Properties

Label 6930.2.a.cj.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $1$
Dimension $3$
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] = [6930,2,Mod(1,6930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.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: \(55.3363286007\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.06418 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.06418 q^{17} -3.38919 q^{19} -1.00000 q^{20} -1.00000 q^{22} +8.58172 q^{23} +1.00000 q^{25} -3.06418 q^{26} +1.00000 q^{28} +4.12836 q^{29} +1.38919 q^{31} +1.00000 q^{32} -5.06418 q^{34} -1.00000 q^{35} +1.06418 q^{37} -3.38919 q^{38} -1.00000 q^{40} -0.610815 q^{41} -3.38919 q^{43} -1.00000 q^{44} +8.58172 q^{46} -6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -3.06418 q^{52} -9.51754 q^{53} +1.00000 q^{55} +1.00000 q^{56} +4.12836 q^{58} -5.38919 q^{59} -4.00000 q^{61} +1.38919 q^{62} +1.00000 q^{64} +3.06418 q^{65} +15.9709 q^{67} -5.06418 q^{68} -1.00000 q^{70} -14.2567 q^{71} -3.38919 q^{73} +1.06418 q^{74} -3.38919 q^{76} -1.00000 q^{77} -12.2567 q^{79} -1.00000 q^{80} -0.610815 q^{82} -13.6459 q^{83} +5.06418 q^{85} -3.38919 q^{86} -1.00000 q^{88} +12.7101 q^{89} -3.06418 q^{91} +8.58172 q^{92} -6.00000 q^{94} +3.38919 q^{95} -2.73917 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.06418 −0.849850 −0.424925 0.905229i \(-0.639700\pi\)
−0.424925 + 0.905229i \(0.639700\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.06418 −1.22824 −0.614122 0.789211i \(-0.710490\pi\)
−0.614122 + 0.789211i \(0.710490\pi\)
\(18\) 0 0
\(19\) −3.38919 −0.777532 −0.388766 0.921336i \(-0.627099\pi\)
−0.388766 + 0.921336i \(0.627099\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 8.58172 1.78941 0.894706 0.446655i \(-0.147385\pi\)
0.894706 + 0.446655i \(0.147385\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.06418 −0.600935
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 4.12836 0.766616 0.383308 0.923621i \(-0.374785\pi\)
0.383308 + 0.923621i \(0.374785\pi\)
\(30\) 0 0
\(31\) 1.38919 0.249505 0.124753 0.992188i \(-0.460186\pi\)
0.124753 + 0.992188i \(0.460186\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.06418 −0.868499
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 1.06418 0.174950 0.0874749 0.996167i \(-0.472120\pi\)
0.0874749 + 0.996167i \(0.472120\pi\)
\(38\) −3.38919 −0.549798
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −0.610815 −0.0953932 −0.0476966 0.998862i \(-0.515188\pi\)
−0.0476966 + 0.998862i \(0.515188\pi\)
\(42\) 0 0
\(43\) −3.38919 −0.516846 −0.258423 0.966032i \(-0.583203\pi\)
−0.258423 + 0.966032i \(0.583203\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 8.58172 1.26531
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −3.06418 −0.424925
\(53\) −9.51754 −1.30733 −0.653667 0.756782i \(-0.726770\pi\)
−0.653667 + 0.756782i \(0.726770\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.12836 0.542080
\(59\) −5.38919 −0.701612 −0.350806 0.936448i \(-0.614092\pi\)
−0.350806 + 0.936448i \(0.614092\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 1.38919 0.176427
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.06418 0.380064
\(66\) 0 0
\(67\) 15.9709 1.95116 0.975578 0.219652i \(-0.0704923\pi\)
0.975578 + 0.219652i \(0.0704923\pi\)
\(68\) −5.06418 −0.614122
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −14.2567 −1.69196 −0.845980 0.533214i \(-0.820984\pi\)
−0.845980 + 0.533214i \(0.820984\pi\)
\(72\) 0 0
\(73\) −3.38919 −0.396674 −0.198337 0.980134i \(-0.563554\pi\)
−0.198337 + 0.980134i \(0.563554\pi\)
\(74\) 1.06418 0.123708
\(75\) 0 0
\(76\) −3.38919 −0.388766
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −12.2567 −1.37899 −0.689494 0.724292i \(-0.742167\pi\)
−0.689494 + 0.724292i \(0.742167\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −0.610815 −0.0674532
\(83\) −13.6459 −1.49783 −0.748916 0.662665i \(-0.769425\pi\)
−0.748916 + 0.662665i \(0.769425\pi\)
\(84\) 0 0
\(85\) 5.06418 0.549287
\(86\) −3.38919 −0.365465
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 12.7101 1.34727 0.673633 0.739066i \(-0.264733\pi\)
0.673633 + 0.739066i \(0.264733\pi\)
\(90\) 0 0
\(91\) −3.06418 −0.321213
\(92\) 8.58172 0.894706
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 3.38919 0.347723
