Properties

Label 1520.2.a.t.1.3
Level $1520$
Weight $2$
Character 1520.1
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
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] = [1520,2,Mod(1,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, 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("1520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.78165\) of defining polynomial
Character \(\chi\) \(=\) 1520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.296842 q^{3} -1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} +O(q^{10})\) \(q-0.296842 q^{3} -1.00000 q^{5} +3.56331 q^{7} -2.91188 q^{9} -5.56331 q^{11} +5.26647 q^{13} +0.296842 q^{15} +1.40632 q^{17} -1.00000 q^{19} -1.05774 q^{21} +6.96962 q^{23} +1.00000 q^{25} +1.75489 q^{27} +1.40632 q^{29} -1.75489 q^{31} +1.65142 q^{33} -3.56331 q^{35} +3.61504 q^{37} -1.56331 q^{39} +4.34858 q^{41} +3.56331 q^{43} +2.91188 q^{45} +8.26046 q^{47} +5.69716 q^{49} -0.417453 q^{51} -7.61504 q^{53} +5.56331 q^{55} +0.296842 q^{57} -9.47519 q^{59} +9.21473 q^{61} -10.3759 q^{63} -5.26647 q^{65} +4.76090 q^{67} -2.06888 q^{69} +14.0689 q^{71} +6.59368 q^{73} -0.296842 q^{75} -19.8238 q^{77} -5.47519 q^{79} +8.21473 q^{81} +4.15699 q^{83} -1.40632 q^{85} -0.417453 q^{87} -9.23009 q^{89} +18.7660 q^{91} +0.520926 q^{93} +1.00000 q^{95} +11.5116 q^{97} +16.1997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.296842 −0.171382 −0.0856908 0.996322i \(-0.527310\pi\)
−0.0856908 + 0.996322i \(0.527310\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.56331 1.34680 0.673402 0.739277i \(-0.264832\pi\)
0.673402 + 0.739277i \(0.264832\pi\)
\(8\) 0 0
\(9\) −2.91188 −0.970628
\(10\) 0 0
\(11\) −5.56331 −1.67740 −0.838700 0.544594i \(-0.816684\pi\)
−0.838700 + 0.544594i \(0.816684\pi\)
\(12\) 0 0
\(13\) 5.26647 1.46065 0.730327 0.683097i \(-0.239368\pi\)
0.730327 + 0.683097i \(0.239368\pi\)
\(14\) 0 0
\(15\) 0.296842 0.0766442
\(16\) 0 0
\(17\) 1.40632 0.341082 0.170541 0.985351i \(-0.445448\pi\)
0.170541 + 0.985351i \(0.445448\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.05774 −0.230817
\(22\) 0 0
\(23\) 6.96962 1.45327 0.726633 0.687025i \(-0.241084\pi\)
0.726633 + 0.687025i \(0.241084\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.75489 0.337730
\(28\) 0 0
\(29\) 1.40632 0.261146 0.130573 0.991439i \(-0.458318\pi\)
0.130573 + 0.991439i \(0.458318\pi\)
\(30\) 0 0
