Properties

Label 7803.2.a.by.1.4
Level $7803$
Weight $2$
Character 7803.1
Self dual yes
Analytic conductor $62.307$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7803 = 3^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7803.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,-6,0,12,3,0,3,-18,0,0,-6,0,-3,-3,0,6,0,0,-3,-6,0,-12,3,0, 6,24,0,9,6,0,0,-42,0,0,-33,0,0,-36,0,15,0,0,-3,18,0,12,-24,0,18,-42,0, -12,-48,0,-3,15,0,-12,-18,0,0,-63,0,30,-24,0,12,0,0,51,21,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.3072686972\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 6 x^{14} - 3 x^{13} + 76 x^{12} - 69 x^{11} - 354 x^{10} + 523 x^{9} + 720 x^{8} - 1437 x^{7} + \cdots - 51 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.01902\) of defining polynomial
Character \(\chi\) \(=\) 7803.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.01902 q^{2} +2.07645 q^{4} -0.707464 q^{5} +4.62694 q^{7} -0.154349 q^{8} +1.42838 q^{10} +1.79601 q^{11} -0.836114 q^{13} -9.34190 q^{14} -3.84126 q^{16} +5.70342 q^{19} -1.46901 q^{20} -3.62619 q^{22} +2.03224 q^{23} -4.49949 q^{25} +1.68813 q^{26} +9.60761 q^{28} -3.20042 q^{29} -3.11899 q^{31} +8.06429 q^{32} -3.27340 q^{35} +5.94714 q^{37} -11.5153 q^{38} +0.109197 q^{40} -10.7502 q^{41} +4.96676 q^{43} +3.72933 q^{44} -4.10314 q^{46} -12.0304 q^{47} +14.4086 q^{49} +9.08458 q^{50} -1.73615 q^{52} -6.29645 q^{53} -1.27061 q^{55} -0.714165 q^{56} +6.46172 q^{58} -14.0008 q^{59} -12.5071 q^{61} +6.29732 q^{62} -8.59945 q^{64} +0.591521 q^{65} +2.58698 q^{67} +6.60906 q^{70} -5.37900 q^{71} -2.99116 q^{73} -12.0074 q^{74} +11.8429 q^{76} +8.31005 q^{77} -10.2295 q^{79} +2.71755 q^{80} +21.7048 q^{82} +9.16954 q^{83} -10.0280 q^{86} -0.277213 q^{88} -3.32288 q^{89} -3.86866 q^{91} +4.21984 q^{92} +24.2897 q^{94} -4.03497 q^{95} -10.6214 q^{97} -29.0913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} + 3 q^{23} + 6 q^{25} + 24 q^{26} + 9 q^{28} + 6 q^{29} - 42 q^{32}+ \cdots - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.01902 −1.42766 −0.713832 0.700317i \(-0.753042\pi\)
−0.713832 + 0.700317i \(0.753042\pi\)
\(3\) 0 0
\(4\) 2.07645 1.03822
\(5\) −0.707464 −0.316388 −0.158194 0.987408i \(-0.550567\pi\)
−0.158194 + 0.987408i \(0.550567\pi\)
\(6\) 0 0
\(7\) 4.62694 1.74882 0.874410 0.485187i \(-0.161249\pi\)
0.874410 + 0.485187i \(0.161249\pi\)
\(8\) −0.154349 −0.0545707
\(9\) 0 0
\(10\) 1.42838 0.451695
\(11\) 1.79601 0.541518 0.270759 0.962647i \(-0.412725\pi\)
0.270759 + 0.962647i \(0.412725\pi\)
\(12\) 0 0
\(13\) −0.836114 −0.231896 −0.115948 0.993255i \(-0.536991\pi\)
−0.115948 + 0.993255i \(0.536991\pi\)
\(14\) −9.34190 −2.49673
\(15\) 0 0
\(16\) −3.84126 −0.960315
\(17\) 0 0
\(18\) 0 0
\(19\) 5.70342 1.30845 0.654227 0.756298i \(-0.272994\pi\)
0.654227 + 0.756298i \(0.272994\pi\)
\(20\) −1.46901 −0.328481
\(21\) 0 0
\(22\) −3.62619 −0.773106
\(23\) 2.03224 0.423751 0.211876 0.977297i \(-0.432043\pi\)
0.211876 + 0.977297i \(0.432043\pi\)
\(24\) 0 0
\(25\) −4.49949 −0.899899
\(26\) 1.68813 0.331070
\(27\) 0 0
\(28\) 9.60761 1.81567
\(29\) −3.20042 −0.594304 −0.297152 0.954830i \(-0.596037\pi\)
−0.297152 + 0.954830i \(0.596037\pi\)
\(30\) 0 0
\(31\) −3.11899 −0.560188 −0.280094 0.959973i \(-0.590366\pi\)
−0.280094 + 0.959973i \(0.590366\pi\)
\(32\) 8.06429 1.42558
\(33\) 0 0
\(34\) 0 0
\(35\) −3.27340 −0.553305
\(36\) 0 0
\(37\) 5.94714 0.977704 0.488852 0.872367i \(-0.337416\pi\)
0.488852 + 0.872367i \(0.337416\pi\)
\(38\) −11.5153 −1.86803
\(39\) 0 0
\(40\) 0.109197 0.0172655
\(41\) −10.7502 −1.67889 −0.839446 0.543443i \(-0.817120\pi\)
−0.839446 + 0.543443i \(0.817120\pi\)
\(42\) 0 0
\(43\) 4.96676 0.757424 0.378712 0.925514i \(-0.376367\pi\)
0.378712 + 0.925514i \(0.376367\pi\)
\(44\) 3.72933 0.562217
\(45\) 0 0
\(46\) −4.10314 −0.604974
\(47\) −12.0304 −1.75482 −0.877409 0.479743i \(-0.840730\pi\)
−0.877409 + 0.479743i \(0.840730\pi\)
\(48\) 0 0
\(49\) 14.4086 2.05837
\(50\) 9.08458 1.28475
\(51\) 0 0
\(52\) −1.73615 −0.240760
\(53\) −6.29645 −0.864884 −0.432442 0.901662i \(-0.642348\pi\)
−0.432442 + 0.901662i \(0.642348\pi\)
\(54\) 0 0
\(55\) −1.27061 −0.171330
\(56\) −0.714165 −0.0954344
\(57\) 0 0
\(58\) 6.46172 0.848466
\(59\) −14.0008 −1.82274 −0.911372 0.411584i \(-0.864976\pi\)
−0.911372 + 0.411584i \(0.864976\pi\)
\(60\) 0 0
\(61\) −12.5071 −1.60136 −0.800682 0.599089i \(-0.795530\pi\)
−0.800682 + 0.599089i \(0.795530\pi\)
\(62\) 6.29732 0.799760
\(63\) 0 0