\(96\) 0 0
\(97\) −2.73917 −0.278121 −0.139060 0.990284i \(-0.544408\pi\)
−0.139060 + 0.990284i \(0.544408\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −14.9067 −1.48327 −0.741637 0.670801i \(-0.765950\pi\)
−0.741637 + 0.670801i \(0.765950\pi\)
\(102\) 0 0
\(103\) 1.38919 0.136881 0.0684403 0.997655i \(-0.478198\pi\)
0.0684403 + 0.997655i \(0.478198\pi\)
\(104\) −3.06418 −0.300467
\(105\) 0 0
\(106\) −9.51754 −0.924425
\(107\) −14.2959 −1.38204 −0.691019 0.722837i \(-0.742838\pi\)
−0.691019 + 0.722837i \(0.742838\pi\)
\(108\) 0 0
\(109\) 5.19253 0.497354 0.248677 0.968586i \(-0.420004\pi\)
0.248677 + 0.968586i \(0.420004\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −5.67499 −0.533858 −0.266929 0.963716i \(-0.586009\pi\)
−0.266929 + 0.963716i \(0.586009\pi\)
\(114\) 0 0
\(115\) −8.58172 −0.800249
\(116\) 4.12836 0.383308
\(117\) 0 0
\(118\) −5.38919 −0.496115
\(119\) −5.06418 −0.464232
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) 1.38919 0.124753
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.61081 0.231672 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.06418 0.268746
\(131\) −10.1284 −0.884919 −0.442459 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(132\) 0 0
\(133\) −3.38919 −0.293880
\(134\) 15.9709 1.37968
\(135\) 0 0
\(136\) −5.06418 −0.434250
\(137\) 2.58172 0.220571 0.110286 0.993900i \(-0.464823\pi\)
0.110286 + 0.993900i \(0.464823\pi\)
\(138\) 0 0
\(139\) −14.7392 −1.25016 −0.625080 0.780561i \(-0.714934\pi\)
−0.625080 + 0.780561i \(0.714934\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −14.2567 −1.19640
\(143\) 3.06418 0.256239
\(144\) 0 0
\(145\) −4.12836 −0.342841
\(146\) −3.38919 −0.280491
\(147\) 0 0
\(148\) 1.06418 0.0874749
\(149\) −4.12836 −0.338208 −0.169104 0.985598i \(-0.554087\pi\)
−0.169104 + 0.985598i \(0.554087\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −3.38919 −0.274899
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −1.38919 −0.111582
\(156\) 0 0
\(157\) −2.16756 −0.172990 −0.0864949 0.996252i \(-0.527567\pi\)
−0.0864949 + 0.996252i \(0.527567\pi\)
\(158\) −12.2567 −0.975092
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 8.58172 0.676334
\(162\) 0 0
\(163\) 5.19253 0.406711 0.203355 0.979105i \(-0.434815\pi\)
0.203355 + 0.979105i \(0.434815\pi\)
\(164\) −0.610815 −0.0476966
\(165\) 0 0
\(166\) −13.6459 −1.05913
\(167\) −17.6750 −1.36773 −0.683866 0.729608i \(-0.739703\pi\)
−0.683866 + 0.729608i \(0.739703\pi\)
\(168\) 0 0
\(169\) −3.61081 −0.277755
\(170\) 5.06418 0.388405
\(171\) 0 0
\(172\) −3.38919 −0.258423
\(173\) 2.25671 0.171575 0.0857873 0.996313i \(-0.472659\pi\)
0.0857873 + 0.996313i \(0.472659\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 12.7101 0.952660
\(179\) −14.8675 −1.11125 −0.555626 0.831433i \(-0.687521\pi\)
−0.555626 + 0.831433i \(0.687521\pi\)
\(180\) 0 0
\(181\) 9.32089 0.692816 0.346408 0.938084i \(-0.387401\pi\)
0.346408 + 0.938084i \(0.387401\pi\)
\(182\) −3.06418 −0.227132
\(183\) 0 0
\(184\) 8.58172 0.632653
\(185\) −1.06418 −0.0782399
\(186\) 0 0
\(187\) 5.06418 0.370329
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 3.38919 0.245877
\(191\) −4.12836 −0.298717 −0.149359 0.988783i \(-0.547721\pi\)
−0.149359 + 0.988783i \(0.547721\pi\)
\(192\) 0 0
\(193\) 7.38919 0.531885 0.265943 0.963989i \(-0.414317\pi\)
0.265943 + 0.963989i \(0.414317\pi\)
\(194\) −2.73917 −0.196661
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −7.64590 −0.544748 −0.272374 0.962192i \(-0.587809\pi\)
−0.272374 + 0.962192i \(0.587809\pi\)
\(198\) 0 0
\(199\) 3.26083 0.231154 0.115577 0.993299i \(-0.463128\pi\)
0.115577 + 0.993299i \(0.463128\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −14.9067 −1.04883
\(203\) 4.12836 0.289754
\(204\) 0 0
\(205\) 0.610815 0.0426611
\(206\) 1.38919 0.0967891
\(207\) 0 0
\(208\) −3.06418 −0.212463
\(209\) 3.38919 0.234435
\(210\) 0 0
\(211\) 10.6209 0.731174 0.365587 0.930777i \(-0.380868\pi\)
0.365587 + 0.930777i \(0.380868\pi\)
\(212\) −9.51754 −0.653667
\(213\) 0 0
\(214\) −14.2959 −0.977248
\(215\) 3.38919 0.231141
\(216\) 0 0
\(217\) 1.38919 0.0943041
\(218\) 5.19253 0.351683
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 15.5175 1.04382
\(222\) 0 0
