Properties

Label 3150.2.d.d.3149.8
Level $3150$
Weight $2$
Character 3150.3149
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(3149,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.3149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.8
Root \(2.73923i\) of defining polynomial
Character \(\chi\) \(=\) 3150.3149
Dual form 3150.2.d.d.3149.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(1.93693 + 1.80230i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(1.93693 + 1.80230i) q^{7} +1.00000 q^{8} -3.87386i q^{11} +1.60461 q^{13} +(1.93693 + 1.80230i) q^{14} +1.00000 q^{16} +8.11650i q^{17} -2.63803i q^{19} -3.87386i q^{22} +5.47847 q^{23} +1.60461 q^{26} +(1.93693 + 1.80230i) q^{28} +5.47847i q^{29} -3.73074i q^{31} +1.00000 q^{32} +8.11650i q^{34} -4.51190i q^{37} -2.63803i q^{38} +1.60461 q^{41} -10.1165i q^{43} -3.87386i q^{44} +5.47847 q^{46} +11.1097i q^{47} +(0.503406 + 6.98188i) q^{49} +1.60461 q^{52} -2.26926 q^{53} +(1.93693 + 1.80230i) q^{56} +5.47847i q^{58} +4.61142 q^{59} +11.8472i q^{61} -3.73074i q^{62} +1.00000 q^{64} -6.90729i q^{67} +8.11650i q^{68} -2.63803i q^{71} +13.7477 q^{73} -4.51190i q^{74} -2.63803i q^{76} +(6.98188 - 7.50341i) q^{77} +8.01698 q^{79} +1.60461 q^{82} -3.20921i q^{83} -10.1165i q^{86} -3.87386i q^{88} -17.8376 q^{89} +(3.10801 + 2.89199i) q^{91} +5.47847 q^{92} +11.1097i q^{94} +8.68768 q^{97} +(0.503406 + 6.98188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 8 q^{13} + 8 q^{16} - 8 q^{23} - 8 q^{26} + 8 q^{32} - 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} - 8 q^{53} + 8 q^{64} + 48 q^{73} - 4 q^{77} - 8 q^{79} - 8 q^{82} + 8 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 1.93693 + 1.80230i 0.732091 + 0.681207i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 3.87386i 1.16801i −0.811749 0.584007i \(-0.801484\pi\)
0.811749 0.584007i \(-0.198516\pi\)
\(12\) 0 0
\(13\) 1.60461 0.445038 0.222519 0.974928i \(-0.428572\pi\)
0.222519 + 0.974928i \(0.428572\pi\)
\(14\) 1.93693 + 1.80230i 0.517667 + 0.481686i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.11650i 1.96854i 0.176667 + 0.984271i \(0.443468\pi\)
−0.176667 + 0.984271i \(0.556532\pi\)
\(18\) 0 0
\(19\) 2.63803i 0.605207i −0.953117 0.302603i \(-0.902144\pi\)
0.953117 0.302603i \(-0.0978557\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.87386i 0.825910i
\(23\) 5.47847 1.14234 0.571170 0.820832i \(-0.306490\pi\)
0.571170 + 0.820832i \(0.306490\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.60461 0.314689
\(27\) 0 0
\(28\) 1.93693 + 1.80230i 0.366046 + 0.340603i
\(29\) 5.47847i 1.01733i 0.860966 + 0.508663i \(0.169860\pi\)
−0.860966 + 0.508663i \(0.830140\pi\)
\(30\) 0 0
\(31\) 3.73074i 0.670061i −0.942207 0.335031i \(-0.891253\pi\)
0.942207 0.335031i \(-0.108747\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 8.11650i 1.39197i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.51190i 0.741751i −0.928683 0.370876i \(-0.879058\pi\)
0.928683 0.370876i \(-0.120942\pi\)
\(38\) 2.63803i 0.427946i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.60461 0.250597 0.125299 0.992119i \(-0.460011\pi\)
0.125299 + 0.992119i \(0.460011\pi\)
\(42\) 0 0
\(43\) 10.1165i 1.54275i −0.636379 0.771376i \(-0.719569\pi\)
0.636379 0.771376i \(-0.280431\pi\)
\(44\) 3.87386i 0.584007i
\(45\) 0 0
\(46\) 5.47847 0.807756
\(47\) 11.1097i 1.62052i 0.586074 + 0.810258i \(0.300673\pi\)
−0.586074 + 0.810258i \(0.699327\pi\)
\(48\) 0 0
\(49\) 0.503406 + 6.98188i 0.0719152 + 0.997411i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.60461 0.222519
\(53\) −2.26926 −0.311706 −0.155853 0.987780i \(-0.549813\pi\)
−0.155853 + 0.987780i \(0.549813\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.93693 + 1.80230i 0.258833 + 0.240843i
\(57\) 0 0
\(58\) 5.47847i 0.719358i
\(59\) 4.61142 0.600356 0.300178 0.953883i \(-0.402954\pi\)
0.300178 + 0.953883i \(0.402954\pi\)
\(60\) 0 0
\(61\) 11.8472i 1.51688i 0.651740 + 0.758442i \(0.274039\pi\)
−0.651740 + 0.758442i \(0.725961\pi\)
\(62\) 3.73074i 0.473805i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.90729i 0.843860i −0.906628 0.421930i \(-0.861353\pi\)
0.906628 0.421930i \(-0.138647\pi\)
\(68\) 8.11650i 0.984271i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.63803i 0.313077i −0.987672 0.156539i \(-0.949966\pi\)
0.987672 0.156539i \(-0.0500335\pi\)
\(72\) 0 0
\(73\) 13.7477 1.60905 0.804525 0.593919i \(-0.202420\pi\)
0.804525 + 0.593919i \(0.202420\pi\)
\(74\) 4.51190i 0.524497i
\(75\) 0 0
\(76\) 2.63803i 0.302603i
\(77\) 6.98188 7.50341i 0.795659 0.855092i
\(78\) 0 0
\(79\) 8.01698 0.901981 0.450990 0.892529i \(-0.351071\pi\)