\(64\) −8.59945 −1.07493
\(65\) 0.591521 0.0733691
\(66\) 0 0
\(67\) 2.58698 0.316050 0.158025 0.987435i \(-0.449487\pi\)
0.158025 + 0.987435i \(0.449487\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 6.60906 0.789933
\(71\) −5.37900 −0.638370 −0.319185 0.947692i \(-0.603409\pi\)
−0.319185 + 0.947692i \(0.603409\pi\)
\(72\) 0 0
\(73\) −2.99116 −0.350089 −0.175044 0.984561i \(-0.556007\pi\)
−0.175044 + 0.984561i \(0.556007\pi\)
\(74\) −12.0074 −1.39583
\(75\) 0 0
\(76\) 11.8429 1.35847
\(77\) 8.31005 0.947019
\(78\) 0 0
\(79\) −10.2295 −1.15091 −0.575453 0.817835i \(-0.695174\pi\)
−0.575453 + 0.817835i \(0.695174\pi\)
\(80\) 2.71755 0.303832
\(81\) 0 0
\(82\) 21.7048 2.39689
\(83\) 9.16954 1.00649 0.503244 0.864145i \(-0.332140\pi\)
0.503244 + 0.864145i \(0.332140\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0280 −1.08135
\(87\) 0 0
\(88\) −0.277213 −0.0295510
\(89\) −3.32288 −0.352225 −0.176112 0.984370i \(-0.556352\pi\)
−0.176112 + 0.984370i \(0.556352\pi\)
\(90\) 0 0
\(91\) −3.86866 −0.405545
\(92\) 4.21984 0.439949
\(93\) 0 0
\(94\) 24.2897 2.50529
\(95\) −4.03497 −0.413979
\(96\) 0 0
\(97\) −10.6214 −1.07844 −0.539218 0.842166i \(-0.681280\pi\)
−0.539218 + 0.842166i \(0.681280\pi\)
\(98\) −29.0913 −2.93867
\(99\) 0 0
\(100\) −9.34296 −0.934296
\(101\) −18.6725 −1.85799 −0.928993 0.370098i \(-0.879324\pi\)
−0.928993 + 0.370098i \(0.879324\pi\)
\(102\) 0 0
\(103\) 5.48731 0.540680 0.270340 0.962765i \(-0.412864\pi\)
0.270340 + 0.962765i \(0.412864\pi\)
\(104\) 0.129054 0.0126548
\(105\) 0 0
\(106\) 12.7127 1.23476
\(107\) −9.79042 −0.946476 −0.473238 0.880935i \(-0.656915\pi\)
−0.473238 + 0.880935i \(0.656915\pi\)
\(108\) 0 0
\(109\) 2.33350 0.223509 0.111755 0.993736i \(-0.464353\pi\)
0.111755 + 0.993736i \(0.464353\pi\)
\(110\) 2.56540 0.244601
\(111\) 0 0
\(112\) −17.7733 −1.67942
\(113\) −6.17484 −0.580880 −0.290440 0.956893i \(-0.593802\pi\)
−0.290440 + 0.956893i \(0.593802\pi\)
\(114\) 0 0
\(115\) −1.43774 −0.134070
\(116\) −6.64551 −0.617020
\(117\) 0 0
\(118\) 28.2678 2.60226
\(119\) 0 0
\(120\) 0 0
\(121\) −7.77434 −0.706758
\(122\) 25.2520 2.28621
\(123\) 0 0
\(124\) −6.47643 −0.581600
\(125\) 6.72055 0.601104
\(126\) 0 0
\(127\) −13.1806 −1.16959 −0.584795 0.811181i \(-0.698825\pi\)
−0.584795 + 0.811181i \(0.698825\pi\)
\(128\) 1.23389 0.109062
\(129\) 0 0
\(130\) −1.19429 −0.104746
\(131\) −12.1634 −1.06272 −0.531360 0.847146i \(-0.678319\pi\)
−0.531360 + 0.847146i \(0.678319\pi\)
\(132\) 0 0
\(133\) 26.3894 2.28825
\(134\) −5.22316 −0.451212
\(135\) 0 0
\(136\) 0 0
\(137\) −19.1327 −1.63462 −0.817309 0.576200i \(-0.804535\pi\)
−0.817309 + 0.576200i \(0.804535\pi\)
\(138\) 0 0
\(139\) −10.7175 −0.909048 −0.454524 0.890735i \(-0.650191\pi\)
−0.454524 + 0.890735i \(0.650191\pi\)
\(140\) −6.79704 −0.574454
\(141\) 0 0
\(142\) 10.8603 0.911377
\(143\) −1.50167 −0.125576
\(144\) 0 0
\(145\) 2.26418 0.188030
\(146\) 6.03922 0.499809
\(147\) 0 0
\(148\) 12.3489 1.01508
\(149\) 14.1781 1.16151 0.580757 0.814077i \(-0.302757\pi\)
0.580757 + 0.814077i \(0.302757\pi\)
\(150\) 0 0
\(151\) 10.1345 0.824733 0.412367 0.911018i \(-0.364702\pi\)
0.412367 + 0.911018i \(0.364702\pi\)
\(152\) −0.880319 −0.0714033
\(153\) 0 0
\(154\) −16.7782 −1.35202
\(155\) 2.20658 0.177236
\(156\) 0 0
\(157\) −7.24708 −0.578380 −0.289190 0.957272i \(-0.593386\pi\)
−0.289190 + 0.957272i \(0.593386\pi\)
\(158\) 20.6535 1.64311
\(159\) 0 0
\(160\) −5.70519 −0.451035
\(161\) 9.40306 0.741065
\(162\) 0 0
\(163\) −25.1848 −1.97263 −0.986313 0.164884i \(-0.947275\pi\)
−0.986313 + 0.164884i \(0.947275\pi\)
\(164\) −22.3221 −1.74307
\(165\) 0 0
\(166\) −18.5135 −1.43693
\(167\) −5.38433 −0.416652 −0.208326 0.978059i \(-0.566801\pi\)
−0.208326 + 0.978059i \(0.566801\pi\)
\(168\) 0 0
\(169\) −12.3009 −0.946224
\(170\) 0 0
\(171\) 0 0
\(172\) 10.3132 0.786376
\(173\) 6.99021 0.531456 0.265728 0.964048i \(-0.414388\pi\)
0.265728 + 0.964048i \(0.414388\pi\)
\(174\) 0 0
\(175\) −20.8189 −1.57376
\(176\) −6.89896 −0.520028
\(177\) 0 0
\(178\) 6.70897 0.502858
\(179\) 13.9188 1.04034 0.520168 0.854064i \(-0.325869\pi\)
0.520168 + 0.854064i \(0.325869\pi\)
\(180\) 0 0
\(181\) 9.81987 0.729905 0.364953 0.931026i \(-0.381085\pi\)
0.364953 + 0.931026i \(0.381085\pi\)
\(182\) 7.81090 0.578982
\(183\) 0 0
\(184\) −0.313675 −0.0231244
\(185\) −4.20739 −0.309333
\(186\) 0 0
\(187\) 0 0
\(188\) −24.9805 −1.82189
\(189\) 0 0
\(190\) 8.14668 0.591022