\(223\) −0.906726 −0.0607189 −0.0303594 0.999539i \(-0.509665\pi\)
−0.0303594 + 0.999539i \(0.509665\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −5.67499 −0.377495
\(227\) 11.7743 0.781485 0.390742 0.920500i \(-0.372218\pi\)
0.390742 + 0.920500i \(0.372218\pi\)
\(228\) 0 0
\(229\) −9.06418 −0.598978 −0.299489 0.954100i \(-0.596816\pi\)
−0.299489 + 0.954100i \(0.596816\pi\)
\(230\) −8.58172 −0.565862
\(231\) 0 0
\(232\) 4.12836 0.271040
\(233\) 13.0351 0.853957 0.426978 0.904262i \(-0.359578\pi\)
0.426978 + 0.904262i \(0.359578\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −5.38919 −0.350806
\(237\) 0 0
\(238\) −5.06418 −0.328262
\(239\) −11.7142 −0.757728 −0.378864 0.925452i \(-0.623685\pi\)
−0.378864 + 0.925452i \(0.623685\pi\)
\(240\) 0 0
\(241\) 3.64590 0.234853 0.117426 0.993082i \(-0.462536\pi\)
0.117426 + 0.993082i \(0.462536\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 10.3851 0.660786
\(248\) 1.38919 0.0882134
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 22.5526 1.42351 0.711754 0.702428i \(-0.247901\pi\)
0.711754 + 0.702428i \(0.247901\pi\)
\(252\) 0 0
\(253\) −8.58172 −0.539528
\(254\) 2.61081 0.163817
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.26083 −0.452918 −0.226459 0.974021i \(-0.572715\pi\)
−0.226459 + 0.974021i \(0.572715\pi\)
\(258\) 0 0
\(259\) 1.06418 0.0661248
\(260\) 3.06418 0.190032
\(261\) 0 0
\(262\) −10.1284 −0.625732
\(263\) 1.87164 0.115411 0.0577053 0.998334i \(-0.481622\pi\)
0.0577053 + 0.998334i \(0.481622\pi\)
\(264\) 0 0
\(265\) 9.51754 0.584658
\(266\) −3.38919 −0.207804
\(267\) 0 0
\(268\) 15.9709 0.975578
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 5.47834 0.332785 0.166393 0.986060i \(-0.446788\pi\)
0.166393 + 0.986060i \(0.446788\pi\)
\(272\) −5.06418 −0.307061
\(273\) 0 0
\(274\) 2.58172 0.155967
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −6.29591 −0.378285 −0.189142 0.981950i \(-0.560571\pi\)
−0.189142 + 0.981950i \(0.560571\pi\)
\(278\) −14.7392 −0.883997
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 8.80747 0.525409 0.262705 0.964876i \(-0.415385\pi\)
0.262705 + 0.964876i \(0.415385\pi\)
\(282\) 0 0
\(283\) −3.67499 −0.218456 −0.109228 0.994017i \(-0.534838\pi\)
−0.109228 + 0.994017i \(0.534838\pi\)
\(284\) −14.2567 −0.845980
\(285\) 0 0
\(286\) 3.06418 0.181189
\(287\) −0.610815 −0.0360552
\(288\) 0 0
\(289\) 8.64590 0.508582
\(290\) −4.12836 −0.242425
\(291\) 0 0
\(292\) −3.38919 −0.198337
\(293\) −13.6851 −0.799492 −0.399746 0.916626i \(-0.630902\pi\)
−0.399746 + 0.916626i \(0.630902\pi\)
\(294\) 0 0
\(295\) 5.38919 0.313771
\(296\) 1.06418 0.0618541
\(297\) 0 0
\(298\) −4.12836 −0.239149
\(299\) −26.2959 −1.52073
\(300\) 0 0
\(301\) −3.38919 −0.195349
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −3.38919 −0.194383
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 7.10338 0.405411 0.202706 0.979240i \(-0.435027\pi\)
0.202706 + 0.979240i \(0.435027\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −1.38919 −0.0789004
\(311\) 22.2276 1.26041 0.630206 0.776428i \(-0.282970\pi\)
0.630206 + 0.776428i \(0.282970\pi\)
\(312\) 0 0
\(313\) 21.6851 1.22571 0.612857 0.790194i \(-0.290020\pi\)
0.612857 + 0.790194i \(0.290020\pi\)
\(314\) −2.16756 −0.122322
\(315\) 0 0
\(316\) −12.2567 −0.689494
\(317\) 6.42427 0.360823 0.180411 0.983591i \(-0.442257\pi\)
0.180411 + 0.983591i \(0.442257\pi\)
\(318\) 0 0
\(319\) −4.12836 −0.231144
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 8.58172 0.478240
\(323\) 17.1634 0.954999
\(324\) 0 0
\(325\) −3.06418 −0.169970
\(326\) 5.19253 0.287588
\(327\) 0 0
\(328\) −0.610815 −0.0337266
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −23.0351 −1.26612 −0.633061 0.774102i \(-0.718202\pi\)
−0.633061 + 0.774102i \(0.718202\pi\)
\(332\) −13.6459 −0.748916
\(333\) 0 0
\(334\) −17.6750 −0.967133
\(335\) −15.9709 −0.872584
\(336\) 0 0
\(337\) −18.9067 −1.02992 −0.514958 0.857216i \(-0.672192\pi\)
−0.514958 + 0.857216i \(0.672192\pi\)
\(338\) −3.61081 −0.196402
\(339\) 0 0
\(340\) 5.06418 0.274644
\(341\) −1.38919 −0.0752286
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −3.38919 −0.182733
\(345\) 0 0
\(346\) 2.25671 0.121322
\(347\) 5.96080 0.319992 0.159996 0.987118i \(-0.448852\pi\)
0.159996 + 0.987118i \(0.448852\pi\)