0.450990 + 0.892529i \(0.351071\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.60461 0.177199
\(83\) 3.20921i 0.352257i −0.984367 0.176128i \(-0.943643\pi\)
0.984367 0.176128i \(-0.0563574\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.1165i 1.09089i
\(87\) 0 0
\(88\) 3.87386i 0.412955i
\(89\) −17.8376 −1.89078 −0.945392 0.325937i \(-0.894320\pi\)
−0.945392 + 0.325937i \(0.894320\pi\)
\(90\) 0 0
\(91\) 3.10801 + 2.89199i 0.325808 + 0.303163i
\(92\) 5.47847 0.571170
\(93\) 0 0
\(94\) 11.1097i 1.14588i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.68768 0.882100 0.441050 0.897482i \(-0.354606\pi\)
0.441050 + 0.897482i \(0.354606\pi\)
\(98\) 0.503406 + 6.98188i 0.0508517 + 0.705276i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.21603 −0.618518 −0.309259 0.950978i \(-0.600081\pi\)
−0.309259 + 0.950978i \(0.600081\pi\)
\(102\) 0 0
\(103\) −12.8807 −1.26917 −0.634585 0.772853i \(-0.718829\pi\)
−0.634585 + 0.772853i \(0.718829\pi\)
\(104\) 1.60461 0.157345
\(105\) 0 0
\(106\) −2.26926 −0.220410
\(107\) 19.9638 1.92997 0.964984 0.262308i \(-0.0844835\pi\)
0.964984 + 0.262308i \(0.0844835\pi\)
\(108\) 0 0
\(109\) 15.0238 1.43902 0.719509 0.694483i \(-0.244367\pi\)
0.719509 + 0.694483i \(0.244367\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.93693 + 1.80230i 0.183023 + 0.170302i
\(113\) −10.2330 −0.962640 −0.481320 0.876545i \(-0.659843\pi\)
−0.481320 + 0.876545i \(0.659843\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.47847i 0.508663i
\(117\) 0 0
\(118\) 4.61142 0.424515
\(119\) −14.6284 + 15.7211i −1.34098 + 1.44115i
\(120\) 0 0
\(121\) −4.00681 −0.364256
\(122\) 11.8472i 1.07260i
\(123\) 0 0
\(124\) 3.73074i 0.335031i
\(125\) 0 0
\(126\) 0 0
\(127\) 2.81382i 0.249686i 0.992177 + 0.124843i \(0.0398427\pi\)
−0.992177 + 0.124843i \(0.960157\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 8.62840 0.753867 0.376933 0.926240i \(-0.376979\pi\)
0.376933 + 0.926240i \(0.376979\pi\)
\(132\) 0 0
\(133\) 4.75454 5.10969i 0.412271 0.443066i
\(134\) 6.90729i 0.596699i
\(135\) 0 0
\(136\) 8.11650i 0.695984i
\(137\) 0.723932 0.0618496 0.0309248 0.999522i \(-0.490155\pi\)
0.0309248 + 0.999522i \(0.490155\pi\)
\(138\) 0 0
\(139\) 2.84043i 0.240923i 0.992718 + 0.120461i \(0.0384374\pi\)
−0.992718 + 0.120461i \(0.961563\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.63803i 0.221379i
\(143\) 6.21603i 0.519810i
\(144\) 0 0
\(145\) 0 0
\(146\) 13.7477 1.13777
\(147\) 0 0
\(148\) 4.51190i 0.370876i
\(149\) 3.00681i 0.246328i 0.992386 + 0.123164i \(0.0393041\pi\)
−0.992386 + 0.123164i \(0.960696\pi\)
\(150\) 0 0
\(151\) −15.2262 −1.23909 −0.619545 0.784961i \(-0.712683\pi\)
−0.619545 + 0.784961i \(0.712683\pi\)
\(152\) 2.63803i 0.213973i
\(153\) 0 0
\(154\) 6.98188 7.50341i 0.562616 0.604642i
\(155\) 0 0
\(156\) 0 0
\(157\) −2.64767 −0.211307 −0.105653 0.994403i \(-0.533693\pi\)
−0.105653 + 0.994403i \(0.533693\pi\)
\(158\) 8.01698 0.637797
\(159\) 0 0
\(160\) 0 0
\(161\) 10.6114 + 9.87386i 0.836297 + 0.778169i
\(162\) 0 0
\(163\) 9.32572i 0.730446i −0.930920 0.365223i \(-0.880993\pi\)
0.930920 0.365223i \(-0.119007\pi\)
\(164\) 1.60461 0.125299
\(165\) 0 0
\(166\) 3.20921i 0.249083i
\(167\) 17.3257i 1.34070i 0.742043 + 0.670352i \(0.233857\pi\)
−0.742043 + 0.670352i \(0.766143\pi\)
\(168\) 0 0
\(169\) −10.4252 −0.801941
\(170\) 0 0
\(171\) 0 0
\(172\) 10.1165i 0.771376i
\(173\) 2.99319i 0.227568i −0.993506 0.113784i \(-0.963703\pi\)
0.993506 0.113784i \(-0.0362972\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.87386i 0.292003i
\(177\) 0 0
\(178\) −17.8376 −1.33699
\(179\) 7.38858i 0.552248i 0.961122 + 0.276124i \(0.0890501\pi\)
−0.961122 + 0.276124i \(0.910950\pi\)
\(180\) 0 0
\(181\) 11.1097i 0.825777i −0.910782 0.412888i \(-0.864520\pi\)
0.910782 0.412888i \(-0.135480\pi\)
\(182\) 3.10801 + 2.89199i 0.230381 + 0.214368i
\(183\) 0 0
\(184\) 5.47847 0.403878
\(185\) 0 0
\(186\) 0 0
\(187\) 31.4422 2.29928
\(188\) 11.1097i 0.810258i
\(189\) 0 0
\(190\) 0 0
\(191\) 9.36197i 0.677408i −0.940893 0.338704i \(-0.890011\pi\)
0.940893 0.338704i \(-0.109989\pi\)
\(192\) 0 0
\(193\) 17.4422i 1.25552i 0.778408 + 0.627759i \(0.216028\pi\)
−0.778408 + 0.627759i \(0.783972\pi\)
\(194\) 8.68768 0.623739
\(195\) 0 0
\(196\) 0.503406 + 6.98188i 0.0359576 + 0.498705i
\(197\) −5.27607 −0.375904 −0.187952 0.982178i \(-0.560185\pi\)
−0.187952 + 0.982178i \(0.560185\pi\)
\(198\) 0 0
\(199\) 12.2160i 0.865971i 0.901401 + 0.432986i \(0.142540\pi\)
−0.901401 + 0.432986i \(0.857460\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.21603 −0.437358
\(203\) −9.87386 + 10.6114i −0.693009 + 0.744776i
\(204\) 0 0
\(205\) 0 0
\(206\) −12.8807 −0.897439
