319 0.973115i \(-0.426023\pi\)
0.230319 + 0.973115i \(0.426023\pi\)
\(168\) 0 0
\(169\) −9.52773 −0.732902
\(170\) 0 0
\(171\) 4.98394 0.381131
\(172\) 0 0
\(173\) 0.925392 0.0703563 0.0351781 0.999381i \(-0.488800\pi\)
0.0351781 + 0.999381i \(0.488800\pi\)
\(174\) 0 0
\(175\) 23.3094 1.76203
\(176\) 0 0
\(177\) 41.4134 3.11282
\(178\) 0 0
\(179\) 18.9602 1.41716 0.708578 0.705633i \(-0.249337\pi\)
0.708578 + 0.705633i \(0.249337\pi\)
\(180\) 0 0
\(181\) 6.27828 0.466661 0.233331 0.972397i \(-0.425038\pi\)
0.233331 + 0.972397i \(0.425038\pi\)
\(182\) 0 0
\(183\) −2.92718 −0.216383
\(184\) 0 0
\(185\) 1.21884 0.0896109
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −26.7654 −1.94690
\(190\) 0 0
\(191\) −18.2273 −1.31888 −0.659440 0.751758i \(-0.729206\pi\)
−0.659440 + 0.751758i \(0.729206\pi\)
\(192\) 0 0
\(193\) 5.51799 0.397194 0.198597 0.980081i \(-0.436362\pi\)
0.198597 + 0.980081i \(0.436362\pi\)
\(194\) 0 0
\(195\) 1.80893 0.129540
\(196\) 0 0
\(197\) −24.5897 −1.75195 −0.875973 0.482359i \(-0.839780\pi\)
−0.875973 + 0.482359i \(0.839780\pi\)
\(198\) 0 0
\(199\) −17.1425 −1.21520 −0.607601 0.794243i \(-0.707868\pi\)
−0.607601 + 0.794243i \(0.707868\pi\)
\(200\) 0 0
\(201\) 22.4100 1.58068
\(202\) 0 0
\(203\) −44.4560 −3.12020
\(204\) 0 0
\(205\) 0.970128 0.0677567
\(206\) 0 0
\(207\) −5.30218 −0.368527
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.10198 −0.282392 −0.141196 0.989982i \(-0.545095\pi\)
−0.141196 + 0.989982i \(0.545095\pi\)
\(212\) 0 0
\(213\) −32.2420 −2.20919
\(214\) 0 0
\(215\) 2.78099 0.189662
\(216\) 0 0
\(217\) −14.1842 −0.962888
\(218\) 0 0
\(219\) 46.0745 3.11343
\(220\) 0 0
\(221\) −10.6115 −0.713806
\(222\) 0 0
\(223\) 26.2530 1.75803 0.879015 0.476793i \(-0.158201\pi\)
0.879015 + 0.476793i \(0.158201\pi\)
\(224\) 0 0
\(225\) −24.3314 −1.62209
\(226\) 0 0
\(227\) 2.54225 0.168735 0.0843677 0.996435i \(-0.473113\pi\)
0.0843677 + 0.996435i \(0.473113\pi\)
\(228\) 0 0
\(229\) −7.23905 −0.478370 −0.239185 0.970974i \(-0.576880\pi\)
−0.239185 + 0.970974i \(0.576880\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.6807 −1.61689 −0.808444 0.588573i \(-0.799690\pi\)
−0.808444 + 0.588573i \(0.799690\pi\)
\(234\) 0 0
\(235\) −1.03329 −0.0674043
\(236\) 0 0
\(237\) 37.7099 2.44952
\(238\) 0 0
\(239\) 22.3130 1.44331 0.721654 0.692254i \(-0.243382\pi\)
0.721654 + 0.692254i \(0.243382\pi\)
\(240\) 0 0
\(241\) −9.08973 −0.585521 −0.292760 0.956186i \(-0.594574\pi\)
−0.292760 + 0.956186i \(0.594574\pi\)
\(242\) 0 0
\(243\) −14.3087 −0.917904
\(244\) 0 0
\(245\) −5.42720 −0.346731
\(246\) 0 0
\(247\) −1.86340 −0.118565
\(248\) 0 0
\(249\) 9.62795 0.610146
\(250\) 0 0
\(251\) −2.11865 −0.133728 −0.0668640 0.997762i \(-0.521299\pi\)
−0.0668640 + 0.997762i \(0.521299\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −5.52822 −0.346191
\(256\) 0 0
\(257\) −4.83262 −0.301450 −0.150725 0.988576i \(-0.548161\pi\)
−0.150725 + 0.988576i \(0.548161\pi\)
\(258\) 0 0
\(259\) 16.9386 1.05251
\(260\) 0 0
\(261\) 46.4051 2.87241
\(262\) 0 0
\(263\) −10.7931 −0.665529 −0.332764 0.943010i \(-0.607981\pi\)
−0.332764 + 0.943010i \(0.607981\pi\)
\(264\) 0 0
\(265\) 1.49048 0.0915596
\(266\) 0 0