\(31\) −1.75489 −0.315188 −0.157594 0.987504i \(-0.550374\pi\)
−0.157594 + 0.987504i \(0.550374\pi\)
\(32\) 0 0
\(33\) 1.65142 0.287476
\(34\) 0 0
\(35\) −3.56331 −0.602309
\(36\) 0 0
\(37\) 3.61504 0.594309 0.297155 0.954829i \(-0.403962\pi\)
0.297155 + 0.954829i \(0.403962\pi\)
\(38\) 0 0
\(39\) −1.56331 −0.250329
\(40\) 0 0
\(41\) 4.34858 0.679134 0.339567 0.940582i \(-0.389720\pi\)
0.339567 + 0.940582i \(0.389720\pi\)
\(42\) 0 0
\(43\) 3.56331 0.543399 0.271700 0.962382i \(-0.412414\pi\)
0.271700 + 0.962382i \(0.412414\pi\)
\(44\) 0 0
\(45\) 2.91188 0.434078
\(46\) 0 0
\(47\) 8.26046 1.20491 0.602456 0.798152i \(-0.294189\pi\)
0.602456 + 0.798152i \(0.294189\pi\)
\(48\) 0 0
\(49\) 5.69716 0.813879
\(50\) 0 0
\(51\) −0.417453 −0.0584552
\(52\) 0 0
\(53\) −7.61504 −1.04601 −0.523003 0.852331i \(-0.675188\pi\)
−0.523003 + 0.852331i \(0.675188\pi\)
\(54\) 0 0
\(55\) 5.56331 0.750156
\(56\) 0 0
\(57\) 0.296842 0.0393177
\(58\) 0 0
\(59\) −9.47519 −1.23356 −0.616782 0.787134i \(-0.711564\pi\)
−0.616782 + 0.787134i \(0.711564\pi\)
\(60\) 0 0
\(61\) 9.21473 1.17983 0.589913 0.807467i \(-0.299162\pi\)
0.589913 + 0.807467i \(0.299162\pi\)
\(62\) 0 0
\(63\) −10.3759 −1.30725
\(64\) 0 0
\(65\) −5.26647 −0.653225
\(66\) 0 0
\(67\) 4.76090 0.581636 0.290818 0.956778i \(-0.406073\pi\)
0.290818 + 0.956778i \(0.406073\pi\)
\(68\) 0 0
\(69\) −2.06888 −0.249063
\(70\) 0 0
\(71\) 14.0689 1.66967 0.834834 0.550502i \(-0.185563\pi\)
0.834834 + 0.550502i \(0.185563\pi\)
\(72\) 0 0
\(73\) 6.59368 0.771732 0.385866 0.922555i \(-0.373903\pi\)
0.385866 + 0.922555i \(0.373903\pi\)
\(74\) 0 0
\(75\) −0.296842 −0.0342763
\(76\) 0 0
\(77\) −19.8238 −2.25913
\(78\) 0 0
\(79\) −5.47519 −0.616007 −0.308004 0.951385i \(-0.599661\pi\)
−0.308004 + 0.951385i \(0.599661\pi\)
\(80\) 0 0
\(81\) 8.21473 0.912748
\(82\) 0 0
\(83\) 4.15699 0.456289 0.228144 0.973627i \(-0.426734\pi\)
0.228144 + 0.973627i \(0.426734\pi\)
\(84\) 0 0
\(85\) −1.40632 −0.152536
\(86\) 0 0
\(87\) −0.417453 −0.0447557
\(88\) 0 0
\(89\) −9.23009 −0.978387 −0.489194 0.872175i \(-0.662709\pi\)
−0.489194 + 0.872175i \(0.662709\pi\)
\(90\) 0 0
\(91\) 18.7660 1.96721
\(92\) 0 0
\(93\) 0.520926 0.0540175
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 11.5116 1.16882 0.584411 0.811457i \(-0.301325\pi\)
0.584411 + 0.811457i \(0.301325\pi\)
\(98\) 0 0
\(99\) 16.1997 1.62813
\(100\) 0 0
\(101\) −11.8511 −1.17923 −0.589616 0.807684i \(-0.700721\pi\)