\(207\) 0 0
\(208\) 1.60461 0.111259
\(209\) −10.2194 −0.706889
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.26926 −0.155853
\(213\) 0 0
\(214\) 19.9638 1.36469
\(215\) 0 0
\(216\) 0 0
\(217\) 6.72393 7.22619i 0.456450 0.490546i
\(218\) 15.0238 1.01754
\(219\) 0 0
\(220\) 0 0
\(221\) 13.0238i 0.876075i
\(222\) 0 0
\(223\) 3.65784 0.244947 0.122473 0.992472i \(-0.460917\pi\)
0.122473 + 0.992472i \(0.460917\pi\)
\(224\) 1.93693 + 1.80230i 0.129417 + 0.120421i
\(225\) 0 0
\(226\) −10.2330 −0.680689
\(227\) 4.23301i 0.280955i 0.990084 + 0.140477i \(0.0448637\pi\)
−0.990084 + 0.140477i \(0.955136\pi\)
\(228\) 0 0
\(229\) 7.59497i 0.501890i 0.968001 + 0.250945i \(0.0807413\pi\)
−0.968001 + 0.250945i \(0.919259\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.47847i 0.359679i
\(233\) 9.94677 0.651634 0.325817 0.945433i \(-0.394361\pi\)
0.325817 + 0.945433i \(0.394361\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.61142 0.300178
\(237\) 0 0
\(238\) −14.6284 + 15.7211i −0.948218 + 1.01905i
\(239\) 9.36197i 0.605575i −0.953058 0.302788i \(-0.902083\pi\)
0.953058 0.302788i \(-0.0979173\pi\)
\(240\) 0 0
\(241\) 23.0408i 1.48419i −0.670296 0.742093i \(-0.733833\pi\)
0.670296 0.742093i \(-0.266167\pi\)
\(242\) −4.00681 −0.257568
\(243\) 0 0
\(244\) 11.8472i 0.758442i
\(245\) 0 0
\(246\) 0 0
\(247\) 4.23301i 0.269340i
\(248\) 3.73074i 0.236902i
\(249\) 0 0
\(250\) 0 0
\(251\) −23.8376 −1.50462 −0.752308 0.658811i \(-0.771060\pi\)
−0.752308 + 0.658811i \(0.771060\pi\)
\(252\) 0 0
\(253\) 21.2228i 1.33427i
\(254\) 2.81382i 0.176555i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 29.7441i 1.85539i −0.373341 0.927694i \(-0.621788\pi\)
0.373341 0.927694i \(-0.378212\pi\)
\(258\) 0 0
\(259\) 8.13181 8.73923i 0.505286 0.543030i
\(260\) 0 0
\(261\) 0 0
\(262\) 8.62840 0.533064
\(263\) −22.7545 −1.40310 −0.701552 0.712618i \(-0.747509\pi\)
−0.701552 + 0.712618i \(0.747509\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.75454 5.10969i 0.291519 0.313295i
\(267\) 0 0
\(268\) 6.90729i 0.421930i
\(269\) −0.202401 −0.0123406 −0.00617030 0.999981i \(-0.501964\pi\)
−0.00617030 + 0.999981i \(0.501964\pi\)
\(270\) 0 0
\(271\) 11.4785i 0.697267i −0.937259 0.348634i \(-0.886646\pi\)
0.937259 0.348634i \(-0.113354\pi\)
\(272\) 8.11650i 0.492135i
\(273\) 0 0
\(274\) 0.723932 0.0437343
\(275\) 0 0
\(276\) 0 0
\(277\) 23.7211i 1.42526i −0.701538 0.712632i \(-0.747503\pi\)
0.701538 0.712632i \(-0.252497\pi\)
\(278\) 2.84043i 0.170358i
\(279\) 0 0
\(280\) 0 0
\(281\) 5.57194i 0.332394i 0.986093 + 0.166197i \(0.0531488\pi\)
−0.986093 + 0.166197i \(0.946851\pi\)
\(282\) 0 0
\(283\) 32.1798 1.91289 0.956445 0.291914i \(-0.0942921\pi\)
0.956445 + 0.291914i \(0.0942921\pi\)
\(284\) 2.63803i 0.156539i
\(285\) 0 0
\(286\) 6.21603i 0.367561i
\(287\) 3.10801 + 2.89199i 0.183460 + 0.170709i
\(288\) 0 0
\(289\) −48.8776 −2.87515
\(290\) 0 0
\(291\) 0 0
\(292\) 13.7477 0.804525
\(293\) 15.8146i 0.923898i −0.886907 0.461949i \(-0.847150\pi\)
0.886907 0.461949i \(-0.152850\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.51190i 0.262249i
\(297\) 0 0
\(298\) 3.00681i 0.174180i
\(299\) 8.79079 0.508384
\(300\) 0 0
\(301\) 18.2330 19.5950i 1.05093 1.12944i
\(302\) −15.2262 −0.876169
\(303\) 0 0
\(304\) 2.63803i 0.151302i
\(305\) 0 0
\(306\) 0 0
\(307\) −6.72393 −0.383755 −0.191878 0.981419i \(-0.561458\pi\)
−0.191878 + 0.981419i \(0.561458\pi\)
\(308\) 6.98188 7.50341i 0.397829 0.427546i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.02379 −0.0580540 −0.0290270 0.999579i \(-0.509241\pi\)
−0.0290270 + 0.999579i \(0.509241\pi\)
\(312\) 0 0
\(313\) 23.2568 1.31455 0.657276 0.753650i \(-0.271709\pi\)
0.657276 + 0.753650i \(0.271709\pi\)
\(314\) −2.64767 −0.149417
\(315\) 0 0
\(316\) 8.01698 0.450990
\(317\) 4.46830 0.250965 0.125482 0.992096i \(-0.459952\pi\)
0.125482 + 0.992096i \(0.459952\pi\)
\(318\) 0 0
\(319\) 21.2228 1.18825
\(320\) 0 0
\(321\) 0 0
\(322\) 10.6114 + 9.87386i 0.591351 + 0.550249i
\(323\) 21.4116 1.19137
\(324\) 0 0
\(325\) 0 0
\(326\) 9.32572i 0.516504i
\(327\) 0 0
\(328\) 1.60461 0.0885996
\(329\) −20.0230 + 21.5187i −1.10391 + 1.18636i
\(330\) 0 0
\(331\) −14.2160 −0.781383 −0.390692 0.920522i \(-0.627764\pi\)
−0.390692 + 0.920522i \(0.627764\pi\)
\(332\) 3.20921i 0.176128i
\(333\) 0 0
\(334\) 17.3257i 0.948021i
\(335\) 0 0
\(336\) 0 0
\(337\) 23.4286i 1.27624i −0.769938 0.638118i \(-0.779713\pi\)
0.769938 0.638118i \(-0.220287\pi\)
\(338\) −10.4252 −0.567058
\(339\) 0 0
\(340\) 0 0
\(341\) −14.4524 −0.782641
\(342\) 0 0
\(343\) −11.6084 + 14.4307i −0.626794 + 0.779185i
\(344\) 10.1165i 0.545445i
\(345\) 0 0