\(191\) −2.14803 −0.155426 −0.0777131 0.996976i \(-0.524762\pi\)
−0.0777131 + 0.996976i \(0.524762\pi\)
\(192\) 0 0
\(193\) 0.0412451 0.00296889 0.00148444 0.999999i \(-0.499527\pi\)
0.00148444 + 0.999999i \(0.499527\pi\)
\(194\) 21.4447 1.53964
\(195\) 0 0
\(196\) 29.9187 2.13705
\(197\) 26.1372 1.86220 0.931099 0.364767i \(-0.118851\pi\)
0.931099 + 0.364767i \(0.118851\pi\)
\(198\) 0 0
\(199\) −17.1302 −1.21433 −0.607163 0.794577i \(-0.707693\pi\)
−0.607163 + 0.794577i \(0.707693\pi\)
\(200\) 0.694494 0.0491081
\(201\) 0 0
\(202\) 37.7002 2.65258
\(203\) −14.8082 −1.03933
\(204\) 0 0
\(205\) 7.60535 0.531181
\(206\) −11.0790 −0.771910
\(207\) 0 0
\(208\) 3.21173 0.222694
\(209\) 10.2434 0.708552
\(210\) 0 0
\(211\) 7.09038 0.488122 0.244061 0.969760i \(-0.421520\pi\)
0.244061 + 0.969760i \(0.421520\pi\)
\(212\) −13.0742 −0.897943
\(213\) 0 0
\(214\) 19.7671 1.35125
\(215\) −3.51381 −0.239640
\(216\) 0 0
\(217\) −14.4314 −0.979668
\(218\) −4.71139 −0.319096
\(219\) 0 0
\(220\) −2.63836 −0.177879
\(221\) 0 0
\(222\) 0 0
\(223\) 21.8855 1.46556 0.732781 0.680465i \(-0.238222\pi\)
0.732781 + 0.680465i \(0.238222\pi\)
\(224\) 37.3130 2.49308
\(225\) 0 0
\(226\) 12.4671 0.829301
\(227\) 19.1300 1.26970 0.634850 0.772635i \(-0.281062\pi\)
0.634850 + 0.772635i \(0.281062\pi\)
\(228\) 0 0
\(229\) −17.0884 −1.12923 −0.564616 0.825354i \(-0.690976\pi\)
−0.564616 + 0.825354i \(0.690976\pi\)
\(230\) 2.90282 0.191406
\(231\) 0 0
\(232\) 0.493983 0.0324316
\(233\) 19.9326 1.30583 0.652915 0.757431i \(-0.273546\pi\)
0.652915 + 0.757431i \(0.273546\pi\)
\(234\) 0 0
\(235\) 8.51109 0.555202
\(236\) −29.0718 −1.89242
\(237\) 0 0
\(238\) 0 0
\(239\) 0.546129 0.0353261 0.0176631 0.999844i \(-0.494377\pi\)
0.0176631 + 0.999844i \(0.494377\pi\)
\(240\) 0 0
\(241\) 18.8594 1.21484 0.607421 0.794380i \(-0.292204\pi\)
0.607421 + 0.794380i \(0.292204\pi\)
\(242\) 15.6966 1.00901
\(243\) 0 0
\(244\) −25.9703 −1.66258
\(245\) −10.1936 −0.651244
\(246\) 0 0
\(247\) −4.76871 −0.303426
\(248\) 0.481414 0.0305698
\(249\) 0 0
\(250\) −13.5689 −0.858175
\(251\) −4.22900 −0.266932 −0.133466 0.991053i \(-0.542611\pi\)
−0.133466 + 0.991053i \(0.542611\pi\)
\(252\) 0 0
\(253\) 3.64993 0.229469
\(254\) 26.6119 1.66978
\(255\) 0 0
\(256\) 14.7076 0.919227
\(257\) 15.2767 0.952933 0.476467 0.879193i \(-0.341917\pi\)
0.476467 + 0.879193i \(0.341917\pi\)
\(258\) 0 0
\(259\) 27.5171 1.70983
\(260\) 1.22826 0.0761736
\(261\) 0 0
\(262\) 24.5581 1.51721
\(263\) −31.0995 −1.91768 −0.958838 0.283953i \(-0.908354\pi\)
−0.958838 + 0.283953i \(0.908354\pi\)
\(264\) 0 0
\(265\) 4.45451 0.273638
\(266\) −53.2808 −3.26685
\(267\) 0 0
\(268\) 5.37172 0.328130
\(269\) 16.1906 0.987157 0.493578 0.869701i \(-0.335689\pi\)
0.493578 + 0.869701i \(0.335689\pi\)
\(270\) 0 0
\(271\) −0.190222 −0.0115552 −0.00577759 0.999983i \(-0.501839\pi\)
−0.00577759 + 0.999983i \(0.501839\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 38.6293 2.33368
\(275\) −8.08115 −0.487312
\(276\) 0 0
\(277\) −15.3476 −0.922150 −0.461075 0.887361i \(-0.652536\pi\)
−0.461075 + 0.887361i \(0.652536\pi\)
\(278\) 21.6389 1.29781
\(279\) 0 0
\(280\) 0.505246 0.0301942
\(281\) 9.65856 0.576181 0.288091 0.957603i \(-0.406980\pi\)
0.288091 + 0.957603i \(0.406980\pi\)
\(282\) 0 0
\(283\) 15.6189 0.928447 0.464224 0.885718i \(-0.346333\pi\)
0.464224 + 0.885718i \(0.346333\pi\)
\(284\) −11.1692 −0.662771
\(285\) 0 0
\(286\) 3.03191 0.179281
\(287\) −49.7404 −2.93608
\(288\) 0 0
\(289\) 0 0
\(290\) −4.57144 −0.268444
\(291\) 0 0
\(292\) −6.21099 −0.363471
\(293\) −21.5424 −1.25852 −0.629260 0.777195i \(-0.716642\pi\)
−0.629260 + 0.777195i \(0.716642\pi\)
\(294\) 0 0
\(295\) 9.90503 0.576693
\(296\) −0.917937 −0.0533540
\(297\) 0 0
\(298\) −28.6259 −1.65825
\(299\) −1.69919 −0.0982664
\(300\) 0 0
\(301\) 22.9809 1.32460
\(302\) −20.4618 −1.17744
\(303\) 0 0
\(304\) −21.9083 −1.25653
\(305\) 8.84829 0.506652
\(306\) 0 0
\(307\) 13.6192 0.777289 0.388645 0.921388i \(-0.372943\pi\)
0.388645 + 0.921388i \(0.372943\pi\)
\(308\) 17.2554 0.983217
\(309\) 0 0
\(310\) −4.45512 −0.253034
\(311\) −0.570591 −0.0323552 −0.0161776 0.999869i \(-0.505150\pi\)
−0.0161776 + 0.999869i \(0.505150\pi\)
\(312\) 0 0
\(313\) 22.1153 1.25003 0.625015 0.780612i \(-0.285093\pi\)
0.625015 + 0.780612i \(0.285093\pi\)
\(314\) 14.6320 0.825732
\(315\) 0 0
\(316\) −21.2410 −1.19490
\(317\) −4.38226 −0.246132 −0.123066 0.992398i \(-0.539273\pi\)
−0.123066 + 0.992398i \(0.539273\pi\)