\(267\) 0.159057 0.00973415
\(268\) 0 0
\(269\) −16.0956 −0.981365 −0.490683 0.871338i \(-0.663253\pi\)
−0.490683 + 0.871338i \(0.663253\pi\)
\(270\) 0 0
\(271\) 22.2262 1.35014 0.675072 0.737752i \(-0.264113\pi\)
0.675072 + 0.737752i \(0.264113\pi\)
\(272\) 0 0
\(273\) 25.1392 1.52150
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.12740 0.0677392 0.0338696 0.999426i \(-0.489217\pi\)
0.0338696 + 0.999426i \(0.489217\pi\)
\(278\) 0 0
\(279\) 14.8061 0.886420
\(280\) 0 0
\(281\) −1.51489 −0.0903705 −0.0451852 0.998979i \(-0.514388\pi\)
−0.0451852 + 0.998979i \(0.514388\pi\)
\(282\) 0 0
\(283\) −5.74917 −0.341753 −0.170876 0.985292i \(-0.554660\pi\)
−0.170876 + 0.985292i \(0.554660\pi\)
\(284\) 0 0
\(285\) −0.970768 −0.0575034
\(286\) 0 0
\(287\) 13.4821 0.795826
\(288\) 0 0
\(289\) 15.4295 0.907615
\(290\) 0 0
\(291\) 32.4315 1.90117
\(292\) 0 0
\(293\) 0.00163269 9.53826e−5 0 4.76913e−5 1.00000i \(-0.499985\pi\)
4.76913e−5 1.00000i \(0.499985\pi\)
\(294\) 0 0
\(295\) −5.03546 −0.293176
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.98238 0.114644
\(300\) 0 0
\(301\) 38.6482 2.22764
\(302\) 0 0
\(303\) −13.1209 −0.753778
\(304\) 0 0
\(305\) 0.355916 0.0203797
\(306\) 0 0
\(307\) 27.0663 1.54475 0.772377 0.635164i \(-0.219068\pi\)
0.772377 + 0.635164i \(0.219068\pi\)
\(308\) 0 0
\(309\) 27.5024 1.56456
\(310\) 0 0
\(311\) 21.8440 1.23866 0.619329 0.785131i \(-0.287405\pi\)
0.619329 + 0.785131i \(0.287405\pi\)
\(312\) 0 0
\(313\) −4.54317 −0.256795 −0.128398 0.991723i \(-0.540983\pi\)
−0.128398 + 0.991723i \(0.540983\pi\)
\(314\) 0 0
\(315\) 8.17554 0.460640
\(316\) 0 0
\(317\) −8.22001 −0.461682 −0.230841 0.972992i \(-0.574148\pi\)
−0.230841 + 0.972992i \(0.574148\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 25.6151 1.42969
\(322\) 0 0
\(323\) 5.69469 0.316861
\(324\) 0 0
\(325\) 9.09706 0.504614
\(326\) 0 0
\(327\) −16.2455 −0.898378
\(328\) 0 0
\(329\) −14.3599 −0.791687
\(330\) 0 0
\(331\) −10.5060 −0.577463 −0.288732 0.957410i \(-0.593234\pi\)
−0.288732 + 0.957410i \(0.593234\pi\)
\(332\) 0 0
\(333\) −17.6812 −0.968926
\(334\) 0 0
\(335\) −2.72483 −0.148874
\(336\) 0 0
\(337\) −21.5797 −1.17552 −0.587760 0.809035i \(-0.699990\pi\)
−0.587760 + 0.809035i \(0.699990\pi\)
\(338\) 0 0
\(339\) 7.59885 0.412713
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −42.0011 −2.26785
\(344\) 0 0
\(345\) 1.03275 0.0556016
\(346\) 0 0
\(347\) 21.5171 1.15510 0.577549 0.816356i \(-0.304009\pi\)
0.577549 + 0.816356i \(0.304009\pi\)
\(348\) 0 0
\(349\) −23.7204 −1.26972 −0.634862 0.772625i \(-0.718943\pi\)
−0.634862 + 0.772625i \(0.718943\pi\)
\(350\) 0 0
\(351\) −10.4459 −0.557559
\(352\) 0 0
\(353\) 13.4757 0.717241 0.358621 0.933483i \(-0.383247\pi\)
0.358621 + 0.933483i \(0.383247\pi\)
\(354\) 0 0
\(355\) 3.92031 0.208069
\(356\) 0 0
\(357\) −76.8273 −4.06613
\(358\) 0 0
\(359\) 14.4579 0.763058 0.381529 0.924357i \(-0.375398\pi\)
0.381529 + 0.924357i \(0.375398\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.60221 −0.293233
\(366\) 0 0
\(367\) −0.692406 −0.0361433 −0.0180716 0.999837i \(-0.505753\pi\)
−0.0180716 + 0.999837i \(0.505753\pi\)
\(368\) 0 0
\(369\) −14.0733 −0.732625
\(370\) 0 0