\(346\) 2.99319i 0.160915i
\(347\) −20.1798 −1.08331 −0.541654 0.840602i \(-0.682202\pi\)
−0.541654 + 0.840602i \(0.682202\pi\)
\(348\) 0 0
\(349\) 27.0372i 1.44727i −0.690184 0.723634i \(-0.742470\pi\)
0.690184 0.723634i \(-0.257530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.87386i 0.206478i
\(353\) 12.3495i 0.657298i −0.944452 0.328649i \(-0.893407\pi\)
0.944452 0.328649i \(-0.106593\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −17.8376 −0.945392
\(357\) 0 0
\(358\) 7.38858i 0.390499i
\(359\) 25.5757i 1.34983i 0.737894 + 0.674917i \(0.235821\pi\)
−0.737894 + 0.674917i \(0.764179\pi\)
\(360\) 0 0
\(361\) 12.0408 0.633725
\(362\) 11.1097i 0.583912i
\(363\) 0 0
\(364\) 3.10801 + 2.89199i 0.162904 + 0.151581i
\(365\) 0 0
\(366\) 0 0
\(367\) −20.8444 −1.08807 −0.544035 0.839062i \(-0.683104\pi\)
−0.544035 + 0.839062i \(0.683104\pi\)
\(368\) 5.47847 0.285585
\(369\) 0 0
\(370\) 0 0
\(371\) −4.39539 4.08989i −0.228197 0.212336i
\(372\) 0 0
\(373\) 7.48810i 0.387719i 0.981029 + 0.193860i \(0.0621006\pi\)
−0.981029 + 0.193860i \(0.937899\pi\)
\(374\) 31.4422 1.62584
\(375\) 0 0
\(376\) 11.1097i 0.572939i
\(377\) 8.79079i 0.452749i
\(378\) 0 0
\(379\) −32.6651 −1.67789 −0.838946 0.544215i \(-0.816827\pi\)
−0.838946 + 0.544215i \(0.816827\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.36197i 0.479000i
\(383\) 0.890309i 0.0454927i 0.999741 + 0.0227463i \(0.00724101\pi\)
−0.999741 + 0.0227463i \(0.992759\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.4422i 0.887786i
\(387\) 0 0
\(388\) 8.68768 0.441050
\(389\) 13.5317i 0.686084i 0.939320 + 0.343042i \(0.111457\pi\)
−0.939320 + 0.343042i \(0.888543\pi\)
\(390\) 0 0
\(391\) 44.4660i 2.24874i
\(392\) 0.503406 + 6.98188i 0.0254258 + 0.352638i
\(393\) 0 0
\(394\) −5.27607 −0.265804
\(395\) 0 0
\(396\) 0 0
\(397\) −25.2991 −1.26973 −0.634863 0.772625i \(-0.718943\pi\)
−0.634863 + 0.772625i \(0.718943\pi\)
\(398\) 12.2160i 0.612334i
\(399\) 0 0
\(400\) 0 0
\(401\) 32.7619i 1.63605i 0.575182 + 0.818025i \(0.304931\pi\)
−0.575182 + 0.818025i \(0.695069\pi\)
\(402\) 0 0
\(403\) 5.98638i 0.298203i
\(404\) −6.21603 −0.309259
\(405\) 0 0
\(406\) −9.87386 + 10.6114i −0.490032 + 0.526636i
\(407\) −17.4785 −0.866376
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −12.8807 −0.634585
\(413\) 8.93200 + 8.31117i 0.439515 + 0.408966i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.60461 0.0786723
\(417\) 0 0
\(418\) −10.2194 −0.499846
\(419\) 4.17937 0.204176 0.102088 0.994775i \(-0.467448\pi\)
0.102088 + 0.994775i \(0.467448\pi\)
\(420\) 0 0
\(421\) −32.6514 −1.59133 −0.795667 0.605735i \(-0.792879\pi\)
−0.795667 + 0.605735i \(0.792879\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −2.26926 −0.110205
\(425\) 0 0
\(426\) 0 0
\(427\) −21.3523 + 22.9473i −1.03331 + 1.11050i
\(428\) 19.9638 0.964984
\(429\) 0 0
\(430\) 0 0
\(431\) 13.3087i 0.641059i 0.947239 + 0.320530i \(0.103861\pi\)
−0.947239 + 0.320530i \(0.896139\pi\)
\(432\) 0 0
\(433\) −6.95358 −0.334168 −0.167084 0.985943i \(-0.553435\pi\)
−0.167084 + 0.985943i \(0.553435\pi\)
\(434\) 6.72393 7.22619i 0.322759 0.346868i
\(435\) 0 0
\(436\) 15.0238 0.719509
\(437\) 14.4524i 0.691352i
\(438\) 0 0
\(439\) 26.9739i 1.28739i −0.765280 0.643697i \(-0.777400\pi\)
0.765280 0.643697i \(-0.222600\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 13.0238i 0.619479i
\(443\) 2.80441 0.133242 0.0666208 0.997778i \(-0.478778\pi\)
0.0666208 + 0.997778i \(0.478778\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.65784 0.173204
\(447\) 0 0
\(448\) 1.93693 + 1.80230i 0.0915114 + 0.0851508i
\(449\) 30.1226i 1.42157i −0.703409 0.710786i \(-0.748340\pi\)
0.703409 0.710786i \(-0.251660\pi\)
\(450\) 0 0
\(451\) 6.21603i 0.292701i
\(452\) −10.2330 −0.481320
\(453\) 0 0
\(454\) 4.23301i 0.198665i
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0340i 1.03071i 0.856978 + 0.515353i \(0.172339\pi\)
−0.856978 + 0.515353i \(0.827661\pi\)
\(458\) 7.59497i 0.354890i
\(459\) 0 0
\(460\) 0 0
\(461\) −15.4116 −0.717790 −0.358895 0.933378i \(-0.616846\pi\)
−0.358895 + 0.933378i \(0.616846\pi\)
\(462\) 0 0
\(463\) 25.0605i 1.16466i −0.812953 0.582329i \(-0.802142\pi\)
0.812953 0.582329i \(-0.197858\pi\)
\(464\) 5.47847i 0.254332i
\(465\) 0 0
\(466\) 9.94677 0.460775
\(467\) 9.62764i 0.445514i 0.974874 + 0.222757i \(0.0715056\pi\)
−0.974874 + 0.222757i \(0.928494\pi\)
\(468\) 0 0
\(469\) 12.4490 13.3789i 0.574843 0.617782i
\(470\) 0 0
\(471\) 0 0
\(472\) 4.61142 0.212258
\(473\) −39.1899 −1.80196
\(474\) 0 0
\(475\) 0 0
\(476\) −14.6284 + 15.7211i −0.670492 + 0.720576i
\(477\) 0 0
\(478\) 9.36197i 0.428206i
\(479\) −37.0238 −1.69166 −0.845830 0.533452i \(-0.820894\pi\)