\(318\) 0 0
\(319\) −5.74800 −0.321826
\(320\) 6.08380 0.340095
\(321\) 0 0
\(322\) −18.9850 −1.05799
\(323\) 0 0
\(324\) 0 0
\(325\) 3.76209 0.208683
\(326\) 50.8487 2.81625
\(327\) 0 0
\(328\) 1.65928 0.0916183
\(329\) −55.6641 −3.06886
\(330\) 0 0
\(331\) −13.5453 −0.744516 −0.372258 0.928129i \(-0.621416\pi\)
−0.372258 + 0.928129i \(0.621416\pi\)
\(332\) 19.0401 1.04496
\(333\) 0 0
\(334\) 10.8711 0.594839
\(335\) −1.83019 −0.0999941
\(336\) 0 0
\(337\) −18.9459 −1.03205 −0.516024 0.856574i \(-0.672588\pi\)
−0.516024 + 0.856574i \(0.672588\pi\)
\(338\) 24.8358 1.35089
\(339\) 0 0
\(340\) 0 0
\(341\) −5.60176 −0.303352
\(342\) 0 0
\(343\) 34.2792 1.85091
\(344\) −0.766616 −0.0413332
\(345\) 0 0
\(346\) −14.1134 −0.758740
\(347\) 22.7527 1.22143 0.610714 0.791852i \(-0.290883\pi\)
0.610714 + 0.791852i \(0.290883\pi\)
\(348\) 0 0
\(349\) 10.7432 0.575068 0.287534 0.957770i \(-0.407165\pi\)
0.287534 + 0.957770i \(0.407165\pi\)
\(350\) 42.0338 2.24680
\(351\) 0 0
\(352\) 14.4836 0.771977
\(353\) 23.4054 1.24575 0.622873 0.782323i \(-0.285965\pi\)
0.622873 + 0.782323i \(0.285965\pi\)
\(354\) 0 0
\(355\) 3.80545 0.201972
\(356\) −6.89979 −0.365688
\(357\) 0 0
\(358\) −28.1023 −1.48525
\(359\) 30.2112 1.59449 0.797243 0.603659i \(-0.206291\pi\)
0.797243 + 0.603659i \(0.206291\pi\)
\(360\) 0 0
\(361\) 13.5290 0.712053
\(362\) −19.8265 −1.04206
\(363\) 0 0
\(364\) −8.03306 −0.421047
\(365\) 2.11614 0.110764
\(366\) 0 0
\(367\) 12.6255 0.659048 0.329524 0.944147i \(-0.393112\pi\)
0.329524 + 0.944147i \(0.393112\pi\)
\(368\) −7.80636 −0.406935
\(369\) 0 0
\(370\) 8.49481 0.441624
\(371\) −29.1333 −1.51253
\(372\) 0 0
\(373\) 12.8657 0.666160 0.333080 0.942899i \(-0.391912\pi\)
0.333080 + 0.942899i \(0.391912\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.85689 0.0957616
\(377\) 2.67592 0.137817
\(378\) 0 0
\(379\) −21.8112 −1.12037 −0.560184 0.828368i \(-0.689269\pi\)
−0.560184 + 0.828368i \(0.689269\pi\)
\(380\) −8.37839 −0.429803
\(381\) 0 0
\(382\) 4.33693 0.221896
\(383\) −12.1588 −0.621284 −0.310642 0.950527i \(-0.600544\pi\)
−0.310642 + 0.950527i \(0.600544\pi\)
\(384\) 0 0
\(385\) −5.87906 −0.299625
\(386\) −0.0832747 −0.00423857
\(387\) 0 0
\(388\) −22.0547 −1.11966
\(389\) 13.2604 0.672327 0.336164 0.941804i \(-0.390870\pi\)
0.336164 + 0.941804i \(0.390870\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.22396 −0.112327
\(393\) 0 0
\(394\) −52.7715 −2.65859
\(395\) 7.23698 0.364132
\(396\) 0 0
\(397\) 25.3045 1.27000 0.634998 0.772514i \(-0.281001\pi\)
0.634998 + 0.772514i \(0.281001\pi\)
\(398\) 34.5862 1.73365
\(399\) 0 0
\(400\) 17.2837 0.864187
\(401\) −19.1195 −0.954782 −0.477391 0.878691i \(-0.658418\pi\)
−0.477391 + 0.878691i \(0.658418\pi\)
\(402\) 0 0
\(403\) 2.60784 0.129906
\(404\) −38.7725 −1.92900
\(405\) 0 0
\(406\) 29.8980 1.48381
\(407\) 10.6811 0.529445
\(408\) 0 0
\(409\) 3.23164 0.159794 0.0798971 0.996803i \(-0.474541\pi\)
0.0798971 + 0.996803i \(0.474541\pi\)
\(410\) −15.3554 −0.758347
\(411\) 0 0
\(412\) 11.3941 0.561347
\(413\) −64.7807 −3.18765
\(414\) 0 0
\(415\) −6.48712 −0.318440
\(416\) −6.74267 −0.330586
\(417\) 0 0
\(418\) −20.6817 −1.01157
\(419\) −29.8593 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(420\) 0 0
\(421\) −20.2363 −0.986255 −0.493127 0.869957i \(-0.664146\pi\)
−0.493127 + 0.869957i \(0.664146\pi\)
\(422\) −14.3156 −0.696874
\(423\) 0 0
\(424\) 0.971852 0.0471973
\(425\) 0 0
\(426\) 0 0
\(427\) −57.8695 −2.80050
\(428\) −20.3293 −0.982654
\(429\) 0 0
\(430\) 7.09445 0.342125
\(431\) 23.7095 1.14205 0.571024 0.820933i \(-0.306546\pi\)
0.571024 + 0.820933i \(0.306546\pi\)
\(432\) 0 0
\(433\) 33.1115 1.59124 0.795619 0.605797i \(-0.207146\pi\)
0.795619 + 0.605797i \(0.207146\pi\)
\(434\) 29.1373 1.39864
\(435\) 0 0
\(436\) 4.84540 0.232052
\(437\) 11.5907 0.554459
\(438\) 0 0
\(439\) 19.9929 0.954207 0.477104 0.878847i \(-0.341687\pi\)
0.477104 + 0.878847i \(0.341687\pi\)
\(440\) 0.196118 0.00934958
\(441\) 0 0
\(442\) 0 0
\(443\) −14.2792 −0.678425 −0.339213 0.940710i \(-0.610161\pi\)
−0.339213 + 0.940710i \(0.610161\pi\)
\(444\) 0 0
\(445\) 2.35082 0.111439
\(446\) −44.1873 −2.09233
\(447\) 0 0
\(448\) −39.7892 −1.87986
\(449\) 29.5108 1.39270 0.696349 0.717703i \(-0.254806\pi\)
0.696349 + 0.717703i \(0.254806\pi\)
\(450\) 0 0
\(451\) −19.3074 −0.909151
\(452\) −12.8217 −0.603083
\(453\) 0 0
\(454\) −38.6238 −1.81271
\(455\) 2.73693 0.128309
\(456\) 0 0