\(348\) 0 0
\(349\) 11.2918 0.604436 0.302218 0.953239i \(-0.402273\pi\)
0.302218 + 0.953239i \(0.402273\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −14.2959 −0.760895 −0.380447 0.924803i \(-0.624230\pi\)
−0.380447 + 0.924803i \(0.624230\pi\)
\(354\) 0 0
\(355\) 14.2567 0.756668
\(356\) 12.7101 0.673633
\(357\) 0 0
\(358\) −14.8675 −0.785773
\(359\) −25.1343 −1.32654 −0.663270 0.748380i \(-0.730832\pi\)
−0.663270 + 0.748380i \(0.730832\pi\)
\(360\) 0 0
\(361\) −7.51342 −0.395443
\(362\) 9.32089 0.489895
\(363\) 0 0
\(364\) −3.06418 −0.160607
\(365\) 3.38919 0.177398
\(366\) 0 0
\(367\) 8.42427 0.439743 0.219872 0.975529i \(-0.429436\pi\)
0.219872 + 0.975529i \(0.429436\pi\)
\(368\) 8.58172 0.447353
\(369\) 0 0
\(370\) −1.06418 −0.0553240
\(371\) −9.51754 −0.494126
\(372\) 0 0
\(373\) −17.6459 −0.913670 −0.456835 0.889551i \(-0.651017\pi\)
−0.456835 + 0.889551i \(0.651017\pi\)
\(374\) 5.06418 0.261862
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.6500 −0.651509
\(378\) 0 0
\(379\) −30.0702 −1.54460 −0.772300 0.635258i \(-0.780894\pi\)
−0.772300 + 0.635258i \(0.780894\pi\)
\(380\) 3.38919 0.173862
\(381\) 0 0
\(382\) −4.12836 −0.211225
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 7.38919 0.376100
\(387\) 0 0
\(388\) −2.73917 −0.139060
\(389\) −29.7743 −1.50961 −0.754807 0.655947i \(-0.772270\pi\)
−0.754807 + 0.655947i \(0.772270\pi\)
\(390\) 0 0
\(391\) −43.4593 −2.19783
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −7.64590 −0.385195
\(395\) 12.2567 0.616702
\(396\) 0 0
\(397\) 30.7392 1.54275 0.771377 0.636378i \(-0.219568\pi\)
0.771377 + 0.636378i \(0.219568\pi\)
\(398\) 3.26083 0.163451
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −4.25671 −0.212042
\(404\) −14.9067 −0.741637
\(405\) 0 0
\(406\) 4.12836 0.204887
\(407\) −1.06418 −0.0527493
\(408\) 0 0
\(409\) −18.9067 −0.934877 −0.467439 0.884025i \(-0.654823\pi\)
−0.467439 + 0.884025i \(0.654823\pi\)
\(410\) 0.610815 0.0301660
\(411\) 0 0
\(412\) 1.38919 0.0684403
\(413\) −5.38919 −0.265184
\(414\) 0 0
\(415\) 13.6459 0.669851
\(416\) −3.06418 −0.150234
\(417\) 0 0
\(418\) 3.38919 0.165770
\(419\) 22.5526 1.10177 0.550884 0.834582i \(-0.314291\pi\)
0.550884 + 0.834582i \(0.314291\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 10.6209 0.517018
\(423\) 0 0
\(424\) −9.51754 −0.462213
\(425\) −5.06418 −0.245649
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −14.2959 −0.691019
\(429\) 0 0
\(430\) 3.38919 0.163441
\(431\) −12.9358 −0.623097 −0.311548 0.950230i \(-0.600848\pi\)
−0.311548 + 0.950230i \(0.600848\pi\)
\(432\) 0 0
\(433\) 13.7743 0.661948 0.330974 0.943640i \(-0.392623\pi\)
0.330974 + 0.943640i \(0.392623\pi\)
\(434\) 1.38919 0.0666830
\(435\) 0 0
\(436\) 5.19253 0.248677
\(437\) −29.0850 −1.39133
\(438\) 0 0
\(439\) 3.83244 0.182913 0.0914563 0.995809i \(-0.470848\pi\)
0.0914563 + 0.995809i \(0.470848\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 15.5175 0.738094
\(443\) 13.0743 0.621178 0.310589 0.950544i \(-0.399474\pi\)
0.310589 + 0.950544i \(0.399474\pi\)
\(444\) 0 0
\(445\) −12.7101 −0.602515
\(446\) −0.906726 −0.0429347
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −8.25671 −0.389658 −0.194829 0.980837i \(-0.562415\pi\)
−0.194829 + 0.980837i \(0.562415\pi\)
\(450\) 0 0
\(451\) 0.610815 0.0287621
\(452\) −5.67499 −0.266929
\(453\) 0 0
\(454\) 11.7743 0.552593
\(455\) 3.06418 0.143651
\(456\) 0 0
\(457\) −21.1242 −0.988150 −0.494075 0.869419i \(-0.664493\pi\)
−0.494075 + 0.869419i \(0.664493\pi\)
\(458\) −9.06418 −0.423541
\(459\) 0 0
\(460\) −8.58172 −0.400125
\(461\) 21.5175 1.00217 0.501086 0.865398i \(-0.332934\pi\)
0.501086 + 0.865398i \(0.332934\pi\)
\(462\) 0 0
\(463\) 38.3851 1.78391 0.891953 0.452129i \(-0.149335\pi\)
0.891953 + 0.452129i \(0.149335\pi\)
\(464\) 4.12836 0.191654
\(465\) 0 0
\(466\) 13.0351 0.603839
\(467\) −18.6108 −0.861206 −0.430603 0.902541i \(-0.641699\pi\)
−0.430603 + 0.902541i \(0.641699\pi\)
\(468\) 0 0
\(469\) 15.9709 0.737468
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) −5.38919 −0.248057
\(473\) 3.38919 0.155835
\(474\) 0 0
\(475\) −3.38919 −0.155506
\(476\) −5.06418 −0.232116
\(477\) 0 0
\(478\) −11.7142 −0.535795
\(479\) 9.55674 0.436659 0.218329 0.975875i \(-0.429939\pi\)