\(371\) 20.7136 1.07540
\(372\) 0 0
\(373\) −15.2174 −0.787929 −0.393964 0.919126i \(-0.628897\pi\)
−0.393964 + 0.919126i \(0.628897\pi\)
\(374\) 0 0
\(375\) 9.59310 0.495385
\(376\) 0 0
\(377\) −17.3500 −0.893571
\(378\) 0 0
\(379\) 30.7668 1.58039 0.790193 0.612859i \(-0.209980\pi\)
0.790193 + 0.612859i \(0.209980\pi\)
\(380\) 0 0
\(381\) 37.2261 1.90715
\(382\) 0 0
\(383\) 31.1031 1.58929 0.794646 0.607073i \(-0.207656\pi\)
0.794646 + 0.607073i \(0.207656\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −40.3427 −2.05074
\(388\) 0 0
\(389\) 16.8274 0.853181 0.426590 0.904445i \(-0.359715\pi\)
0.426590 + 0.904445i \(0.359715\pi\)
\(390\) 0 0
\(391\) −6.05831 −0.306382
\(392\) 0 0
\(393\) −55.5678 −2.80302
\(394\) 0 0
\(395\) −4.58515 −0.230704
\(396\) 0 0
\(397\) 18.3895 0.922943 0.461472 0.887155i \(-0.347322\pi\)
0.461472 + 0.887155i \(0.347322\pi\)
\(398\) 0 0
\(399\) −13.4910 −0.675397
\(400\) 0 0
\(401\) −18.4034 −0.919022 −0.459511 0.888172i \(-0.651975\pi\)
−0.459511 + 0.888172i \(0.651975\pi\)
\(402\) 0 0
\(403\) −5.53574 −0.275755
\(404\) 0 0
\(405\) −0.305031 −0.0151571
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 39.3705 1.94675 0.973373 0.229225i \(-0.0736193\pi\)
0.973373 + 0.229225i \(0.0736193\pi\)
\(410\) 0 0
\(411\) 25.4568 1.25569
\(412\) 0 0
\(413\) −69.9792 −3.44345
\(414\) 0 0
\(415\) −1.17066 −0.0574656
\(416\) 0 0
\(417\) 56.2131 2.75277
\(418\) 0 0
\(419\) 17.2402 0.842237 0.421118 0.907006i \(-0.361638\pi\)
0.421118 + 0.907006i \(0.361638\pi\)
\(420\) 0 0
\(421\) 7.86946 0.383534 0.191767 0.981440i \(-0.438578\pi\)
0.191767 + 0.981440i \(0.438578\pi\)
\(422\) 0 0
\(423\) 14.9895 0.728815
\(424\) 0 0
\(425\) −27.8013 −1.34856
\(426\) 0 0
\(427\) 4.94626 0.239366
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.9971 1.05957 0.529783 0.848133i \(-0.322274\pi\)
0.529783 + 0.848133i \(0.322274\pi\)
\(432\) 0 0
\(433\) 24.7111 1.18754 0.593771 0.804634i \(-0.297639\pi\)
0.593771 + 0.804634i \(0.297639\pi\)
\(434\) 0 0
\(435\) −9.03876 −0.433375
\(436\) 0 0
\(437\) −1.06385 −0.0508909
\(438\) 0 0
\(439\) −27.3096 −1.30342 −0.651708 0.758470i \(-0.725947\pi\)
−0.651708 + 0.758470i \(0.725947\pi\)
\(440\) 0 0
\(441\) 78.7303 3.74906
\(442\) 0 0
\(443\) −38.0147 −1.80613 −0.903066 0.429501i \(-0.858689\pi\)
−0.903066 + 0.429501i \(0.858689\pi\)
\(444\) 0 0
\(445\) −0.0193398 −0.000916795 0
\(446\) 0 0
\(447\) 36.8150 1.74129
\(448\) 0 0
\(449\) 39.1470 1.84746 0.923730 0.383043i \(-0.125124\pi\)
0.923730 + 0.383043i \(0.125124\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.91144 0.230759
\(454\) 0 0
\(455\) −3.05668 −0.143300
\(456\) 0 0
\(457\) −4.72692 −0.221116 −0.110558 0.993870i \(-0.535264\pi\)
−0.110558 + 0.993870i \(0.535264\pi\)
\(458\) 0 0
\(459\) 31.9233 1.49005
\(460\) 0 0
\(461\) 11.1666 0.520079 0.260040 0.965598i \(-0.416264\pi\)
0.260040 + 0.965598i \(0.416264\pi\)
\(462\) 0 0
\(463\) 30.5320 1.41894 0.709470 0.704735i \(-0.248934\pi\)
0.709470 + 0.704735i \(0.248934\pi\)
\(464\) 0 0
\(465\) −2.88393 −0.133739
\(466\) 0 0
\(467\) 11.2216 0.519274 0.259637 0.965706i \(-0.416397\pi\)
0.259637 + 0.965706i \(0.416397\pi\)
\(468\) 0 0
\(469\) −37.8678 −1.74857
\(470\) 0 0