\(457\) −16.3770 −0.766085 −0.383042 0.923731i \(-0.625124\pi\)
−0.383042 + 0.923731i \(0.625124\pi\)
\(458\) 34.5018 1.61216
\(459\) 0 0
\(460\) −2.98538 −0.139194
\(461\) 3.30178 0.153779 0.0768897 0.997040i \(-0.475501\pi\)
0.0768897 + 0.997040i \(0.475501\pi\)
\(462\) 0 0
\(463\) −23.1054 −1.07380 −0.536899 0.843646i \(-0.680405\pi\)
−0.536899 + 0.843646i \(0.680405\pi\)
\(464\) 12.2937 0.570719
\(465\) 0 0
\(466\) −40.2444 −1.86429
\(467\) 39.5702 1.83109 0.915545 0.402216i \(-0.131760\pi\)
0.915545 + 0.402216i \(0.131760\pi\)
\(468\) 0 0
\(469\) 11.9698 0.552714
\(470\) −17.1841 −0.792642
\(471\) 0 0
\(472\) 2.16101 0.0994684
\(473\) 8.92037 0.410159
\(474\) 0 0
\(475\) −25.6625 −1.17748
\(476\) 0 0
\(477\) 0 0
\(478\) −1.10265 −0.0504339
\(479\) −34.0854 −1.55740 −0.778701 0.627396i \(-0.784121\pi\)
−0.778701 + 0.627396i \(0.784121\pi\)
\(480\) 0 0
\(481\) −4.97249 −0.226726
\(482\) −38.0776 −1.73439
\(483\) 0 0
\(484\) −16.1430 −0.733773
\(485\) 7.51423 0.341203
\(486\) 0 0
\(487\) −40.2231 −1.82268 −0.911341 0.411653i \(-0.864951\pi\)
−0.911341 + 0.411653i \(0.864951\pi\)
\(488\) 1.93046 0.0873876
\(489\) 0 0
\(490\) 20.5810 0.929757
\(491\) −21.1648 −0.955156 −0.477578 0.878589i \(-0.658485\pi\)
−0.477578 + 0.878589i \(0.658485\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 9.62813 0.433190
\(495\) 0 0
\(496\) 11.9809 0.537957
\(497\) −24.8883 −1.11639
\(498\) 0 0
\(499\) 24.1391 1.08061 0.540307 0.841468i \(-0.318308\pi\)
0.540307 + 0.841468i \(0.318308\pi\)
\(500\) 13.9549 0.624081
\(501\) 0 0
\(502\) 8.53845 0.381089
\(503\) 19.4903 0.869028 0.434514 0.900665i \(-0.356920\pi\)
0.434514 + 0.900665i \(0.356920\pi\)
\(504\) 0 0
\(505\) 13.2101 0.587843
\(506\) −7.36929 −0.327605
\(507\) 0 0
\(508\) −27.3688 −1.21430
\(509\) −11.2605 −0.499111 −0.249555 0.968361i \(-0.580284\pi\)
−0.249555 + 0.968361i \(0.580284\pi\)
\(510\) 0 0
\(511\) −13.8399 −0.612243
\(512\) −32.1628 −1.42141
\(513\) 0 0
\(514\) −30.8439 −1.36047
\(515\) −3.88207 −0.171065
\(516\) 0 0
\(517\) −21.6068 −0.950266
\(518\) −55.5576 −2.44106
\(519\) 0 0
\(520\) −0.0913008 −0.00400381
\(521\) 25.7599 1.12856 0.564280 0.825583i \(-0.309154\pi\)
0.564280 + 0.825583i \(0.309154\pi\)
\(522\) 0 0
\(523\) −16.3630 −0.715505 −0.357752 0.933816i \(-0.616457\pi\)
−0.357752 + 0.933816i \(0.616457\pi\)
\(524\) −25.2566 −1.10334
\(525\) 0 0
\(526\) 62.7905 2.73780
\(527\) 0 0
\(528\) 0 0
\(529\) −18.8700 −0.820435
\(530\) −8.99375 −0.390664
\(531\) 0 0
\(532\) 54.7962 2.37572
\(533\) 8.98836 0.389329
\(534\) 0 0
\(535\) 6.92637 0.299453
\(536\) −0.399298 −0.0172470
\(537\) 0 0
\(538\) −32.6891 −1.40933
\(539\) 25.8781 1.11465
\(540\) 0 0
\(541\) −0.0654823 −0.00281530 −0.00140765 0.999999i \(-0.500448\pi\)
−0.00140765 + 0.999999i \(0.500448\pi\)
\(542\) 0.384063 0.0164969
\(543\) 0 0
\(544\) 0 0
\(545\) −1.65087 −0.0707155
\(546\) 0 0
\(547\) −9.71350 −0.415319 −0.207660 0.978201i \(-0.566585\pi\)
−0.207660 + 0.978201i \(0.566585\pi\)
\(548\) −39.7281 −1.69710
\(549\) 0 0
\(550\) 16.3160 0.695717
\(551\) −18.2534 −0.777620
\(552\) 0 0
\(553\) −47.3312 −2.01273
\(554\) 30.9872 1.31652
\(555\) 0 0
\(556\) −22.2544 −0.943795
\(557\) −19.1479 −0.811322 −0.405661 0.914024i \(-0.632959\pi\)
−0.405661 + 0.914024i \(0.632959\pi\)
\(558\) 0 0
\(559\) −4.15278 −0.175644
\(560\) 12.5740 0.531347
\(561\) 0 0
\(562\) −19.5008 −0.822593
\(563\) 5.67364 0.239115 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(564\) 0 0
\(565\) 4.36847 0.183783
\(566\) −31.5349 −1.32551
\(567\) 0 0
\(568\) 0.830244 0.0348363
\(569\) 6.32590 0.265195 0.132598 0.991170i \(-0.457668\pi\)
0.132598 + 0.991170i \(0.457668\pi\)
\(570\) 0 0
\(571\) 29.3754 1.22932 0.614661 0.788791i \(-0.289293\pi\)
0.614661 + 0.788791i \(0.289293\pi\)
\(572\) −3.11814 −0.130376
\(573\) 0 0
\(574\) 100.427 4.19174
\(575\) −9.14405 −0.381333
\(576\) 0 0
\(577\) 14.0393 0.584465 0.292233 0.956347i \(-0.405602\pi\)
0.292233 + 0.956347i \(0.405602\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 4.70146 0.195218
\(581\) 42.4269 1.76017
\(582\) 0 0
\(583\) −11.3085 −0.468351
\(584\) 0.461684 0.0191046
\(585\) 0 0
\(586\) 43.4946 1.79674
\(587\) 0.612807 0.0252932 0.0126466 0.999920i \(-0.495974\pi\)
0.0126466 + 0.999920i \(0.495974\pi\)
\(588\) 0 0
\(589\) −17.7889 −0.732980
\(590\) −19.9985 −0.823324
\(591\) 0 0
\(592\) −22.8445 −0.938904
\(593\) 14.3438 0.589028 0.294514 0.955647i \(-0.404842\pi\)