−0.589616 + 0.807684i \(0.700721\pi\)
\(102\) 0 0
\(103\) −1.35458 −0.133471 −0.0667354 0.997771i \(-0.521258\pi\)
−0.0667354 + 0.997771i \(0.521258\pi\)
\(104\) 0 0
\(105\) 1.05774 0.103225
\(106\) 0 0
\(107\) 7.06287 0.682794 0.341397 0.939919i \(-0.389100\pi\)
0.341397 + 0.939919i \(0.389100\pi\)
\(108\) 0 0
\(109\) 10.1844 0.975484 0.487742 0.872988i \(-0.337821\pi\)
0.487742 + 0.872988i \(0.337821\pi\)
\(110\) 0 0
\(111\) −1.07310 −0.101854
\(112\) 0 0
\(113\) 1.86015 0.174988 0.0874940 0.996165i \(-0.472114\pi\)
0.0874940 + 0.996165i \(0.472114\pi\)
\(114\) 0 0
\(115\) −6.96962 −0.649921
\(116\) 0 0
\(117\) −15.3353 −1.41775
\(118\) 0 0
\(119\) 5.01114 0.459370
\(120\) 0 0
\(121\) 19.9504 1.81367
\(122\) 0 0
\(123\) −1.29084 −0.116391
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.89053 0.433964 0.216982 0.976176i \(-0.430379\pi\)
0.216982 + 0.976176i \(0.430379\pi\)
\(128\) 0 0
\(129\) −1.05774 −0.0931287
\(130\) 0 0
\(131\) −2.81263 −0.245741 −0.122870 0.992423i \(-0.539210\pi\)
−0.122870 + 0.992423i \(0.539210\pi\)
\(132\) 0 0
\(133\) −3.56331 −0.308978
\(134\) 0 0
\(135\) −1.75489 −0.151037
\(136\) 0 0
\(137\) 9.23009 0.788579 0.394290 0.918986i \(-0.370991\pi\)
0.394290 + 0.918986i \(0.370991\pi\)
\(138\) 0 0
\(139\) 3.67878 0.312030 0.156015 0.987755i \(-0.450135\pi\)
0.156015 + 0.987755i \(0.450135\pi\)
\(140\) 0 0
\(141\) −2.45205 −0.206500
\(142\) 0 0
\(143\) −29.2990 −2.45010
\(144\) 0 0
\(145\) −1.40632 −0.116788
\(146\) 0 0
\(147\) −1.69115 −0.139484
\(148\) 0 0
\(149\) 7.09925 0.581593 0.290797 0.956785i \(-0.406080\pi\)
0.290797 + 0.956785i \(0.406080\pi\)
\(150\) 0 0
\(151\) 18.3567 1.49385 0.746924 0.664910i \(-0.231530\pi\)
0.746924 + 0.664910i \(0.231530\pi\)
\(152\) 0 0
\(153\) −4.09503 −0.331064
\(154\) 0 0
\(155\) 1.75489 0.140957
\(156\) 0 0
\(157\) 17.2301 1.37511 0.687555 0.726132i \(-0.258684\pi\)
0.687555 + 0.726132i \(0.258684\pi\)
\(158\) 0 0
\(159\) 2.26046 0.179266
\(160\) 0 0
\(161\) 24.8349 1.95726
\(162\) 0 0
\(163\) −10.8662 −0.851103 −0.425551 0.904934i \(-0.639920\pi\)
−0.425551 + 0.904934i \(0.639920\pi\)
\(164\) 0 0
\(165\) −1.65142 −0.128563
\(166\) 0 0
\(167\) −2.82977 −0.218974 −0.109487 0.993988i \(-0.534921\pi\)
−0.109487 + 0.993988i \(0.534921\pi\)
\(168\) 0 0
\(169\) 14.7357 1.13351
\(170\) 0 0
\(171\) 2.91188 0.222677
\(172\) 0 0
\(173\) −9.26647 −0.704516 −0.352258 0.935903i \(-0.614586\pi\)
−0.352258 + 0.935903i \(0.614586\pi\)