\(471\) −19.6521 −0.905522
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.88196 −0.224000
\(476\) 0 0
\(477\) −21.6218 −0.989996
\(478\) 0 0
\(479\) −15.0838 −0.689197 −0.344598 0.938750i \(-0.611985\pi\)
−0.344598 + 0.938750i \(0.611985\pi\)
\(480\) 0 0
\(481\) 6.61069 0.301421
\(482\) 0 0
\(483\) 14.3525 0.653060
\(484\) 0 0
\(485\) −3.94335 −0.179058
\(486\) 0 0
\(487\) −24.3598 −1.10385 −0.551925 0.833894i \(-0.686106\pi\)
−0.551925 + 0.833894i \(0.686106\pi\)
\(488\) 0 0
\(489\) 41.5930 1.88090
\(490\) 0 0
\(491\) −15.4105 −0.695467 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(492\) 0 0
\(493\) 53.0228 2.38803
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 54.4817 2.44384
\(498\) 0 0
\(499\) 11.7733 0.527043 0.263522 0.964653i \(-0.415116\pi\)
0.263522 + 0.964653i \(0.415116\pi\)
\(500\) 0 0
\(501\) 16.8200 0.751464
\(502\) 0 0
\(503\) 21.6073 0.963421 0.481710 0.876330i \(-0.340016\pi\)
0.481710 + 0.876330i \(0.340016\pi\)
\(504\) 0 0
\(505\) 1.59538 0.0709933
\(506\) 0 0
\(507\) −26.9214 −1.19562
\(508\) 0 0
\(509\) −30.4087 −1.34784 −0.673921 0.738804i \(-0.735391\pi\)
−0.673921 + 0.738804i \(0.735391\pi\)
\(510\) 0 0
\(511\) −77.8555 −3.44412
\(512\) 0 0
\(513\) 5.60580 0.247502
\(514\) 0 0
\(515\) −3.34402 −0.147355
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.61478 0.114776
\(520\) 0 0
\(521\) 25.2167 1.10476 0.552381 0.833591i \(-0.313719\pi\)
0.552381 + 0.833591i \(0.313719\pi\)
\(522\) 0 0
\(523\) −27.9677 −1.22294 −0.611472 0.791266i \(-0.709422\pi\)
−0.611472 + 0.791266i \(0.709422\pi\)
\(524\) 0 0
\(525\) 65.8628 2.87449
\(526\) 0 0
\(527\) 16.9176 0.736942
\(528\) 0 0
\(529\) −21.8682 −0.950792
\(530\) 0 0
\(531\) 73.0474 3.16999
\(532\) 0 0
\(533\) 5.26174 0.227911
\(534\) 0 0
\(535\) −3.11454 −0.134653
\(536\) 0 0
\(537\) 53.5738 2.31188
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −28.8814 −1.24171 −0.620855 0.783926i \(-0.713214\pi\)
−0.620855 + 0.783926i \(0.713214\pi\)
\(542\) 0 0
\(543\) 17.7398 0.761289
\(544\) 0 0
\(545\) 1.97529 0.0846122
\(546\) 0 0
\(547\) −16.4332 −0.702632 −0.351316 0.936257i \(-0.614266\pi\)
−0.351316 + 0.936257i \(0.614266\pi\)
\(548\) 0 0
\(549\) −5.16313 −0.220357
\(550\) 0 0
\(551\) 9.31093 0.396659
\(552\) 0 0
\(553\) −63.7211 −2.70970
\(554\) 0 0
\(555\) 3.44394 0.146187
\(556\) 0 0
\(557\) −0.814831 −0.0345255 −0.0172628 0.999851i \(-0.505495\pi\)
−0.0172628 + 0.999851i \(0.505495\pi\)
\(558\) 0 0
\(559\) 15.0834 0.637960
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0662 −1.01427 −0.507134 0.861867i \(-0.669295\pi\)
−0.507134 + 0.861867i \(0.669295\pi\)
\(564\) 0 0
\(565\) −0.923945 −0.0388706
\(566\) 0 0
\(567\) −4.23910 −0.178026
\(568\) 0 0
\(569\) 21.5082 0.901670 0.450835 0.892607i \(-0.351126\pi\)
0.450835 + 0.892607i \(0.351126\pi\)
\(570\) 0 0
\(571\) −14.6812 −0.614388 −0.307194 0.951647i \(-0.599390\pi\)
−0.307194 + 0.951647i \(0.599390\pi\)
\(572\) 0 0
\(573\) −51.5027 −2.15156
\(574\) 0 0
\(575\) 5.19369 0.216592
\(576\) 0 0
\(577\) −22.0271 −0.917001 −0.458501 0.888694i \(-0.651613\pi\)
−0.458501 + 0.888694i \(0.651613\pi\)
\(578\) 0 0
\(579\) 15.5916 0.647963
\(580\) 0 0
\(581\) −16.2690 −0.674954