0.218329 + 0.975875i \(0.429939\pi\)
\(480\) 0 0
\(481\) −3.26083 −0.148681
\(482\) 3.64590 0.166066
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 2.73917 0.124379
\(486\) 0 0
\(487\) 30.7784 1.39470 0.697351 0.716730i \(-0.254362\pi\)
0.697351 + 0.716730i \(0.254362\pi\)
\(488\) −4.00000 −0.181071
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 23.3500 1.05377 0.526885 0.849937i \(-0.323360\pi\)
0.526885 + 0.849937i \(0.323360\pi\)
\(492\) 0 0
\(493\) −20.9067 −0.941592
\(494\) 10.3851 0.467246
\(495\) 0 0
\(496\) 1.38919 0.0623763
\(497\) −14.2567 −0.639501
\(498\) 0 0
\(499\) −40.1985 −1.79953 −0.899766 0.436372i \(-0.856263\pi\)
−0.899766 + 0.436372i \(0.856263\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 22.5526 1.00657
\(503\) −34.1884 −1.52439 −0.762193 0.647350i \(-0.775877\pi\)
−0.762193 + 0.647350i \(0.775877\pi\)
\(504\) 0 0
\(505\) 14.9067 0.663341
\(506\) −8.58172 −0.381504
\(507\) 0 0
\(508\) 2.61081 0.115836
\(509\) −36.3851 −1.61274 −0.806370 0.591412i \(-0.798571\pi\)
−0.806370 + 0.591412i \(0.798571\pi\)
\(510\) 0 0
\(511\) −3.38919 −0.149929
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −7.26083 −0.320261
\(515\) −1.38919 −0.0612148
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 1.06418 0.0467573
\(519\) 0 0
\(520\) 3.06418 0.134373
\(521\) −1.36009 −0.0595866 −0.0297933 0.999556i \(-0.509485\pi\)
−0.0297933 + 0.999556i \(0.509485\pi\)
\(522\) 0 0
\(523\) −1.80335 −0.0788549 −0.0394274 0.999222i \(-0.512553\pi\)
−0.0394274 + 0.999222i \(0.512553\pi\)
\(524\) −10.1284 −0.442459
\(525\) 0 0
\(526\) 1.87164 0.0816076
\(527\) −7.03508 −0.306453
\(528\) 0 0
\(529\) 50.6459 2.20200
\(530\) 9.51754 0.413416
\(531\) 0 0
\(532\) −3.38919 −0.146940
\(533\) 1.87164 0.0810699
\(534\) 0 0
\(535\) 14.2959 0.618066
\(536\) 15.9709 0.689838
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −23.9709 −1.03059 −0.515295 0.857013i \(-0.672318\pi\)
−0.515295 + 0.857013i \(0.672318\pi\)
\(542\) 5.47834 0.235315
\(543\) 0 0
\(544\) −5.06418 −0.217125
\(545\) −5.19253 −0.222424
\(546\) 0 0
\(547\) −11.0743 −0.473502 −0.236751 0.971570i \(-0.576083\pi\)
−0.236751 + 0.971570i \(0.576083\pi\)
\(548\) 2.58172 0.110286
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) −13.9918 −0.596069
\(552\) 0 0
\(553\) −12.2567 −0.521208
\(554\) −6.29591 −0.267488
\(555\) 0 0
\(556\) −14.7392 −0.625080
\(557\) −8.86753 −0.375729 −0.187865 0.982195i \(-0.560157\pi\)
−0.187865 + 0.982195i \(0.560157\pi\)
\(558\) 0 0
\(559\) 10.3851 0.439242
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 8.80747 0.371521
\(563\) 19.2608 0.811747 0.405874 0.913929i \(-0.366967\pi\)
0.405874 + 0.913929i \(0.366967\pi\)
\(564\) 0 0
\(565\) 5.67499 0.238749
\(566\) −3.67499 −0.154471
\(567\) 0 0
\(568\) −14.2567 −0.598198
\(569\) 9.00599 0.377551 0.188775 0.982020i \(-0.439548\pi\)
0.188775 + 0.982020i \(0.439548\pi\)
\(570\) 0 0
\(571\) −16.0993 −0.673733 −0.336867 0.941552i \(-0.609367\pi\)
−0.336867 + 0.941552i \(0.609367\pi\)
\(572\) 3.06418 0.128120
\(573\) 0 0
\(574\) −0.610815 −0.0254949
\(575\) 8.58172 0.357882
\(576\) 0 0
\(577\) 24.1284 1.00448 0.502238 0.864729i \(-0.332510\pi\)
0.502238 + 0.864729i \(0.332510\pi\)
\(578\) 8.64590 0.359622
\(579\) 0 0
\(580\) −4.12836 −0.171421
\(581\) −13.6459 −0.566127
\(582\) 0 0
\(583\) 9.51754 0.394176
\(584\) −3.38919 −0.140245
\(585\) 0 0
\(586\) −13.6851 −0.565326
\(587\) −10.3541 −0.427360 −0.213680 0.976904i \(-0.568545\pi\)
−0.213680 + 0.976904i \(0.568545\pi\)
\(588\) 0 0
\(589\) −4.70821 −0.193998
\(590\) 5.38919 0.221869
\(591\) 0 0
\(592\) 1.06418 0.0437374
\(593\) 8.35597 0.343139 0.171569 0.985172i \(-0.445116\pi\)
0.171569 + 0.985172i \(0.445116\pi\)
\(594\) 0 0
\(595\) 5.06418 0.207611
\(596\) −4.12836 −0.169104
\(597\) 0 0
\(598\) −26.2959 −1.07532
\(599\) 1.60670 0.0656478 0.0328239 0.999461i \(-0.489550\pi\)
0.0328239 + 0.999461i \(0.489550\pi\)
\(600\) 0 0
\(601\) 33.6851 1.37404 0.687022 0.726637i \(-0.258918\pi\)
0.687022 + 0.726637i \(0.258918\pi\)
\(602\) −3.38919 −0.138133
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 1.61493 0.0655481 0.0327741 0.999463i \(-0.489566\pi\)
0.0327741 + 0.999463i \(0.489566\pi\)
\(608\) −3.38919 −0.137450
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 18.3851 0.743780