−0.845830 + 0.533452i \(0.820894\pi\)
\(480\) 0 0
\(481\) 7.23982i 0.330107i
\(482\) 23.0408i 1.04948i
\(483\) 0 0
\(484\) −4.00681 −0.182128
\(485\) 0 0
\(486\) 0 0
\(487\) 11.6386i 0.527394i 0.964606 + 0.263697i \(0.0849419\pi\)
−0.964606 + 0.263697i \(0.915058\pi\)
\(488\) 11.8472i 0.536300i
\(489\) 0 0
\(490\) 0 0
\(491\) 32.1069i 1.44896i −0.689294 0.724481i \(-0.742079\pi\)
0.689294 0.724481i \(-0.257921\pi\)
\(492\) 0 0
\(493\) −44.4660 −2.00265
\(494\) 4.23301i 0.190452i
\(495\) 0 0
\(496\) 3.73074i 0.167515i
\(497\) 4.75454 5.10969i 0.213270 0.229201i
\(498\) 0 0
\(499\) −13.6122 −0.609365 −0.304682 0.952454i \(-0.598550\pi\)
−0.304682 + 0.952454i \(0.598550\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −23.8376 −1.06392
\(503\) 3.07573i 0.137140i −0.997646 0.0685700i \(-0.978156\pi\)
0.997646 0.0685700i \(-0.0218436\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 21.2228i 0.943470i
\(507\) 0 0
\(508\) 2.81382i 0.124843i
\(509\) −14.7908 −0.655590 −0.327795 0.944749i \(-0.606306\pi\)
−0.327795 + 0.944749i \(0.606306\pi\)
\(510\) 0 0
\(511\) 26.6284 + 24.7776i 1.17797 + 1.09610i
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 29.7441i 1.31196i
\(515\) 0 0
\(516\) 0 0
\(517\) 43.0374 1.89278
\(518\) 8.13181 8.73923i 0.357291 0.383980i
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0332 −1.18435 −0.592173 0.805811i \(-0.701730\pi\)
−0.592173 + 0.805811i \(0.701730\pi\)
\(522\) 0 0
\(523\) 3.51472 0.153688 0.0768440 0.997043i \(-0.475516\pi\)
0.0768440 + 0.997043i \(0.475516\pi\)
\(524\) 8.62840 0.376933
\(525\) 0 0
\(526\) −22.7545 −0.992145
\(527\) 30.2806 1.31904
\(528\) 0 0
\(529\) 7.01362 0.304940
\(530\) 0 0
\(531\) 0 0
\(532\) 4.75454 5.10969i 0.206135 0.221533i
\(533\) 2.57476 0.111525
\(534\) 0 0
\(535\) 0 0
\(536\) 6.90729i 0.298350i
\(537\) 0 0
\(538\) −0.202401 −0.00872612
\(539\) 27.0468 1.95013i 1.16499 0.0839979i
\(540\) 0 0
\(541\) −39.6616 −1.70519 −0.852593 0.522576i \(-0.824971\pi\)
−0.852593 + 0.522576i \(0.824971\pi\)
\(542\) 11.4785i 0.493042i
\(543\) 0 0
\(544\) 8.11650i 0.347992i
\(545\) 0 0
\(546\) 0 0
\(547\) 34.3495i 1.46868i 0.678782 + 0.734339i \(0.262508\pi\)
−0.678782 + 0.734339i \(0.737492\pi\)
\(548\) 0.723932 0.0309248
\(549\) 0 0
\(550\) 0 0
\(551\) 14.4524 0.615692
\(552\) 0 0
\(553\) 15.5283 + 14.4490i 0.660332 + 0.614435i
\(554\) 23.7211i 1.00781i
\(555\) 0 0
\(556\) 2.84043i 0.120461i
\(557\) −38.4660 −1.62986 −0.814929 0.579561i \(-0.803224\pi\)
−0.814929 + 0.579561i \(0.803224\pi\)
\(558\) 0 0
\(559\) 16.2330i 0.686583i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.57194i 0.235038i
\(563\) 10.2194i 0.430696i −0.976537 0.215348i \(-0.930911\pi\)
0.976537 0.215348i \(-0.0690885\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.1798 1.35262
\(567\) 0 0
\(568\) 2.63803i 0.110689i
\(569\) 2.48129i 0.104021i −0.998647 0.0520106i \(-0.983437\pi\)
0.998647 0.0520106i \(-0.0165629\pi\)
\(570\) 0 0
\(571\) 20.6345 0.863525 0.431762 0.901987i \(-0.357892\pi\)
0.431762 + 0.901987i \(0.357892\pi\)
\(572\) 6.21603i 0.259905i
\(573\) 0 0
\(574\) 3.10801 + 2.89199i 0.129726 + 0.120709i
\(575\) 0 0
\(576\) 0 0
\(577\) 2.50455 0.104266 0.0521329 0.998640i \(-0.483398\pi\)
0.0521329 + 0.998640i \(0.483398\pi\)
\(578\) −48.8776 −2.03304
\(579\) 0 0
\(580\) 0 0
\(581\) 5.78397 6.21603i 0.239960 0.257884i
\(582\) 0 0
\(583\) 8.79079i 0.364077i
\(584\) 13.7477 0.568885
\(585\) 0 0
\(586\) 15.8146i 0.653294i
\(587\) 31.4422i 1.29776i −0.760891 0.648880i \(-0.775238\pi\)
0.760891 0.648880i \(-0.224762\pi\)
\(588\) 0 0
\(589\) −9.84183 −0.405526
\(590\) 0 0
\(591\) 0 0
\(592\) 4.51190i 0.185438i
\(593\) 27.1267i 1.11396i 0.830526 + 0.556979i \(0.188040\pi\)
−0.830526 + 0.556979i \(0.811960\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00681i 0.123164i
\(597\) 0 0
\(598\) 8.79079 0.359482
\(599\) 33.7747i 1.38000i −0.723810 0.689999i \(-0.757611\pi\)
0.723810 0.689999i \(-0.242389\pi\)
\(600\) 0 0
\(601\) 24.4886i 0.998912i −0.866339 0.499456i \(-0.833533\pi\)
0.866339 0.499456i \(-0.166467\pi\)
\(602\) 18.2330 19.5950i 0.743122 0.798631i
\(603\) 0 0
\(604\) −15.2262 −0.619545
\(605\) 0 0
\(606\) 0 0
\(607\) −27.3331 −1.10941 −0.554707 0.832045i \(-0.687170\pi\)
−0.554707 + 0.832045i \(0.687170\pi\)
\(608\) 2.63803i 0.106986i
\(609\) 0 0
\(610\) 0 0
\(611\) 17.8267i 0.721190i
\(612\) 0 0
\(613\) 11.3163i 0.457061i 0.973537 + 0.228531i \(0.0733921\pi\)
−0.973537 + 0.228531i \(0.926608\pi\)
\(614\) −6.72393 −0.271356
\(615\) 0 0
\(616\) 6.98188 7.50341i 0.281308 0.302321i
\(617\) −10.6843 −0.430135 −0.215067 0.976599i \(-0.568997\pi\)