\(174\) 0 0
\(175\) 3.56331 0.269361
\(176\) 0 0
\(177\) 2.81263 0.211410
\(178\) 0 0
\(179\) 3.59067 0.268379 0.134190 0.990956i \(-0.457157\pi\)
0.134190 + 0.990956i \(0.457157\pi\)
\(180\) 0 0
\(181\) −19.7630 −1.46897 −0.734487 0.678623i \(-0.762577\pi\)
−0.734487 + 0.678623i \(0.762577\pi\)
\(182\) 0 0
\(183\) −2.73532 −0.202200
\(184\) 0 0
\(185\) −3.61504 −0.265783
\(186\) 0 0
\(187\) −7.82377 −0.572131
\(188\) 0 0
\(189\) 6.25323 0.454855
\(190\) 0 0
\(191\) −8.31398 −0.601579 −0.300789 0.953691i \(-0.597250\pi\)
−0.300789 + 0.953691i \(0.597250\pi\)
\(192\) 0 0
\(193\) −22.2514 −1.60169 −0.800847 0.598869i \(-0.795617\pi\)
−0.800847 + 0.598869i \(0.795617\pi\)
\(194\) 0 0
\(195\) 1.56331 0.111951
\(196\) 0 0
\(197\) −8.81263 −0.627874 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(198\) 0 0
\(199\) −21.0659 −1.49332 −0.746660 0.665206i \(-0.768344\pi\)
−0.746660 + 0.665206i \(0.768344\pi\)
\(200\) 0 0
\(201\) −1.41323 −0.0996818
\(202\) 0 0
\(203\) 5.01114 0.351713
\(204\) 0 0
\(205\) −4.34858 −0.303718
\(206\) 0 0
\(207\) −20.2947 −1.41058
\(208\) 0 0
\(209\) 5.56331 0.384822
\(210\) 0 0
\(211\) −5.34556 −0.368004 −0.184002 0.982926i \(-0.558905\pi\)
−0.184002 + 0.982926i \(0.558905\pi\)
\(212\) 0 0
\(213\) −4.17623 −0.286151
\(214\) 0 0
\(215\) −3.56331 −0.243016
\(216\) 0 0
\(217\) −6.25323 −0.424497
\(218\) 0 0
\(219\) −1.95728 −0.132261
\(220\) 0 0
\(221\) 7.40632 0.498203
\(222\) 0 0
\(223\) 3.10947 0.208226 0.104113 0.994565i \(-0.466800\pi\)
0.104113 + 0.994565i \(0.466800\pi\)
\(224\) 0 0
\(225\) −2.91188 −0.194126
\(226\) 0 0
\(227\) −14.4692 −0.960354 −0.480177 0.877172i \(-0.659428\pi\)
−0.480177 + 0.877172i \(0.659428\pi\)
\(228\) 0 0
\(229\) −5.21473 −0.344599 −0.172299 0.985045i \(-0.555120\pi\)
−0.172299 + 0.985045i \(0.555120\pi\)
\(230\) 0 0
\(231\) 5.88452 0.387173
\(232\) 0 0
\(233\) −3.18737 −0.208811 −0.104406 0.994535i \(-0.533294\pi\)
−0.104406 + 0.994535i \(0.533294\pi\)
\(234\) 0 0
\(235\) −8.26046 −0.538853
\(236\) 0 0
\(237\) 1.62527 0.105572
\(238\) 0 0
\(239\) −16.5209 −1.06865 −0.534325 0.845279i \(-0.679434\pi\)
−0.534325 + 0.845279i \(0.679434\pi\)
\(240\) 0 0
\(241\) −12.2271 −0.787615 −0.393807 0.919193i \(-0.628842\pi\)
−0.393807 + 0.919193i \(0.628842\pi\)
\(242\) 0 0
\(243\) −7.70316 −0.494158
\(244\) 0 0
\(245\) −5.69716 −0.363978
\(246\) 0 0
\(247\) −5.26647 −0.335097
\(248\) 0 0
\(249\) −1.23397 −0.0781996