\(612\) 0 0
\(613\) −6.21751 −0.251123 −0.125561 0.992086i \(-0.540073\pi\)
−0.125561 + 0.992086i \(0.540073\pi\)
\(614\) 7.10338 0.286669
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −23.4884 −0.945609 −0.472805 0.881167i \(-0.656758\pi\)
−0.472805 + 0.881167i \(0.656758\pi\)
\(618\) 0 0
\(619\) 29.0452 1.16742 0.583712 0.811961i \(-0.301600\pi\)
0.583712 + 0.811961i \(0.301600\pi\)
\(620\) −1.38919 −0.0557910
\(621\) 0 0
\(622\) 22.2276 0.891246
\(623\) 12.7101 0.509218
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 21.6851 0.866711
\(627\) 0 0
\(628\) −2.16756 −0.0864949
\(629\) −5.38919 −0.214881
\(630\) 0 0
\(631\) 40.0310 1.59361 0.796804 0.604238i \(-0.206522\pi\)
0.796804 + 0.604238i \(0.206522\pi\)
\(632\) −12.2567 −0.487546
\(633\) 0 0
\(634\) 6.42427 0.255140
\(635\) −2.61081 −0.103607
\(636\) 0 0
\(637\) −3.06418 −0.121407
\(638\) −4.12836 −0.163443
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 10.7000 0.422623 0.211312 0.977419i \(-0.432226\pi\)
0.211312 + 0.977419i \(0.432226\pi\)
\(642\) 0 0
\(643\) 22.0702 0.870362 0.435181 0.900343i \(-0.356684\pi\)
0.435181 + 0.900343i \(0.356684\pi\)
\(644\) 8.58172 0.338167
\(645\) 0 0
\(646\) 17.1634 0.675286
\(647\) −33.2918 −1.30884 −0.654418 0.756133i \(-0.727086\pi\)
−0.654418 + 0.756133i \(0.727086\pi\)
\(648\) 0 0
\(649\) 5.38919 0.211544
\(650\) −3.06418 −0.120187
\(651\) 0 0
\(652\) 5.19253 0.203355
\(653\) 8.21751 0.321576 0.160788 0.986989i \(-0.448596\pi\)
0.160788 + 0.986989i \(0.448596\pi\)
\(654\) 0 0
\(655\) 10.1284 0.395748
\(656\) −0.610815 −0.0238483
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) 19.0351 0.741502 0.370751 0.928732i \(-0.379100\pi\)
0.370751 + 0.928732i \(0.379100\pi\)
\(660\) 0 0
\(661\) 6.87763 0.267509 0.133754 0.991015i \(-0.457297\pi\)
0.133754 + 0.991015i \(0.457297\pi\)
\(662\) −23.0351 −0.895284
\(663\) 0 0
\(664\) −13.6459 −0.529563
\(665\) 3.38919 0.131427
\(666\) 0 0
\(667\) 35.4284 1.37179
\(668\) −17.6750 −0.683866
\(669\) 0 0
\(670\) −15.9709 −0.617010
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 46.4552 1.79072 0.895359 0.445346i \(-0.146919\pi\)
0.895359 + 0.445346i \(0.146919\pi\)
\(674\) −18.9067 −0.728260
\(675\) 0 0
\(676\) −3.61081 −0.138877
\(677\) 33.9418 1.30449 0.652245 0.758008i \(-0.273827\pi\)
0.652245 + 0.758008i \(0.273827\pi\)
\(678\) 0 0
\(679\) −2.73917 −0.105120
\(680\) 5.06418 0.194202
\(681\) 0 0
\(682\) −1.38919 −0.0531947
\(683\) −39.7161 −1.51969 −0.759846 0.650103i \(-0.774726\pi\)
−0.759846 + 0.650103i \(0.774726\pi\)
\(684\) 0 0
\(685\) −2.58172 −0.0986424
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −3.38919 −0.129211
\(689\) 29.1634 1.11104
\(690\) 0 0
\(691\) 40.9668 1.55845 0.779225 0.626744i \(-0.215613\pi\)
0.779225 + 0.626744i \(0.215613\pi\)
\(692\) 2.25671 0.0857873
\(693\) 0 0
\(694\) 5.96080 0.226269
\(695\) 14.7392 0.559089
\(696\) 0 0
\(697\) 3.09327 0.117166
\(698\) 11.2918 0.427401
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 6.57161 0.248206 0.124103 0.992269i \(-0.460395\pi\)
0.124103 + 0.992269i \(0.460395\pi\)
\(702\) 0 0
\(703\) −3.60670 −0.136029
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −14.2959 −0.538034
\(707\) −14.9067 −0.560625
\(708\) 0 0
\(709\) 11.4783 0.431078 0.215539 0.976495i \(-0.430849\pi\)
0.215539 + 0.976495i \(0.430849\pi\)
\(710\) 14.2567 0.535045
\(711\) 0 0
\(712\) 12.7101 0.476330
\(713\) 11.9216 0.446467
\(714\) 0 0
\(715\) −3.06418 −0.114594
\(716\) −14.8675 −0.555626
\(717\) 0 0
\(718\) −25.1343 −0.938005
\(719\) −39.1925 −1.46163 −0.730817 0.682573i \(-0.760861\pi\)
−0.730817 + 0.682573i \(0.760861\pi\)
\(720\) 0 0
\(721\) 1.38919 0.0517360
\(722\) −7.51342 −0.279621
\(723\) 0 0
\(724\) 9.32089 0.346408
\(725\) 4.12836 0.153323
\(726\) 0 0
\(727\) 2.83656 0.105202 0.0526011 0.998616i \(-0.483249\pi\)
0.0526011 + 0.998616i \(0.483249\pi\)
\(728\) −3.06418 −0.113566
\(729\) 0 0
\(730\) 3.38919 0.125439
\(731\) 17.1634 0.634813
\(732\) 0 0
\(733\) −37.3911 −1.38107 −0.690535 0.723299i \(-0.742625\pi\)
−0.690535 + 0.723299i \(0.742625\pi\)
\(734\) 8.42427 0.310945
\(735\) 0 0
\(736\) 8.58172 0.316326
\(737\) −15.9709 −0.588296
\(738\) 0 0
\(739\) −5.39929 −0.198616 −0.0993081 0.995057i \(-0.531663\pi\)