−0.215067 + 0.976599i \(0.568997\pi\)
\(618\) 0 0
\(619\) 9.56437i 0.384424i −0.981353 0.192212i \(-0.938434\pi\)
0.981353 0.192212i \(-0.0615662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.02379 −0.0410504
\(623\) −34.5502 32.1488i −1.38423 1.28801i
\(624\) 0 0
\(625\) 0 0
\(626\) 23.2568 0.929529
\(627\) 0 0
\(628\) −2.64767 −0.105653
\(629\) 36.6208 1.46017
\(630\) 0 0
\(631\) 17.1956 0.684546 0.342273 0.939601i \(-0.388803\pi\)
0.342273 + 0.939601i \(0.388803\pi\)
\(632\) 8.01698 0.318898
\(633\) 0 0
\(634\) 4.46830 0.177459
\(635\) 0 0
\(636\) 0 0
\(637\) 0.807769 + 11.2032i 0.0320050 + 0.443885i
\(638\) 21.2228 0.840220
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5289i 0.652851i 0.945223 + 0.326426i \(0.105844\pi\)
−0.945223 + 0.326426i \(0.894156\pi\)
\(642\) 0 0
\(643\) −37.4286 −1.47604 −0.738020 0.674779i \(-0.764239\pi\)
−0.738020 + 0.674779i \(0.764239\pi\)
\(644\) 10.6114 + 9.87386i 0.418148 + 0.389085i
\(645\) 0 0
\(646\) 21.4116 0.842429
\(647\) 5.52812i 0.217333i 0.994078 + 0.108666i \(0.0346580\pi\)
−0.994078 + 0.108666i \(0.965342\pi\)
\(648\) 0 0
\(649\) 17.8640i 0.701223i
\(650\) 0 0
\(651\) 0 0
\(652\) 9.32572i 0.365223i
\(653\) 18.5159 0.724583 0.362291 0.932065i \(-0.381995\pi\)
0.362291 + 0.932065i \(0.381995\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.60461 0.0626494
\(657\) 0 0
\(658\) −20.0230 + 21.5187i −0.780579 + 0.838887i
\(659\) 43.3693i 1.68943i −0.535217 0.844714i \(-0.679770\pi\)
0.535217 0.844714i \(-0.320230\pi\)
\(660\) 0 0
\(661\) 36.1142i 1.40468i 0.711842 + 0.702340i \(0.247861\pi\)
−0.711842 + 0.702340i \(0.752139\pi\)
\(662\) −14.2160 −0.552522
\(663\) 0 0
\(664\) 3.20921i 0.124542i
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0136i 1.16213i
\(668\) 17.3257i 0.670352i
\(669\) 0 0
\(670\) 0 0
\(671\) 45.8946 1.77174
\(672\) 0 0
\(673\) 33.8743i 1.30576i −0.757463 0.652879i \(-0.773561\pi\)
0.757463 0.652879i \(-0.226439\pi\)
\(674\) 23.4286i 0.902436i
\(675\) 0 0
\(676\) −10.4252 −0.400971
\(677\) 4.38446i 0.168509i 0.996444 + 0.0842543i \(0.0268508\pi\)
−0.996444 + 0.0842543i \(0.973149\pi\)
\(678\) 0 0
\(679\) 16.8274 + 15.6578i 0.645778 + 0.600893i
\(680\) 0 0
\(681\) 0 0
\(682\) −14.4524 −0.553411
\(683\) 1.95013 0.0746195 0.0373098 0.999304i \(-0.488121\pi\)
0.0373098 + 0.999304i \(0.488121\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −11.6084 + 14.4307i −0.443211 + 0.550967i
\(687\) 0 0
\(688\) 10.1165i 0.385688i
\(689\) −3.64126 −0.138721
\(690\) 0 0
\(691\) 12.1471i 0.462098i 0.972942 + 0.231049i \(0.0742157\pi\)
−0.972942 + 0.231049i \(0.925784\pi\)
\(692\) 2.99319i 0.113784i
\(693\) 0 0
\(694\) −20.1798 −0.766014
\(695\) 0 0
\(696\) 0 0
\(697\) 13.0238i 0.493311i
\(698\) 27.0372i 1.02337i
\(699\) 0 0
\(700\) 0 0
\(701\) 11.0600i 0.417732i −0.977944 0.208866i \(-0.933023\pi\)
0.977944 0.208866i \(-0.0669773\pi\)
\(702\) 0 0
\(703\) −11.9025 −0.448913
\(704\) 3.87386i 0.146002i
\(705\) 0 0
\(706\) 12.3495i 0.464780i
\(707\) −12.0400 11.2032i −0.452811 0.421338i
\(708\) 0 0
\(709\) −14.2330 −0.534532 −0.267266 0.963623i \(-0.586120\pi\)
−0.267266 + 0.963623i \(0.586120\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −17.8376 −0.668493
\(713\) 20.4388i 0.765438i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.38858i 0.276124i
\(717\) 0 0
\(718\) 25.5757i 0.954477i
\(719\) 20.4660 0.763254 0.381627 0.924317i \(-0.375364\pi\)
0.381627 + 0.924317i \(0.375364\pi\)
\(720\) 0 0
\(721\) −24.9490 23.2149i −0.929149 0.864567i
\(722\) 12.0408 0.448111
\(723\) 0 0
\(724\) 11.1097i 0.412888i
\(725\) 0 0
\(726\) 0 0
\(727\) −25.5717 −0.948402 −0.474201 0.880417i \(-0.657263\pi\)
−0.474201 + 0.880417i \(0.657263\pi\)
\(728\) 3.10801 + 2.89199i 0.115191 + 0.107184i
\(729\) 0 0
\(730\) 0 0
\(731\) 82.1106 3.03697
\(732\) 0 0
\(733\) −22.6624 −0.837053 −0.418527 0.908204i \(-0.637453\pi\)
−0.418527 + 0.908204i \(0.637453\pi\)
\(734\) −20.8444 −0.769382
\(735\) 0 0
\(736\) 5.47847 0.201939
\(737\) −26.7579 −0.985640
\(738\) 0 0
\(739\) 44.8675 1.65048 0.825238 0.564785i \(-0.191041\pi\)
0.825238 + 0.564785i \(0.191041\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.39539 4.08989i −0.161360 0.150145i
\(743\) −5.37536 −0.197203 −0.0986015 0.995127i \(-0.531437\pi\)
−0.0986015 + 0.995127i \(0.531437\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.48810i 0.274159i
\(747\) 0 0
\(748\) 31.4422 1.14964
\(749\) 38.6684 + 35.9807i 1.41291 + 1.31471i
\(750\) 0 0
\(751\) 13.6276 0.497280 0.248640 0.968596i \(-0.420016\pi\)
0.248640 + 0.968596i \(0.420016\pi\)
\(752\) 11.1097i 0.405129i
\(753\) 0 0
\(754\) 8.79079i 0.320142i
\(755\) 0 0
\(756\) 0 0