0.294514 + 0.955647i \(0.404842\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 29.4401 1.20591
\(597\) 0 0
\(598\) 3.43069 0.140291
\(599\) −35.6635 −1.45717 −0.728585 0.684955i \(-0.759822\pi\)
−0.728585 + 0.684955i \(0.759822\pi\)
\(600\) 0 0
\(601\) 32.7304 1.33510 0.667550 0.744565i \(-0.267343\pi\)
0.667550 + 0.744565i \(0.267343\pi\)
\(602\) −46.3990 −1.89108
\(603\) 0 0
\(604\) 21.0437 0.856258
\(605\) 5.50006 0.223609
\(606\) 0 0
\(607\) −7.90851 −0.320997 −0.160498 0.987036i \(-0.551310\pi\)
−0.160498 + 0.987036i \(0.551310\pi\)
\(608\) 45.9940 1.86530
\(609\) 0 0
\(610\) −17.8649 −0.723328
\(611\) 10.0588 0.406936
\(612\) 0 0
\(613\) 29.6052 1.19574 0.597872 0.801592i \(-0.296013\pi\)
0.597872 + 0.801592i \(0.296013\pi\)
\(614\) −27.4975 −1.10971
\(615\) 0 0
\(616\) −1.28265 −0.0516795
\(617\) −1.93021 −0.0777074 −0.0388537 0.999245i \(-0.512371\pi\)
−0.0388537 + 0.999245i \(0.512371\pi\)
\(618\) 0 0
\(619\) −26.0074 −1.04532 −0.522662 0.852540i \(-0.675061\pi\)
−0.522662 + 0.852540i \(0.675061\pi\)
\(620\) 4.58184 0.184011
\(621\) 0 0
\(622\) 1.15204 0.0461924
\(623\) −15.3748 −0.615978
\(624\) 0 0
\(625\) 17.7429 0.709717
\(626\) −44.6513 −1.78462
\(627\) 0 0
\(628\) −15.0482 −0.600488
\(629\) 0 0
\(630\) 0 0
\(631\) −34.5543 −1.37558 −0.687792 0.725908i \(-0.741420\pi\)
−0.687792 + 0.725908i \(0.741420\pi\)
\(632\) 1.57891 0.0628057
\(633\) 0 0
\(634\) 8.84788 0.351394
\(635\) 9.32480 0.370043
\(636\) 0 0
\(637\) −12.0473 −0.477329
\(638\) 11.6053 0.459460
\(639\) 0 0
\(640\) −0.872934 −0.0345058
\(641\) 32.4230 1.28063 0.640316 0.768111i \(-0.278803\pi\)
0.640316 + 0.768111i \(0.278803\pi\)
\(642\) 0 0
\(643\) −1.37140 −0.0540827 −0.0270413 0.999634i \(-0.508609\pi\)
−0.0270413 + 0.999634i \(0.508609\pi\)
\(644\) 19.5250 0.769391
\(645\) 0 0
\(646\) 0 0
\(647\) 5.77601 0.227078 0.113539 0.993534i \(-0.463781\pi\)
0.113539 + 0.993534i \(0.463781\pi\)
\(648\) 0 0
\(649\) −25.1455 −0.987049
\(650\) −7.59575 −0.297930
\(651\) 0 0
\(652\) −52.2949 −2.04803
\(653\) 0.860592 0.0336776 0.0168388 0.999858i \(-0.494640\pi\)
0.0168388 + 0.999858i \(0.494640\pi\)
\(654\) 0 0
\(655\) 8.60516 0.336231
\(656\) 41.2942 1.61227
\(657\) 0 0
\(658\) 112.387 4.38130
\(659\) −12.2049 −0.475435 −0.237718 0.971334i \(-0.576399\pi\)
−0.237718 + 0.971334i \(0.576399\pi\)
\(660\) 0 0
\(661\) −39.5790 −1.53944 −0.769722 0.638380i \(-0.779605\pi\)
−0.769722 + 0.638380i \(0.779605\pi\)
\(662\) 27.3482 1.06292
\(663\) 0 0
\(664\) −1.41531 −0.0549247
\(665\) −18.6696 −0.723974
\(666\) 0 0
\(667\) −6.50403 −0.251837
\(668\) −11.1803 −0.432578
\(669\) 0 0
\(670\) 3.69520 0.142758
\(671\) −22.4628 −0.867169
\(672\) 0 0
\(673\) −0.737727 −0.0284373 −0.0142186 0.999899i \(-0.504526\pi\)
−0.0142186 + 0.999899i \(0.504526\pi\)
\(674\) 38.2521 1.47342
\(675\) 0 0
\(676\) −25.5422 −0.982392
\(677\) 36.0059 1.38382 0.691910 0.721984i \(-0.256769\pi\)
0.691910 + 0.721984i \(0.256769\pi\)
\(678\) 0 0
\(679\) −49.1444 −1.88599
\(680\) 0 0
\(681\) 0 0
\(682\) 11.3101 0.433085
\(683\) −4.20984 −0.161085 −0.0805425 0.996751i \(-0.525665\pi\)
−0.0805425 + 0.996751i \(0.525665\pi\)
\(684\) 0 0
\(685\) 13.5357 0.517172
\(686\) −69.2105 −2.64247
\(687\) 0 0
\(688\) −19.0786 −0.727366
\(689\) 5.26455 0.200563
\(690\) 0 0
\(691\) 20.9026 0.795171 0.397585 0.917565i \(-0.369848\pi\)
0.397585 + 0.917565i \(0.369848\pi\)
\(692\) 14.5148 0.551770
\(693\) 0 0
\(694\) −45.9381 −1.74379
\(695\) 7.58225 0.287611
\(696\) 0 0
\(697\) 0 0
\(698\) −21.6907 −0.821003
\(699\) 0 0
\(700\) −43.2294 −1.63392
\(701\) −1.69253 −0.0639260 −0.0319630 0.999489i \(-0.510176\pi\)
−0.0319630 + 0.999489i \(0.510176\pi\)
\(702\) 0 0
\(703\) 33.9191 1.27928
\(704\) −15.4447 −0.582095
\(705\) 0 0
\(706\) −47.2561 −1.77851
\(707\) −86.3967 −3.24928
\(708\) 0 0
\(709\) −28.9497 −1.08723 −0.543614 0.839336i \(-0.682944\pi\)
−0.543614 + 0.839336i \(0.682944\pi\)
\(710\) −7.68328 −0.288348
\(711\) 0 0
\(712\) 0.512884 0.0192211
\(713\) −6.33854 −0.237380
\(714\) 0 0
\(715\) 1.06238 0.0397307
\(716\) 28.9016 1.08010
\(717\) 0 0
\(718\) −60.9970 −2.27639
\(719\) 12.7690 0.476204 0.238102 0.971240i \(-0.423475\pi\)
0.238102 + 0.971240i \(0.423475\pi\)
\(720\) 0 0
\(721\) 25.3895 0.945553
\(722\) −27.3154 −1.01657
\(723\) 0 0
\(724\) 20.3904 0.757805
\(725\) 14.4003 0.534813
\(726\) 0 0
\(727\) 16.5039 0.612096 0.306048 0.952016i \(-0.400993\pi\)
0.306048 + 0.952016i \(0.400993\pi\)
\(728\) 0.597124 0.0221309
\(729\) 0 0
\(730\) −4.27253 −0.158133