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.19070 0.131919
\(586\) 0 0
\(587\) 27.9648 1.15423 0.577116 0.816663i \(-0.304178\pi\)
0.577116 + 0.816663i \(0.304178\pi\)
\(588\) 0 0
\(589\) 2.97077 0.122408
\(590\) 0 0
\(591\) −69.4805 −2.85804
\(592\) 0 0
\(593\) 14.4726 0.594317 0.297159 0.954828i \(-0.403961\pi\)
0.297159 + 0.954828i \(0.403961\pi\)
\(594\) 0 0
\(595\) 9.34144 0.382962
\(596\) 0 0
\(597\) −48.4377 −1.98242
\(598\) 0 0
\(599\) 12.7403 0.520554 0.260277 0.965534i \(-0.416186\pi\)
0.260277 + 0.965534i \(0.416186\pi\)
\(600\) 0 0
\(601\) 0.00889305 0.000362755 0 0.000181378 1.00000i \(-0.499942\pi\)
0.000181378 1.00000i \(0.499942\pi\)
\(602\) 0 0
\(603\) 39.5281 1.60971
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −10.0651 −0.408530 −0.204265 0.978916i \(-0.565480\pi\)
−0.204265 + 0.978916i \(0.565480\pi\)
\(608\) 0 0
\(609\) −125.614 −5.09014
\(610\) 0 0
\(611\) −5.60430 −0.226726
\(612\) 0 0
\(613\) 17.0238 0.687584 0.343792 0.939046i \(-0.388288\pi\)
0.343792 + 0.939046i \(0.388288\pi\)
\(614\) 0 0
\(615\) 2.74118 0.110535
\(616\) 0 0
\(617\) −39.2503 −1.58016 −0.790078 0.613006i \(-0.789960\pi\)
−0.790078 + 0.613006i \(0.789960\pi\)
\(618\) 0 0
\(619\) −49.2905 −1.98115 −0.990577 0.136960i \(-0.956267\pi\)
−0.990577 + 0.136960i \(0.956267\pi\)
\(620\) 0 0
\(621\) −5.96374 −0.239317
\(622\) 0 0
\(623\) −0.268771 −0.0107681
\(624\) 0 0
\(625\) 23.2434 0.929736
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.2027 −0.805535
\(630\) 0 0
\(631\) −32.9741 −1.31268 −0.656340 0.754465i \(-0.727896\pi\)
−0.656340 + 0.754465i \(0.727896\pi\)
\(632\) 0 0
\(633\) −11.5905 −0.460681
\(634\) 0 0
\(635\) −4.52633 −0.179622
\(636\) 0 0
\(637\) −29.4358 −1.16629
\(638\) 0 0
\(639\) −56.8704 −2.24976
\(640\) 0 0
\(641\) −5.23839 −0.206904 −0.103452 0.994634i \(-0.532989\pi\)
−0.103452 + 0.994634i \(0.532989\pi\)
\(642\) 0 0
\(643\) 12.3440 0.486799 0.243400 0.969926i \(-0.421737\pi\)
0.243400 + 0.969926i \(0.421737\pi\)
\(644\) 0 0
\(645\) 7.85793 0.309406
\(646\) 0 0
\(647\) −31.0492 −1.22067 −0.610334 0.792144i \(-0.708965\pi\)
−0.610334 + 0.792144i \(0.708965\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −40.0788 −1.57081
\(652\) 0 0
\(653\) −24.2141 −0.947570 −0.473785 0.880640i \(-0.657113\pi\)
−0.473785 + 0.880640i \(0.657113\pi\)
\(654\) 0 0
\(655\) 6.75650 0.263998
\(656\) 0 0
\(657\) 81.2691 3.17061
\(658\) 0 0
\(659\) −36.4592 −1.42025 −0.710124 0.704076i \(-0.751361\pi\)
−0.710124 + 0.704076i \(0.751361\pi\)
\(660\) 0 0
\(661\) 47.0828 1.83131 0.915655 0.401965i \(-0.131673\pi\)
0.915655 + 0.401965i \(0.131673\pi\)
\(662\) 0 0
\(663\) −29.9837 −1.16447
\(664\) 0 0
\(665\) 1.64038 0.0636111
\(666\) 0 0
\(667\) −9.90545 −0.383541
\(668\) 0 0
\(669\) 74.1801 2.86797
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33.4857 −1.29078 −0.645390 0.763853i \(-0.723305\pi\)
−0.645390 + 0.763853i \(0.723305\pi\)
\(674\) 0 0
\(675\) −27.3673 −1.05337
\(676\) 0 0
\(677\) 39.8223 1.53050 0.765249 0.643735i \(-0.222616\pi\)
0.765249 + 0.643735i \(0.222616\pi\)
\(678\) 0 0
\(679\) −54.8018 −2.10310
\(680\) 0 0
\(681\) 7.18336 0.275267
\(682\) 0 0
\(683\) 1.36966 0.0524085 0.0262043 0.999657i \(-0.491658\pi\)