−0.0993081 + 0.995057i \(0.531663\pi\)
\(740\) −1.06418 −0.0391200
\(741\) 0 0
\(742\) −9.51754 −0.349400
\(743\) 20.2567 0.743147 0.371573 0.928404i \(-0.378818\pi\)
0.371573 + 0.928404i \(0.378818\pi\)
\(744\) 0 0
\(745\) 4.12836 0.151251
\(746\) −17.6459 −0.646062
\(747\) 0 0
\(748\) 5.06418 0.185165
\(749\) −14.2959 −0.522361
\(750\) 0 0
\(751\) 14.0392 0.512298 0.256149 0.966637i \(-0.417546\pi\)
0.256149 + 0.966637i \(0.417546\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −12.6500 −0.460686
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −22.4843 −0.817207 −0.408603 0.912712i \(-0.633984\pi\)
−0.408603 + 0.912712i \(0.633984\pi\)
\(758\) −30.0702 −1.09220
\(759\) 0 0
\(760\) 3.38919 0.122939
\(761\) −6.42427 −0.232880 −0.116440 0.993198i \(-0.537148\pi\)
−0.116440 + 0.993198i \(0.537148\pi\)
\(762\) 0 0
\(763\) 5.19253 0.187982
\(764\) −4.12836 −0.149359
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 16.5134 0.596265
\(768\) 0 0
\(769\) −36.6418 −1.32134 −0.660668 0.750678i \(-0.729727\pi\)
−0.660668 + 0.750678i \(0.729727\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 7.38919 0.265943
\(773\) 1.26083 0.0453489 0.0226744 0.999743i \(-0.492782\pi\)
0.0226744 + 0.999743i \(0.492782\pi\)
\(774\) 0 0
\(775\) 1.38919 0.0499010
\(776\) −2.73917 −0.0983305
\(777\) 0 0
\(778\) −29.7743 −1.06746
\(779\) 2.07016 0.0741713
\(780\) 0 0
\(781\) 14.2567 0.510145
\(782\) −43.4593 −1.55410
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 2.16756 0.0773634
\(786\) 0 0
\(787\) 46.3168 1.65101 0.825507 0.564391i \(-0.190889\pi\)
0.825507 + 0.564391i \(0.190889\pi\)
\(788\) −7.64590 −0.272374
\(789\) 0 0
\(790\) 12.2567 0.436074
\(791\) −5.67499 −0.201779
\(792\) 0 0
\(793\) 12.2567 0.435249
\(794\) 30.7392 1.09089
\(795\) 0 0
\(796\) 3.26083 0.115577
\(797\) −30.9959 −1.09793 −0.548965 0.835845i \(-0.684978\pi\)
−0.548965 + 0.835845i \(0.684978\pi\)
\(798\) 0 0
\(799\) 30.3851 1.07495
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 3.38919 0.119602
\(804\) 0 0
\(805\) −8.58172 −0.302466
\(806\) −4.25671 −0.149936
\(807\) 0 0
\(808\) −14.9067 −0.524417
\(809\) 29.2627 1.02882 0.514411 0.857544i \(-0.328011\pi\)
0.514411 + 0.857544i \(0.328011\pi\)
\(810\) 0 0
\(811\) −38.9377 −1.36729 −0.683644 0.729816i \(-0.739606\pi\)
−0.683644 + 0.729816i \(0.739606\pi\)
\(812\) 4.12836 0.144877
\(813\) 0 0
\(814\) −1.06418 −0.0372994
\(815\) −5.19253 −0.181887
\(816\) 0 0
\(817\) 11.4866 0.401864
\(818\) −18.9067 −0.661058
\(819\) 0 0
\(820\) 0.610815 0.0213306
\(821\) −29.5485 −1.03125 −0.515625 0.856814i \(-0.672440\pi\)
−0.515625 + 0.856814i \(0.672440\pi\)
\(822\) 0 0
\(823\) 23.8634 0.831826 0.415913 0.909404i \(-0.363462\pi\)
0.415913 + 0.909404i \(0.363462\pi\)
\(824\) 1.38919 0.0483946
\(825\) 0 0
\(826\) −5.38919 −0.187514
\(827\) 44.0310 1.53111 0.765553 0.643372i \(-0.222465\pi\)
0.765553 + 0.643372i \(0.222465\pi\)
\(828\) 0 0
\(829\) −47.7844 −1.65962 −0.829810 0.558047i \(-0.811551\pi\)
−0.829810 + 0.558047i \(0.811551\pi\)
\(830\) 13.6459 0.473656
\(831\) 0 0
\(832\) −3.06418 −0.106231
\(833\) −5.06418 −0.175463
\(834\) 0 0
\(835\) 17.6750 0.611668
\(836\) 3.38919 0.117217
\(837\) 0 0
\(838\) 22.5526 0.779067
\(839\) 12.6709 0.437447 0.218724 0.975787i \(-0.429811\pi\)
0.218724 + 0.975787i \(0.429811\pi\)
\(840\) 0 0
\(841\) −11.9567 −0.412299
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) 10.6209 0.365587
\(845\) 3.61081 0.124216
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −9.51754 −0.326834
\(849\) 0 0
\(850\) −5.06418 −0.173700
\(851\) 9.13247 0.313057
\(852\) 0 0
\(853\) −20.8776 −0.714836 −0.357418 0.933944i \(-0.616343\pi\)
−0.357418 + 0.933944i \(0.616343\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −14.2959 −0.488624
\(857\) −3.19253 −0.109055 −0.0545274 0.998512i \(-0.517365\pi\)
−0.0545274 + 0.998512i \(0.517365\pi\)
\(858\) 0 0
\(859\) 21.3601 0.728797 0.364398 0.931243i \(-0.381275\pi\)
0.364398 + 0.931243i \(0.381275\pi\)
\(860\) 3.38919 0.115570
\(861\) 0 0
\(862\) −12.9358 −0.440596
\(863\) −55.5586 −1.89124 −0.945619 0.325278i \(-0.894542\pi\)
−0.945619 + 0.325278i \(0.894542\pi\)
\(864\) 0 0
\(865\) −2.25671 −0.0767305
\(866\) 13.7743 0.468068
\(867\) 0 0
\(868\) 1.38919 0.0471520
\(869\) 12.2567 0.415780
\(870\) 0 0