\(731\) 0 0
\(732\) 0 0
\(733\) −11.1444 −0.411627 −0.205814 0.978591i \(-0.565984\pi\)
−0.205814 + 0.978591i \(0.565984\pi\)
\(734\) −25.4913 −0.940899
\(735\) 0 0
\(736\) 16.3886 0.604090
\(737\) 4.64625 0.171147
\(738\) 0 0
\(739\) −18.6191 −0.684913 −0.342456 0.939534i \(-0.611259\pi\)
−0.342456 + 0.939534i \(0.611259\pi\)
\(740\) −8.73642 −0.321157
\(741\) 0 0
\(742\) 58.8208 2.15938
\(743\) −3.29586 −0.120913 −0.0604566 0.998171i \(-0.519256\pi\)
−0.0604566 + 0.998171i \(0.519256\pi\)
\(744\) 0 0
\(745\) −10.0305 −0.367489
\(746\) −25.9761 −0.951052
\(747\) 0 0
\(748\) 0 0
\(749\) −45.2997 −1.65522
\(750\) 0 0
\(751\) 28.4579 1.03844 0.519222 0.854639i \(-0.326222\pi\)
0.519222 + 0.854639i \(0.326222\pi\)
\(752\) 46.2120 1.68518
\(753\) 0 0
\(754\) −5.40274 −0.196756
\(755\) −7.16979 −0.260935
\(756\) 0 0
\(757\) −33.9857 −1.23523 −0.617615 0.786480i \(-0.711901\pi\)
−0.617615 + 0.786480i \(0.711901\pi\)
\(758\) 44.0373 1.59951
\(759\) 0 0
\(760\) 0.622794 0.0225911
\(761\) 18.8297 0.682576 0.341288 0.939959i \(-0.389137\pi\)
0.341288 + 0.939959i \(0.389137\pi\)
\(762\) 0 0
\(763\) 10.7970 0.390877
\(764\) −4.46028 −0.161367
\(765\) 0 0
\(766\) 24.5488 0.886984
\(767\) 11.7062 0.422688
\(768\) 0 0
\(769\) 13.8929 0.500990 0.250495 0.968118i \(-0.419407\pi\)
0.250495 + 0.968118i \(0.419407\pi\)
\(770\) 11.8700 0.427764
\(771\) 0 0
\(772\) 0.0856433 0.00308237
\(773\) −0.699415 −0.0251562 −0.0125781 0.999921i \(-0.504004\pi\)
−0.0125781 + 0.999921i \(0.504004\pi\)
\(774\) 0 0
\(775\) 14.0339 0.504113
\(776\) 1.63940 0.0588510
\(777\) 0 0
\(778\) −26.7730 −0.959858
\(779\) −61.3127 −2.19675
\(780\) 0 0
\(781\) −9.66075 −0.345689
\(782\) 0 0
\(783\) 0 0
\(784\) −55.3472 −1.97669
\(785\) 5.12705 0.182992
\(786\) 0 0
\(787\) 31.7384 1.13135 0.565676 0.824628i \(-0.308615\pi\)
0.565676 + 0.824628i \(0.308615\pi\)
\(788\) 54.2725 1.93338
\(789\) 0 0
\(790\) −14.6116 −0.519858
\(791\) −28.5706 −1.01585
\(792\) 0 0
\(793\) 10.4573 0.371351
\(794\) −51.0903 −1.81313
\(795\) 0 0
\(796\) −35.5699 −1.26074
\(797\) 14.4932 0.513374 0.256687 0.966495i \(-0.417369\pi\)
0.256687 + 0.966495i \(0.417369\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −36.2852 −1.28288
\(801\) 0 0
\(802\) 38.6027 1.36311
\(803\) −5.37217 −0.189580
\(804\) 0 0
\(805\) −6.65233 −0.234464
\(806\) −5.26528 −0.185461
\(807\) 0 0
\(808\) 2.88209 0.101392
\(809\) −12.3944 −0.435764 −0.217882 0.975975i \(-0.569915\pi\)
−0.217882 + 0.975975i \(0.569915\pi\)
\(810\) 0 0
\(811\) 6.89310 0.242049 0.121025 0.992649i \(-0.461382\pi\)
0.121025 + 0.992649i \(0.461382\pi\)
\(812\) −30.7484 −1.07906
\(813\) 0 0
\(814\) −21.5655 −0.755869
\(815\) 17.8173 0.624114
\(816\) 0 0
\(817\) 28.3275 0.991055
\(818\) −6.52475 −0.228133
\(819\) 0 0
\(820\) 15.7921 0.551484
\(821\) 2.01308 0.0702570 0.0351285 0.999383i \(-0.488816\pi\)
0.0351285 + 0.999383i \(0.488816\pi\)
\(822\) 0 0
\(823\) 0.588567 0.0205161 0.0102581 0.999947i \(-0.496735\pi\)
0.0102581 + 0.999947i \(0.496735\pi\)
\(824\) −0.846962 −0.0295053
\(825\) 0 0
\(826\) 130.794 4.55089
\(827\) 13.9282 0.484330 0.242165 0.970235i \(-0.422142\pi\)
0.242165 + 0.970235i \(0.422142\pi\)
\(828\) 0 0
\(829\) −4.52363 −0.157112 −0.0785562 0.996910i \(-0.525031\pi\)
−0.0785562 + 0.996910i \(0.525031\pi\)
\(830\) 13.0976 0.454625
\(831\) 0 0
\(832\) 7.19012 0.249273
\(833\) 0 0
\(834\) 0 0
\(835\) 3.80922 0.131823
\(836\) 21.2699 0.735636
\(837\) 0 0
\(838\) 60.2865 2.08256
\(839\) 25.8744 0.893284 0.446642 0.894713i \(-0.352620\pi\)
0.446642 + 0.894713i \(0.352620\pi\)
\(840\) 0 0
\(841\) −18.7573 −0.646803
\(842\) 40.8574 1.40804
\(843\) 0 0
\(844\) 14.7228 0.506780
\(845\) 8.70245 0.299373
\(846\) 0 0
\(847\) −35.9714 −1.23599
\(848\) 24.1863 0.830561
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0860 0.414303
\(852\) 0 0
\(853\) 13.6688 0.468012 0.234006 0.972235i \(-0.424816\pi\)
0.234006 + 0.972235i \(0.424816\pi\)
\(854\) 116.840 3.99817
\(855\) 0 0
\(856\) 1.51114 0.0516499
\(857\) 48.7835 1.66641 0.833207 0.552961i \(-0.186502\pi\)
0.833207 + 0.552961i \(0.186502\pi\)
\(858\) 0 0
\(859\) −44.1426 −1.50612 −0.753062 0.657949i \(-0.771424\pi\)
−0.753062 + 0.657949i \(0.771424\pi\)
\(860\) −7.29623 −0.248800
\(861\) 0 0
\(862\) −47.8700 −1.63046
\(863\) −47.6997 −1.62372 −0.811858 0.583855i \(-0.801544\pi\)
−0.811858 + 0.583855i \(0.801544\pi\)
\(864\) 0 0
\(865\) −4.94532 −0.168146
\(866\) −66.8529 −2.27175
\(867\) 0 0
\(868\) −29.9661 −1.01711