\(250\) 0 0
\(251\) −4.52093 −0.285358 −0.142679 0.989769i \(-0.545572\pi\)
−0.142679 + 0.989769i \(0.545572\pi\)
\(252\) 0 0
\(253\) −38.7742 −2.43771
\(254\) 0 0
\(255\) 0.417453 0.0261420
\(256\) 0 0
\(257\) 15.9290 0.993625 0.496813 0.867858i \(-0.334504\pi\)
0.496813 + 0.867858i \(0.334504\pi\)
\(258\) 0 0
\(259\) 12.8815 0.800418
\(260\) 0 0
\(261\) −4.09503 −0.253476
\(262\) 0 0
\(263\) −0.854147 −0.0526689 −0.0263345 0.999653i \(-0.508383\pi\)
−0.0263345 + 0.999653i \(0.508383\pi\)
\(264\) 0 0
\(265\) 7.61504 0.467788
\(266\) 0 0
\(267\) 2.73988 0.167678
\(268\) 0 0
\(269\) 10.3913 0.633569 0.316784 0.948498i \(-0.397397\pi\)
0.316784 + 0.948498i \(0.397397\pi\)
\(270\) 0 0
\(271\) 8.19971 0.498097 0.249048 0.968491i \(-0.419882\pi\)
0.249048 + 0.968491i \(0.419882\pi\)
\(272\) 0 0
\(273\) −5.57054 −0.337145
\(274\) 0 0
\(275\) −5.56331 −0.335480
\(276\) 0 0
\(277\) 14.6484 0.880137 0.440069 0.897964i \(-0.354954\pi\)
0.440069 + 0.897964i \(0.354954\pi\)
\(278\) 0 0
\(279\) 5.11005 0.305931
\(280\) 0 0
\(281\) −31.6129 −1.88587 −0.942935 0.332977i \(-0.891947\pi\)
−0.942935 + 0.332977i \(0.891947\pi\)
\(282\) 0 0
\(283\) −24.7326 −1.47020 −0.735101 0.677957i \(-0.762865\pi\)
−0.735101 + 0.677957i \(0.762865\pi\)
\(284\) 0 0
\(285\) −0.296842 −0.0175834
\(286\) 0 0
\(287\) 15.4953 0.914660
\(288\) 0 0
\(289\) −15.0223 −0.883663
\(290\) 0 0
\(291\) −3.41712 −0.200315
\(292\) 0 0
\(293\) 30.2857 1.76931 0.884655 0.466245i \(-0.154394\pi\)
0.884655 + 0.466245i \(0.154394\pi\)
\(294\) 0 0
\(295\) 9.47519 0.551667
\(296\) 0 0
\(297\) −9.76302 −0.566508
\(298\) 0 0
\(299\) 36.7053 2.12272
\(300\) 0 0
\(301\) 12.6972 0.731852
\(302\) 0 0
\(303\) 3.51791 0.202099
\(304\) 0 0
\(305\) −9.21473 −0.527634
\(306\) 0 0
\(307\) 23.0629 1.31627 0.658134 0.752901i \(-0.271346\pi\)
0.658134 + 0.752901i \(0.271346\pi\)
\(308\) 0 0
\(309\) 0.402096 0.0228744
\(310\) 0 0
\(311\) 10.3152 0.584921 0.292460 0.956278i \(-0.405526\pi\)
0.292460 + 0.956278i \(0.405526\pi\)
\(312\) 0 0
\(313\) −4.95038 −0.279812 −0.139906 0.990165i \(-0.544680\pi\)
−0.139906 + 0.990165i \(0.544680\pi\)
\(314\) 0 0
\(315\) 10.3759 0.584618
\(316\) 0 0
\(317\) −14.2433 −0.799985 −0.399992 0.916518i \(-0.630987\pi\)
−0.399992 + 0.916518i \(0.630987\pi\)
\(318\) 0 0
\(319\) −7.82377 −0.438047
\(320\) 0 0
\(321\) −2.09656 −0.117018
\(322\) 0 0
\(323\) −1.40632 −0.0782495
\(324\) 0 0
\(325\) 5.26647 0.292131