0.0262043 + 0.999657i \(0.491658\pi\)
\(684\) 0 0
\(685\) −3.09529 −0.118265
\(686\) 0 0
\(687\) −20.4546 −0.780390
\(688\) 0 0
\(689\) 8.08400 0.307976
\(690\) 0 0
\(691\) 29.5333 1.12350 0.561751 0.827307i \(-0.310128\pi\)
0.561751 + 0.827307i \(0.310128\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.83496 −0.259265
\(696\) 0 0
\(697\) −16.0802 −0.609082
\(698\) 0 0
\(699\) −69.7376 −2.63772
\(700\) 0 0
\(701\) −40.5434 −1.53130 −0.765651 0.643256i \(-0.777583\pi\)
−0.765651 + 0.643256i \(0.777583\pi\)
\(702\) 0 0
\(703\) −3.54764 −0.133802
\(704\) 0 0
\(705\) −2.91964 −0.109960
\(706\) 0 0
\(707\) 22.1714 0.833841
\(708\) 0 0
\(709\) 10.1180 0.379991 0.189995 0.981785i \(-0.439153\pi\)
0.189995 + 0.981785i \(0.439153\pi\)
\(710\) 0 0
\(711\) 66.5150 2.49451
\(712\) 0 0
\(713\) −3.16046 −0.118360
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 63.0473 2.35455
\(718\) 0 0
\(719\) −10.5326 −0.392799 −0.196400 0.980524i \(-0.562925\pi\)
−0.196400 + 0.980524i \(0.562925\pi\)
\(720\) 0 0
\(721\) −46.4728 −1.73074
\(722\) 0 0
\(723\) −25.6838 −0.955191
\(724\) 0 0
\(725\) −45.4556 −1.68818
\(726\) 0 0
\(727\) −12.8599 −0.476947 −0.238474 0.971149i \(-0.576647\pi\)
−0.238474 + 0.971149i \(0.576647\pi\)
\(728\) 0 0
\(729\) −43.0940 −1.59608
\(730\) 0 0
\(731\) −46.0959 −1.70492
\(732\) 0 0
\(733\) 13.3272 0.492252 0.246126 0.969238i \(-0.420842\pi\)
0.246126 + 0.969238i \(0.420842\pi\)
\(734\) 0 0
\(735\) −15.3350 −0.565641
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 19.7362 0.726009 0.363004 0.931787i \(-0.381751\pi\)
0.363004 + 0.931787i \(0.381751\pi\)
\(740\) 0 0
\(741\) −5.26521 −0.193422
\(742\) 0 0
\(743\) −10.9311 −0.401022 −0.200511 0.979691i \(-0.564260\pi\)
−0.200511 + 0.979691i \(0.564260\pi\)
\(744\) 0 0
\(745\) −4.47634 −0.164000
\(746\) 0 0
\(747\) 16.9824 0.621352
\(748\) 0 0
\(749\) −43.2837 −1.58155
\(750\) 0 0
\(751\) −0.00491790 −0.000179457 0 −8.97284e−5 1.00000i \(-0.500029\pi\)
−8.97284e−5 1.00000i \(0.500029\pi\)
\(752\) 0 0
\(753\) −5.98643 −0.218158
\(754\) 0 0
\(755\) −0.597182 −0.0217337
\(756\) 0 0
\(757\) 3.46494 0.125935 0.0629677 0.998016i \(-0.479943\pi\)
0.0629677 + 0.998016i \(0.479943\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −10.8298 −0.392580 −0.196290 0.980546i \(-0.562889\pi\)
−0.196290 + 0.980546i \(0.562889\pi\)
\(762\) 0 0
\(763\) 27.4512 0.993800
\(764\) 0 0
\(765\) −9.75101 −0.352549
\(766\) 0 0
\(767\) −27.3111 −0.986146
\(768\) 0 0
\(769\) 5.55885 0.200457 0.100229 0.994964i \(-0.468043\pi\)
0.100229 + 0.994964i \(0.468043\pi\)
\(770\) 0 0
\(771\) −13.6550 −0.491772
\(772\) 0 0
\(773\) −20.6807 −0.743834 −0.371917 0.928266i \(-0.621299\pi\)
−0.371917 + 0.928266i \(0.621299\pi\)
\(774\) 0 0
\(775\) −14.5032 −0.520970
\(776\) 0 0
\(777\) 47.8614 1.71702
\(778\) 0 0
\(779\) −2.82372 −0.101170
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 52.1952 1.86530
\(784\) 0 0
\(785\) 2.38950 0.0852851
\(786\) 0 0
\(787\) −7.09548 −0.252926 −0.126463 0.991971i \(-0.540363\pi\)
−0.126463 + 0.991971i \(0.540363\pi\)
\(788\) 0 0
\(789\) −30.4967 −1.08571
\(790\) 0 0
\(791\) −12.8403 −0.456549
\(792\) 0 0
\(793\) 1.93040 0.0685504