\(869\) −18.3723 −0.623237
\(870\) 0 0
\(871\) −2.16301 −0.0732908
\(872\) −0.360174 −0.0121970
\(873\) 0 0
\(874\) −23.4019 −0.791582
\(875\) 31.0956 1.05122
\(876\) 0 0
\(877\) 16.4688 0.556111 0.278056 0.960565i \(-0.410310\pi\)
0.278056 + 0.960565i \(0.410310\pi\)
\(878\) −40.3660 −1.36229
\(879\) 0 0
\(880\) 4.88076 0.164530
\(881\) 39.5566 1.33270 0.666348 0.745640i \(-0.267856\pi\)
0.666348 + 0.745640i \(0.267856\pi\)
\(882\) 0 0
\(883\) −25.0310 −0.842360 −0.421180 0.906977i \(-0.638384\pi\)
−0.421180 + 0.906977i \(0.638384\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28.8300 0.968563
\(887\) 44.1968 1.48398 0.741992 0.670409i \(-0.233881\pi\)
0.741992 + 0.670409i \(0.233881\pi\)
\(888\) 0 0
\(889\) −60.9859 −2.04540
\(890\) −4.74635 −0.159098
\(891\) 0 0
\(892\) 45.4441 1.52158
\(893\) −68.6146 −2.29610
\(894\) 0 0
\(895\) −9.84702 −0.329150
\(896\) 5.70915 0.190729
\(897\) 0 0
\(898\) −59.5828 −1.98831
\(899\) 9.98210 0.332922
\(900\) 0 0
\(901\) 0 0
\(902\) 38.9821 1.29796
\(903\) 0 0
\(904\) 0.953081 0.0316990
\(905\) −6.94720 −0.230933
\(906\) 0 0
\(907\) −31.1826 −1.03540 −0.517700 0.855562i \(-0.673212\pi\)
−0.517700 + 0.855562i \(0.673212\pi\)
\(908\) 39.7224 1.31823
\(909\) 0 0
\(910\) −5.52593 −0.183183
\(911\) −37.0795 −1.22850 −0.614249 0.789112i \(-0.710541\pi\)
−0.614249 + 0.789112i \(0.710541\pi\)
\(912\) 0 0
\(913\) 16.4686 0.545031
\(914\) 33.0656 1.09371
\(915\) 0 0
\(916\) −35.4831 −1.17240
\(917\) −56.2793 −1.85851
\(918\) 0 0
\(919\) 31.8291 1.04995 0.524973 0.851119i \(-0.324076\pi\)
0.524973 + 0.851119i \(0.324076\pi\)
\(920\) 0.221914 0.00731627
\(921\) 0 0
\(922\) −6.66637 −0.219545
\(923\) 4.49746 0.148036
\(924\) 0 0
\(925\) −26.7591 −0.879835
\(926\) 46.6503 1.53302
\(927\) 0 0
\(928\) −25.8091 −0.847226
\(929\) 15.8917 0.521390 0.260695 0.965421i \(-0.416048\pi\)
0.260695 + 0.965421i \(0.416048\pi\)
\(930\) 0 0
\(931\) 82.1784 2.69329
\(932\) 41.3891 1.35574
\(933\) 0 0
\(934\) −79.8931 −2.61418
\(935\) 0 0
\(936\) 0 0
\(937\) 16.2472 0.530773 0.265386 0.964142i \(-0.414501\pi\)
0.265386 + 0.964142i \(0.414501\pi\)
\(938\) −24.1673 −0.789090
\(939\) 0 0
\(940\) 17.6728 0.576424
\(941\) −3.98317 −0.129848 −0.0649238 0.997890i \(-0.520680\pi\)
−0.0649238 + 0.997890i \(0.520680\pi\)
\(942\) 0 0
\(943\) −21.8469 −0.711433
\(944\) 53.7806 1.75041
\(945\) 0 0
\(946\) −18.0104 −0.585569
\(947\) −32.1167 −1.04365 −0.521825 0.853052i \(-0.674749\pi\)
−0.521825 + 0.853052i \(0.674749\pi\)
\(948\) 0 0
\(949\) 2.50095 0.0811844
\(950\) 51.8132 1.68104
\(951\) 0 0
\(952\) 0 0
\(953\) −35.5808 −1.15258 −0.576288 0.817246i \(-0.695499\pi\)
−0.576288 + 0.817246i \(0.695499\pi\)
\(954\) 0 0
\(955\) 1.51966 0.0491749
\(956\) 1.13401 0.0366764
\(957\) 0 0
\(958\) 68.8191 2.22344
\(959\) −88.5260 −2.85865
\(960\) 0 0
\(961\) −21.2719 −0.686189
\(962\) 10.0396 0.323689
\(963\) 0 0
\(964\) 39.1606 1.26128
\(965\) −0.0291794 −0.000939319 0
\(966\) 0 0
\(967\) −49.6209 −1.59570 −0.797851 0.602855i \(-0.794030\pi\)
−0.797851 + 0.602855i \(0.794030\pi\)
\(968\) 1.19996 0.0385683
\(969\) 0 0
\(970\) −15.1714 −0.487124
\(971\) 5.02788 0.161352 0.0806762 0.996740i \(-0.474292\pi\)
0.0806762 + 0.996740i \(0.474292\pi\)
\(972\) 0 0
\(973\) −49.5893 −1.58976
\(974\) 81.2112 2.60218
\(975\) 0 0
\(976\) 48.0429 1.53782
\(977\) −26.6435 −0.852401 −0.426201 0.904629i \(-0.640148\pi\)
−0.426201 + 0.904629i \(0.640148\pi\)
\(978\) 0 0
\(979\) −5.96794 −0.190736
\(980\) −21.1664 −0.676137
\(981\) 0 0
\(982\) 42.7323 1.36364
\(983\) 25.0411 0.798687 0.399343 0.916801i \(-0.369238\pi\)
0.399343 + 0.916801i \(0.369238\pi\)
\(984\) 0 0
\(985\) −18.4911 −0.589176
\(986\) 0 0
\(987\) 0 0
\(988\) −9.90198 −0.315024
\(989\) 10.0937 0.320960
\(990\) 0 0
\(991\) 25.4643 0.808901 0.404450 0.914560i \(-0.367463\pi\)
0.404450 + 0.914560i \(0.367463\pi\)
\(992\) −25.1525 −0.798592
\(993\) 0 0
\(994\) 50.2501 1.59384
\(995\) 12.1190 0.384198
\(996\) 0 0
\(997\) −31.7829 −1.00657 −0.503287 0.864119i \(-0.667876\pi\)
−0.503287 + 0.864119i \(0.667876\pi\)
\(998\) −48.7374 −1.54275
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7803.2.a.by.1.4 yes 15
3.2 odd 2 7803.2.a.bz.1.12 yes 15
17.16 even 2 7803.2.a.bx.1.4 15
51.50 odd 2 7803.2.a.ca.1.12 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7803.2.a.bx.1.4 15 17.16 even 2
7803.2.a.by.1.4 yes 15 1.1 even 1 trivial
7803.2.a.bz.1.12 yes 15 3.2 odd 2
7803.2.a.ca.1.12 yes 15 51.50 odd 2