\(326\) 0 0
\(327\) −3.02314 −0.167180
\(328\) 0 0
\(329\) 29.4346 1.62278
\(330\) 0 0
\(331\) −2.04272 −0.112278 −0.0561390 0.998423i \(-0.517879\pi\)
−0.0561390 + 0.998423i \(0.517879\pi\)
\(332\) 0 0
\(333\) −10.5266 −0.576854
\(334\) 0 0
\(335\) −4.76090 −0.260116
\(336\) 0 0
\(337\) 34.6951 1.88996 0.944980 0.327128i \(-0.106081\pi\)
0.944980 + 0.327128i \(0.106081\pi\)
\(338\) 0 0
\(339\) −0.552170 −0.0299898
\(340\) 0 0
\(341\) 9.76302 0.528697
\(342\) 0 0
\(343\) −4.64243 −0.250668
\(344\) 0 0
\(345\) 2.06888 0.111385
\(346\) 0 0
\(347\) −7.35280 −0.394719 −0.197359 0.980331i \(-0.563237\pi\)
−0.197359 + 0.980331i \(0.563237\pi\)
\(348\) 0 0
\(349\) −31.7630 −1.70024 −0.850118 0.526593i \(-0.823469\pi\)
−0.850118 + 0.526593i \(0.823469\pi\)
\(350\) 0 0
\(351\) 9.24209 0.493306
\(352\) 0 0
\(353\) −7.52179 −0.400345 −0.200172 0.979761i \(-0.564150\pi\)
−0.200172 + 0.979761i \(0.564150\pi\)
\(354\) 0 0
\(355\) −14.0689 −0.746698
\(356\) 0 0
\(357\) −1.48751 −0.0787276
\(358\) 0 0
\(359\) 30.3982 1.60436 0.802178 0.597085i \(-0.203674\pi\)
0.802178 + 0.597085i \(0.203674\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −5.92211 −0.310830
\(364\) 0 0
\(365\) −6.59368 −0.345129
\(366\) 0 0
\(367\) −3.91577 −0.204401 −0.102201 0.994764i \(-0.532588\pi\)
−0.102201 + 0.994764i \(0.532588\pi\)
\(368\) 0 0
\(369\) −12.6626 −0.659186
\(370\) 0 0
\(371\) −27.1347 −1.40877
\(372\) 0 0
\(373\) 26.7759 1.38641 0.693203 0.720743i \(-0.256199\pi\)
0.693203 + 0.720743i \(0.256199\pi\)
\(374\) 0 0
\(375\) 0.296842 0.0153288
\(376\) 0 0
\(377\) 7.40632 0.381445
\(378\) 0 0
\(379\) −14.9504 −0.767950 −0.383975 0.923344i \(-0.625445\pi\)
−0.383975 + 0.923344i \(0.625445\pi\)
\(380\) 0 0
\(381\) −1.45171 −0.0743735
\(382\) 0 0
\(383\) 27.9910 1.43027 0.715136 0.698985i \(-0.246365\pi\)
0.715136 + 0.698985i \(0.246365\pi\)
\(384\) 0 0
\(385\) 19.8238 1.01031
\(386\) 0 0
\(387\) −10.3759 −0.527439
\(388\) 0 0
\(389\) −35.2036 −1.78489 −0.892447 0.451152i \(-0.851013\pi\)
−0.892447 + 0.451152i \(0.851013\pi\)
\(390\) 0 0
\(391\) 9.80150 0.495683
\(392\) 0 0
\(393\) 0.834907 0.0421155
\(394\) 0 0
\(395\) 5.47519 0.275487
\(396\) 0 0
\(397\) −35.9735 −1.80546 −0.902730 0.430208i \(-0.858440\pi\)
−0.902730 + 0.430208i \(0.858440\pi\)
\(398\) 0 0
\(399\) 1.05774 0.0529532
\(400\) 0 0
\(401\) 23.2421 1.16065 0.580327 0.814383i \(-0.302925\pi\)
0.580327 + 0.814383i \(0.302925\pi\)