\(757\) 4.54586i 0.165222i 0.996582 + 0.0826110i \(0.0263259\pi\)
−0.996582 + 0.0826110i \(0.973674\pi\)
\(758\) −32.6651 −1.18645
\(759\) 0 0
\(760\) 0 0
\(761\) −15.6522 −0.567392 −0.283696 0.958914i \(-0.591561\pi\)
−0.283696 + 0.958914i \(0.591561\pi\)
\(762\) 0 0
\(763\) 29.1001 + 27.0774i 1.05349 + 0.980269i
\(764\) 9.36197i 0.338704i
\(765\) 0 0
\(766\) 0.890309i 0.0321682i
\(767\) 7.39951 0.267181
\(768\) 0 0
\(769\) 17.1922i 0.619968i −0.950742 0.309984i \(-0.899676\pi\)
0.950742 0.309984i \(-0.100324\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17.4422i 0.627759i
\(773\) 42.6990i 1.53578i 0.640584 + 0.767889i \(0.278693\pi\)
−0.640584 + 0.767889i \(0.721307\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.68768 0.311870
\(777\) 0 0
\(778\) 13.5317i 0.485135i
\(779\) 4.23301i 0.151663i
\(780\) 0 0
\(781\) −10.2194 −0.365678
\(782\) 44.4660i 1.59010i
\(783\) 0 0
\(784\) 0.503406 + 6.98188i 0.0179788 + 0.249353i
\(785\) 0 0
\(786\) 0 0
\(787\) −30.3992 −1.08361 −0.541806 0.840503i \(-0.682259\pi\)
−0.541806 + 0.840503i \(0.682259\pi\)
\(788\) −5.27607 −0.187952
\(789\) 0 0
\(790\) 0 0
\(791\) −19.8206 18.4430i −0.704741 0.655757i
\(792\) 0 0
\(793\) 19.0102i 0.675071i
\(794\) −25.2991 −0.897831
\(795\) 0 0
\(796\) 12.2160i 0.432986i
\(797\) 2.99319i 0.106024i −0.998594 0.0530121i \(-0.983118\pi\)
0.998594 0.0530121i \(-0.0168822\pi\)
\(798\) 0 0
\(799\) −90.1718 −3.19005
\(800\) 0 0
\(801\) 0 0
\(802\) 32.7619i 1.15686i
\(803\) 53.2568i 1.87939i
\(804\) 0 0
\(805\) 0 0
\(806\) 5.98638i 0.210861i
\(807\) 0 0
\(808\) −6.21603 −0.218679
\(809\) 25.0334i 0.880128i 0.897966 + 0.440064i \(0.145044\pi\)
−0.897966 + 0.440064i \(0.854956\pi\)
\(810\) 0 0
\(811\) 19.4782i 0.683974i 0.939705 + 0.341987i \(0.111100\pi\)
−0.939705 + 0.341987i \(0.888900\pi\)
\(812\) −9.87386 + 10.6114i −0.346505 + 0.372388i
\(813\) 0 0
\(814\) −17.4785 −0.612620
\(815\) 0 0
\(816\) 0 0
\(817\) −26.6877 −0.933684
\(818\) 12.0000i 0.419570i
\(819\) 0 0
\(820\) 0 0
\(821\) 45.3787i 1.58373i −0.610697 0.791864i \(-0.709111\pi\)
0.610697 0.791864i \(-0.290889\pi\)
\(822\) 0 0
\(823\) 25.8512i 0.901117i 0.892747 + 0.450559i \(0.148775\pi\)
−0.892747 + 0.450559i \(0.851225\pi\)
\(824\) −12.8807 −0.448720
\(825\) 0 0
\(826\) 8.93200 + 8.31117i 0.310784 + 0.289183i
\(827\) 19.3429 0.672619 0.336310 0.941751i \(-0.390821\pi\)
0.336310 + 0.941751i \(0.390821\pi\)
\(828\) 0 0
\(829\) 43.4764i 1.51000i −0.655726 0.754999i \(-0.727637\pi\)
0.655726 0.754999i \(-0.272363\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.60461 0.0556297
\(833\) −56.6684 + 4.08590i −1.96344 + 0.141568i
\(834\) 0 0
\(835\) 0 0
\(836\) −10.2194 −0.353445
\(837\) 0 0
\(838\) 4.17937 0.144374
\(839\) 24.2670 0.837789 0.418894 0.908035i \(-0.362418\pi\)
0.418894 + 0.908035i \(0.362418\pi\)
\(840\) 0 0
\(841\) −1.01362 −0.0349526
\(842\) −32.6514 −1.12524
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) −7.76092 7.22149i −0.266668 0.248133i
\(848\) −2.26926 −0.0779266
\(849\) 0 0
\(850\) 0 0
\(851\) 24.7183i 0.847332i
\(852\) 0 0
\(853\) −16.2439 −0.556182 −0.278091 0.960555i \(-0.589702\pi\)
−0.278091 + 0.960555i \(0.589702\pi\)
\(854\) −21.3523 + 22.9473i −0.730662 + 0.785241i
\(855\) 0 0
\(856\) 19.9638 0.682347
\(857\) 40.5825i 1.38627i −0.720807 0.693136i \(-0.756228\pi\)
0.720807 0.693136i \(-0.243772\pi\)
\(858\) 0 0
\(859\) 5.05310i 0.172410i −0.996277 0.0862048i \(-0.972526\pi\)
0.996277 0.0862048i \(-0.0274739\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 13.3087i 0.453297i
\(863\) 48.3289 1.64514 0.822568 0.568666i \(-0.192540\pi\)
0.822568 + 0.568666i \(0.192540\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −6.95358 −0.236292
\(867\) 0 0
\(868\) 6.72393 7.22619i 0.228225 0.245273i
\(869\) 31.0567i 1.05353i
\(870\) 0 0
\(871\) 11.0835i 0.375549i
\(872\) 15.0238 0.508770
\(873\) 0 0
\(874\) 14.4524i 0.488859i
\(875\) 0 0
\(876\) 0 0
\(877\) 55.3963i 1.87060i 0.353854 + 0.935301i \(0.384871\pi\)
−0.353854 + 0.935301i \(0.615129\pi\)
\(878\) 26.9739i 0.910326i
\(879\) 0 0
\(880\) 0 0
\(881\) −49.8716 −1.68022 −0.840108 0.542419i \(-0.817509\pi\)
−0.840108 + 0.542419i \(0.817509\pi\)
\(882\) 0 0
\(883\) 43.3393i 1.45848i −0.684255 0.729242i \(-0.739873\pi\)
0.684255 0.729242i \(-0.260127\pi\)
\(884\) 13.0238i 0.438038i
\(885\) 0 0
\(886\) 2.80441 0.0942160
\(887\) 33.1539i 1.11320i 0.830781 + 0.556600i \(0.187894\pi\)
−0.830781 + 0.556600i \(0.812106\pi\)
\(888\) 0 0
\(889\) −5.07136 + 5.45018i −0.170088 + 0.182793i
\(890\) 0 0
\(891\) 0 0
\(892\) 3.65784 0.122473
\(893\) 29.3077 0.980746
\(894\) 0 0
\(895\) 0 0
\(896\) 1.93693 + 1.80230i 0.0647083 + 0.0602107i