\(871\) −48.9377 −1.65819
\(872\) 5.19253 0.175841
\(873\) 0 0
\(874\) −29.0850 −0.983816
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −18.2175 −0.615162 −0.307581 0.951522i \(-0.599519\pi\)
−0.307581 + 0.951522i \(0.599519\pi\)
\(878\) 3.83244 0.129339
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 24.7101 0.832504 0.416252 0.909249i \(-0.363343\pi\)
0.416252 + 0.909249i \(0.363343\pi\)
\(882\) 0 0
\(883\) 11.3791 0.382937 0.191468 0.981499i \(-0.438675\pi\)
0.191468 + 0.981499i \(0.438675\pi\)
\(884\) 15.5175 0.521911
\(885\) 0 0
\(886\) 13.0743 0.439239
\(887\) 43.7452 1.46882 0.734409 0.678707i \(-0.237459\pi\)
0.734409 + 0.678707i \(0.237459\pi\)
\(888\) 0 0
\(889\) 2.61081 0.0875639
\(890\) −12.7101 −0.426043
\(891\) 0 0
\(892\) −0.906726 −0.0303594
\(893\) 20.3351 0.680489
\(894\) 0 0
\(895\) 14.8675 0.496967
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −8.25671 −0.275530
\(899\) 5.73505 0.191275
\(900\) 0 0
\(901\) 48.1985 1.60573
\(902\) 0.610815 0.0203379
\(903\) 0 0
\(904\) −5.67499 −0.188747
\(905\) −9.32089 −0.309837
\(906\) 0 0
\(907\) 19.0642 0.633016 0.316508 0.948590i \(-0.397490\pi\)
0.316508 + 0.948590i \(0.397490\pi\)
\(908\) 11.7743 0.390742
\(909\) 0 0
\(910\) 3.06418 0.101577
\(911\) 40.7784 1.35105 0.675524 0.737338i \(-0.263918\pi\)
0.675524 + 0.737338i \(0.263918\pi\)
\(912\) 0 0
\(913\) 13.6459 0.451613
\(914\) −21.1242 −0.698728
\(915\) 0 0
\(916\) −9.06418 −0.299489
\(917\) −10.1284 −0.334468
\(918\) 0 0
\(919\) 43.6269 1.43912 0.719559 0.694431i \(-0.244344\pi\)
0.719559 + 0.694431i \(0.244344\pi\)
\(920\) −8.58172 −0.282931
\(921\) 0 0
\(922\) 21.5175 0.708642
\(923\) 43.6851 1.43791
\(924\) 0 0
\(925\) 1.06418 0.0349899
\(926\) 38.3851 1.26141
\(927\) 0 0
\(928\) 4.12836 0.135520
\(929\) 12.1385 0.398250 0.199125 0.979974i \(-0.436190\pi\)
0.199125 + 0.979974i \(0.436190\pi\)
\(930\) 0 0
\(931\) −3.38919 −0.111076
\(932\) 13.0351 0.426978
\(933\) 0 0
\(934\) −18.6108 −0.608964
\(935\) −5.06418 −0.165616
\(936\) 0 0
\(937\) 25.1242 0.820773 0.410387 0.911912i \(-0.365394\pi\)
0.410387 + 0.911912i \(0.365394\pi\)
\(938\) 15.9709 0.521469
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 36.3851 1.18612 0.593060 0.805159i \(-0.297920\pi\)
0.593060 + 0.805159i \(0.297920\pi\)
\(942\) 0 0
\(943\) −5.24184 −0.170698
\(944\) −5.38919 −0.175403
\(945\) 0 0
\(946\) 3.38919 0.110192
\(947\) 46.5526 1.51276 0.756378 0.654134i \(-0.226967\pi\)
0.756378 + 0.654134i \(0.226967\pi\)
\(948\) 0 0
\(949\) 10.3851 0.337114
\(950\) −3.38919 −0.109960
\(951\) 0 0
\(952\) −5.06418 −0.164131
\(953\) 30.6500 0.992851 0.496426 0.868079i \(-0.334646\pi\)
0.496426 + 0.868079i \(0.334646\pi\)
\(954\) 0 0
\(955\) 4.12836 0.133590
\(956\) −11.7142 −0.378864
\(957\) 0 0
\(958\) 9.55674 0.308764
\(959\) 2.58172 0.0833680
\(960\) 0 0
\(961\) −29.0702 −0.937747
\(962\) −3.26083 −0.105133
\(963\) 0 0
\(964\) 3.64590 0.117426
\(965\) −7.38919 −0.237866
\(966\) 0 0
\(967\) 2.61081 0.0839581 0.0419791 0.999118i \(-0.486634\pi\)
0.0419791 + 0.999118i \(0.486634\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 2.73917 0.0879495
\(971\) −60.4243 −1.93911 −0.969554 0.244880i \(-0.921252\pi\)
−0.969554 + 0.244880i \(0.921252\pi\)
\(972\) 0 0
\(973\) −14.7392 −0.472516
\(974\) 30.7784 0.986203
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 44.9668 1.43861 0.719307 0.694692i \(-0.244459\pi\)
0.719307 + 0.694692i \(0.244459\pi\)
\(978\) 0 0
\(979\) −12.7101 −0.406216
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 23.3500 0.745128
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 7.64590 0.243619
\(986\) −20.9067 −0.665806
\(987\) 0 0
\(988\) 10.3851 0.330393
\(989\) −29.0850 −0.924850
\(990\) 0 0
\(991\) 20.2257 0.642492 0.321246 0.946996i \(-0.395898\pi\)
0.321246 + 0.946996i \(0.395898\pi\)
\(992\) 1.38919 0.0441067
\(993\) 0 0
\(994\) −14.2567 −0.452195
\(995\) −3.26083 −0.103375
\(996\) 0 0
\(997\) 17.8425 0.565079 0.282540 0.959256i \(-0.408823\pi\)
0.282540 + 0.959256i \(0.408823\pi\)
\(998\) −40.1985 −1.27246
\(999\) 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 6930.2.a.cj.1.1 yes 3
3.2 odd 2 6930.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.ci.1.1 3 3.2 odd 2
6930.2.a.cj.1.1 yes 3 1.1 even 1 trivial