\(794\) 0 0
\(795\) 4.21149 0.149366
\(796\) 0 0
\(797\) −21.9429 −0.777258 −0.388629 0.921394i \(-0.627051\pi\)
−0.388629 + 0.921394i \(0.627051\pi\)
\(798\) 0 0
\(799\) 17.1271 0.605914
\(800\) 0 0
\(801\) 0.280555 0.00991293
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.74512 −0.0615074
\(806\) 0 0
\(807\) −45.4795 −1.60095
\(808\) 0 0
\(809\) −3.89093 −0.136798 −0.0683990 0.997658i \(-0.521789\pi\)
−0.0683990 + 0.997658i \(0.521789\pi\)
\(810\) 0 0
\(811\) 8.09828 0.284369 0.142185 0.989840i \(-0.454587\pi\)
0.142185 + 0.989840i \(0.454587\pi\)
\(812\) 0 0
\(813\) 62.8019 2.20256
\(814\) 0 0
\(815\) −5.05729 −0.177149
\(816\) 0 0
\(817\) −8.09454 −0.283192
\(818\) 0 0
\(819\) 44.3421 1.54944
\(820\) 0 0
\(821\) 29.0408 1.01353 0.506765 0.862084i \(-0.330841\pi\)
0.506765 + 0.862084i \(0.330841\pi\)
\(822\) 0 0
\(823\) −46.6098 −1.62471 −0.812357 0.583160i \(-0.801816\pi\)
−0.812357 + 0.583160i \(0.801816\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −55.5230 −1.93072 −0.965361 0.260917i \(-0.915975\pi\)
−0.965361 + 0.260917i \(0.915975\pi\)
\(828\) 0 0
\(829\) 16.7515 0.581802 0.290901 0.956753i \(-0.406045\pi\)
0.290901 + 0.956753i \(0.406045\pi\)
\(830\) 0 0
\(831\) 3.18558 0.110507
\(832\) 0 0
\(833\) 89.9578 3.11685
\(834\) 0 0
\(835\) −2.04515 −0.0707754
\(836\) 0 0
\(837\) 16.6535 0.575630
\(838\) 0 0
\(839\) 40.4855 1.39771 0.698857 0.715261i \(-0.253692\pi\)
0.698857 + 0.715261i \(0.253692\pi\)
\(840\) 0 0
\(841\) 57.6934 1.98943
\(842\) 0 0
\(843\) −4.28044 −0.147426
\(844\) 0 0
\(845\) 3.27338 0.112608
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16.2448 −0.557519
\(850\) 0 0
\(851\) 3.77417 0.129377
\(852\) 0 0
\(853\) −51.7573 −1.77214 −0.886068 0.463555i \(-0.846574\pi\)
−0.886068 + 0.463555i \(0.846574\pi\)
\(854\) 0 0
\(855\) −1.71230 −0.0585594
\(856\) 0 0
\(857\) 27.8604 0.951693 0.475846 0.879528i \(-0.342142\pi\)
0.475846 + 0.879528i \(0.342142\pi\)
\(858\) 0 0
\(859\) −51.6139 −1.76104 −0.880521 0.474007i \(-0.842807\pi\)
−0.880521 + 0.474007i \(0.842807\pi\)
\(860\) 0 0
\(861\) 38.0950 1.29827
\(862\) 0 0
\(863\) 1.36733 0.0465444 0.0232722 0.999729i \(-0.492592\pi\)
0.0232722 + 0.999729i \(0.492592\pi\)
\(864\) 0 0
\(865\) −0.317931 −0.0108100
\(866\) 0 0
\(867\) 43.5973 1.48064
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −14.7788 −0.500761
\(872\) 0 0
\(873\) 57.2046 1.93608
\(874\) 0 0
\(875\) −16.2102 −0.548003
\(876\) 0 0
\(877\) 3.25271 0.109836 0.0549181 0.998491i \(-0.482510\pi\)
0.0549181 + 0.998491i \(0.482510\pi\)
\(878\) 0 0
\(879\) 0.00461330 0.000155603 0
\(880\) 0 0
\(881\) 9.48028 0.319399 0.159699 0.987166i \(-0.448948\pi\)
0.159699 + 0.987166i \(0.448948\pi\)
\(882\) 0 0
\(883\) 33.3488 1.12228 0.561138 0.827722i \(-0.310364\pi\)
0.561138 + 0.827722i \(0.310364\pi\)
\(884\) 0 0
\(885\) −14.2281 −0.478273
\(886\) 0 0
\(887\) −38.2336 −1.28376 −0.641880 0.766805i \(-0.721845\pi\)
−0.641880 + 0.766805i \(0.721845\pi\)
\(888\) 0 0
\(889\) −62.9036 −2.10972
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.00756 0.100644
\(894\) 0 0
\(895\) −6.51405 −0.217741
\(896\) 0 0
\(897\) 5.60140 0.187025
\(898\) 0 0
\(899\) 27.6606 0.922533
\(900\) 0 0