\(897\) 0 0
\(898\) 30.1226i 1.00520i
\(899\) 20.4388 0.681671
\(900\) 0 0
\(901\) 18.4184i 0.613607i
\(902\) 6.21603i 0.206971i
\(903\) 0 0
\(904\) −10.2330 −0.340345
\(905\) 0 0
\(906\) 0 0
\(907\) 50.5553i 1.67866i −0.543622 0.839330i \(-0.682948\pi\)
0.543622 0.839330i \(-0.317052\pi\)
\(908\) 4.23301i 0.140477i
\(909\) 0 0
\(910\) 0 0
\(911\) 21.6755i 0.718140i −0.933311 0.359070i \(-0.883094\pi\)
0.933311 0.359070i \(-0.116906\pi\)
\(912\) 0 0
\(913\) −12.4321 −0.411441
\(914\) 22.0340i 0.728819i
\(915\) 0 0
\(916\) 7.59497i 0.250945i
\(917\) 16.7126 + 15.5510i 0.551899 + 0.513539i
\(918\) 0 0
\(919\) 25.6616 0.846498 0.423249 0.906013i \(-0.360890\pi\)
0.423249 + 0.906013i \(0.360890\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −15.4116 −0.507554
\(923\) 4.23301i 0.139331i
\(924\) 0 0
\(925\) 0 0
\(926\) 25.0605i 0.823538i
\(927\) 0 0
\(928\) 5.47847i 0.179840i
\(929\) 31.6182 1.03736 0.518680 0.854968i \(-0.326424\pi\)
0.518680 + 0.854968i \(0.326424\pi\)
\(930\) 0 0
\(931\) 18.4184 1.32800i 0.603640 0.0435235i
\(932\) 9.94677 0.325817
\(933\) 0 0
\(934\) 9.62764i 0.315026i
\(935\) 0 0
\(936\) 0 0
\(937\) −38.7013 −1.26432 −0.632158 0.774839i \(-0.717831\pi\)
−0.632158 + 0.774839i \(0.717831\pi\)
\(938\) 12.4490 13.3789i 0.406475 0.436838i
\(939\) 0 0
\(940\) 0 0
\(941\) −32.9932 −1.07555 −0.537774 0.843089i \(-0.680734\pi\)
−0.537774 + 0.843089i \(0.680734\pi\)
\(942\) 0 0
\(943\) 8.79079 0.286267
\(944\) 4.61142 0.150089
\(945\) 0 0
\(946\) −39.1899 −1.27418
\(947\) 31.8452 1.03483 0.517415 0.855735i \(-0.326894\pi\)
0.517415 + 0.855735i \(0.326894\pi\)
\(948\) 0 0
\(949\) 22.0597 0.716088
\(950\) 0 0
\(951\) 0 0
\(952\) −14.6284 + 15.7211i −0.474109 + 0.509524i
\(953\) 45.9003 1.48686 0.743428 0.668817i \(-0.233199\pi\)
0.743428 + 0.668817i \(0.233199\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.36197i 0.302788i
\(957\) 0 0
\(958\) −37.0238 −1.19618
\(959\) 1.40221 + 1.30474i 0.0452796 + 0.0421324i
\(960\) 0 0
\(961\) 17.0816 0.551018
\(962\) 7.23982i 0.233421i
\(963\) 0 0
\(964\) 23.0408i 0.742093i
\(965\) 0 0
\(966\) 0 0
\(967\) 41.2662i 1.32703i 0.748162 + 0.663516i \(0.230936\pi\)
−0.748162 + 0.663516i \(0.769064\pi\)
\(968\) −4.00681 −0.128784
\(969\) 0 0
\(970\) 0 0
\(971\) −11.0008 −0.353031 −0.176516 0.984298i \(-0.556483\pi\)
−0.176516 + 0.984298i \(0.556483\pi\)
\(972\) 0 0
\(973\) −5.11933 + 5.50173i −0.164118 + 0.176377i
\(974\) 11.6386i 0.372924i
\(975\) 0 0
\(976\) 11.8472i 0.379221i
\(977\) 2.48528 0.0795112 0.0397556 0.999209i \(-0.487342\pi\)
0.0397556 + 0.999209i \(0.487342\pi\)
\(978\) 0 0
\(979\) 69.1005i 2.20846i
\(980\) 0 0
\(981\) 0 0
\(982\) 32.1069i 1.02457i
\(983\) 31.7408i 1.01237i 0.862424 + 0.506187i \(0.168945\pi\)
−0.862424 + 0.506187i \(0.831055\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −44.4660 −1.41609
\(987\) 0 0
\(988\) 4.23301i 0.134670i
\(989\) 55.4230i 1.76235i
\(990\) 0 0
\(991\) 57.3368 1.82136 0.910682 0.413108i \(-0.135557\pi\)
0.910682 + 0.413108i \(0.135557\pi\)
\(992\) 3.73074i 0.118451i
\(993\) 0 0
\(994\) 4.75454 5.10969i 0.150805 0.162070i
\(995\) 0 0
\(996\) 0 0
\(997\) 52.6760 1.66827 0.834133 0.551564i \(-0.185969\pi\)
0.834133 + 0.551564i \(0.185969\pi\)
\(998\) −13.6122 −0.430886
\(999\) 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 3150.2.d.d.3149.8 8
3.2 odd 2 3150.2.d.a.3149.8 8
5.2 odd 4 630.2.b.a.251.6 yes 8
5.3 odd 4 3150.2.b.e.251.3 8
5.4 even 2 3150.2.d.c.3149.1 8
7.6 odd 2 3150.2.d.f.3149.2 8
15.2 even 4 630.2.b.b.251.2 yes 8
15.8 even 4 3150.2.b.f.251.7 8
15.14 odd 2 3150.2.d.f.3149.1 8
20.7 even 4 5040.2.f.f.881.5 8
21.20 even 2 3150.2.d.c.3149.2 8
35.13 even 4 3150.2.b.f.251.3 8
35.27 even 4 630.2.b.b.251.6 yes 8
35.34 odd 2 3150.2.d.a.3149.7 8
60.47 odd 4 5040.2.f.i.881.5 8
105.62 odd 4 630.2.b.a.251.2 8
105.83 odd 4 3150.2.b.e.251.7 8
105.104 even 2 inner 3150.2.d.d.3149.7 8
140.27 odd 4 5040.2.f.i.881.6 8
420.167 even 4 5040.2.f.f.881.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.b.a.251.2 8 105.62 odd 4
630.2.b.a.251.6 yes 8 5.2 odd 4
630.2.b.b.251.2 yes 8 15.2 even 4
630.2.b.b.251.6 yes 8 35.27 even 4
3150.2.b.e.251.3 8 5.3 odd 4
3150.2.b.e.251.7 8 105.83 odd 4
3150.2.b.f.251.3 8 35.13 even 4
3150.2.b.f.251.7 8 15.8 even 4
3150.2.d.a.3149.7 8 35.34 odd 2
3150.2.d.a.3149.8 8 3.2 odd 2
3150.2.d.c.3149.1 8 5.4 even 2
3150.2.d.c.3149.2 8 21.20 even 2
3150.2.d.d.3149.7 8 105.104 even 2 inner
3150.2.d.d.3149.8 8 1.1 even 1 trivial
3150.2.d.f.3149.1 8 15.14 odd 2
3150.2.d.f.3149.2 8 7.6 odd 2
5040.2.f.f.881.5 8 20.7 even 4
5040.2.f.f.881.6 8 420.167 even 4
5040.2.f.i.881.5 8 60.47 odd 4
5040.2.f.i.881.6 8 140.27 odd 4