\(901\) −24.7053 −0.823052
\(902\) 0 0
\(903\) 109.204 3.63408
\(904\) 0 0
\(905\) −2.15699 −0.0717007
\(906\) 0 0
\(907\) −52.9937 −1.75963 −0.879814 0.475318i \(-0.842333\pi\)
−0.879814 + 0.475318i \(0.842333\pi\)
\(908\) 0 0
\(909\) −23.1435 −0.767621
\(910\) 0 0
\(911\) 11.4793 0.380325 0.190162 0.981753i \(-0.439099\pi\)
0.190162 + 0.981753i \(0.439099\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1.00567 0.0332464
\(916\) 0 0
\(917\) 93.8969 3.10075
\(918\) 0 0
\(919\) 50.7962 1.67561 0.837806 0.545968i \(-0.183838\pi\)
0.837806 + 0.545968i \(0.183838\pi\)
\(920\) 0 0
\(921\) 76.4781 2.52004
\(922\) 0 0
\(923\) 21.2628 0.699874
\(924\) 0 0
\(925\) 17.3195 0.569461
\(926\) 0 0
\(927\) 48.5104 1.59329
\(928\) 0 0
\(929\) 25.5521 0.838339 0.419169 0.907908i \(-0.362321\pi\)
0.419169 + 0.907908i \(0.362321\pi\)
\(930\) 0 0
\(931\) 15.7968 0.517719
\(932\) 0 0
\(933\) 61.7221 2.02069
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.0081 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(938\) 0 0
\(939\) −12.8371 −0.418924
\(940\) 0 0
\(941\) 10.2690 0.334761 0.167380 0.985892i \(-0.446469\pi\)
0.167380 + 0.985892i \(0.446469\pi\)
\(942\) 0 0
\(943\) 3.00403 0.0978245
\(944\) 0 0
\(945\) 9.19562 0.299134
\(946\) 0 0
\(947\) −23.4610 −0.762379 −0.381190 0.924497i \(-0.624486\pi\)
−0.381190 + 0.924497i \(0.624486\pi\)
\(948\) 0 0
\(949\) −30.3850 −0.986338
\(950\) 0 0
\(951\) −23.2264 −0.753166
\(952\) 0 0
\(953\) −38.9755 −1.26254 −0.631271 0.775563i \(-0.717466\pi\)
−0.631271 + 0.775563i \(0.717466\pi\)
\(954\) 0 0
\(955\) 6.26222 0.202641
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −43.0161 −1.38906
\(960\) 0 0
\(961\) −22.1745 −0.715308
\(962\) 0 0
\(963\) 45.1814 1.45595
\(964\) 0 0
\(965\) −1.89578 −0.0610273
\(966\) 0 0
\(967\) 51.3942 1.65273 0.826363 0.563137i \(-0.190406\pi\)
0.826363 + 0.563137i \(0.190406\pi\)
\(968\) 0 0
\(969\) 16.0908 0.516912
\(970\) 0 0
\(971\) 4.66223 0.149618 0.0748090 0.997198i \(-0.476165\pi\)
0.0748090 + 0.997198i \(0.476165\pi\)
\(972\) 0 0
\(973\) −94.9874 −3.04515
\(974\) 0 0
\(975\) 25.7045 0.823204
\(976\) 0 0
\(977\) −51.7991 −1.65720 −0.828600 0.559841i \(-0.810862\pi\)
−0.828600 + 0.559841i \(0.810862\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −28.6548 −0.914877
\(982\) 0 0
\(983\) 51.7747 1.65136 0.825678 0.564142i \(-0.190793\pi\)
0.825678 + 0.564142i \(0.190793\pi\)
\(984\) 0 0
\(985\) 8.44814 0.269180
\(986\) 0 0
\(987\) −40.5751 −1.29152
\(988\) 0 0
\(989\) 8.61140 0.273826
\(990\) 0 0
\(991\) −21.2223 −0.674148 −0.337074 0.941478i \(-0.609437\pi\)
−0.337074 + 0.941478i \(0.609437\pi\)
\(992\) 0 0
\(993\) −29.6857 −0.942047
\(994\) 0 0
\(995\) 5.88955 0.186711
\(996\) 0 0
\(997\) −51.7325 −1.63839 −0.819193 0.573518i \(-0.805578\pi\)
−0.819193 + 0.573518i \(0.805578\pi\)
\(998\) 0 0
\(999\) −19.8874 −0.629209
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 9196.2.a.n.1.5 6
11.10 odd 2 836.2.a.e.1.5 6
33.32 even 2 7524.2.a.q.1.5 6
44.43 even 2 3344.2.a.w.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.e.1.5 6 11.10 odd 2
3344.2.a.w.1.2 6 44.43 even 2
7524.2.a.q.1.5 6 33.32 even 2
9196.2.a.n.1.5 6 1.1 even 1 trivial