\(402\) 0 0
\(403\) −9.24209 −0.460381
\(404\) 0 0
\(405\) −8.21473 −0.408193
\(406\) 0 0
\(407\) −20.1116 −0.996895
\(408\) 0 0
\(409\) −31.8926 −1.57699 −0.788495 0.615041i \(-0.789139\pi\)
−0.788495 + 0.615041i \(0.789139\pi\)
\(410\) 0 0
\(411\) −2.73988 −0.135148
\(412\) 0 0
\(413\) −33.7630 −1.66137
\(414\) 0 0
\(415\) −4.15699 −0.204059
\(416\) 0 0
\(417\) −1.09202 −0.0534763
\(418\) 0 0
\(419\) −31.8238 −1.55469 −0.777346 0.629073i \(-0.783435\pi\)
−0.777346 + 0.629073i \(0.783435\pi\)
\(420\) 0 0
\(421\) 0.348578 0.0169887 0.00849433 0.999964i \(-0.497296\pi\)
0.00849433 + 0.999964i \(0.497296\pi\)
\(422\) 0 0
\(423\) −24.0535 −1.16952
\(424\) 0 0
\(425\) 1.40632 0.0682164
\(426\) 0 0
\(427\) 32.8349 1.58899
\(428\) 0 0
\(429\) 8.69716 0.419903
\(430\) 0 0
\(431\) −29.2764 −1.41019 −0.705097 0.709111i \(-0.749096\pi\)
−0.705097 + 0.709111i \(0.749096\pi\)
\(432\) 0 0
\(433\) −0.883290 −0.0424482 −0.0212241 0.999775i \(-0.506756\pi\)
−0.0212241 + 0.999775i \(0.506756\pi\)
\(434\) 0 0
\(435\) 0.417453 0.0200154
\(436\) 0 0
\(437\) −6.96962 −0.333402
\(438\) 0 0
\(439\) 13.8584 0.661424 0.330712 0.943732i \(-0.392711\pi\)
0.330712 + 0.943732i \(0.392711\pi\)
\(440\) 0 0
\(441\) −16.5895 −0.789974
\(442\) 0 0
\(443\) 13.5753 0.644982 0.322491 0.946572i \(-0.395480\pi\)
0.322491 + 0.946572i \(0.395480\pi\)
\(444\) 0 0
\(445\) 9.23009 0.437548
\(446\) 0 0
\(447\) −2.10735 −0.0996745
\(448\) 0 0
\(449\) −15.6334 −0.737785 −0.368893 0.929472i \(-0.620263\pi\)
−0.368893 + 0.929472i \(0.620263\pi\)
\(450\) 0 0
\(451\) −24.1925 −1.13918
\(452\) 0 0
\(453\) −5.44904 −0.256018
\(454\) 0 0
\(455\) −18.7660 −0.879765
\(456\) 0 0
\(457\) 5.30284 0.248057 0.124028 0.992279i \(-0.460419\pi\)
0.124028 + 0.992279i \(0.460419\pi\)
\(458\) 0 0
\(459\) 2.46794 0.115193
\(460\) 0 0
\(461\) 0.374734 0.0174531 0.00872656 0.999962i \(-0.497222\pi\)
0.00872656 + 0.999962i \(0.497222\pi\)
\(462\) 0 0
\(463\) −6.65564 −0.309314 −0.154657 0.987968i \(-0.549427\pi\)
−0.154657 + 0.987968i \(0.549427\pi\)
\(464\) 0 0
\(465\) −0.520926 −0.0241574
\(466\) 0 0
\(467\) 0.854147 0.0395252 0.0197626 0.999805i \(-0.493709\pi\)
0.0197626 + 0.999805i \(0.493709\pi\)
\(468\) 0 0
\(469\) 16.9645 0.783349
\(470\) 0 0
\(471\) −5.11461 −0.235669
\(472\) 0 0
\(473\) −19.8238 −0.911498
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 22.1741 1.01528
\(478\) 0 0
\(479\) 17.0731 0.780090 0.390045 0.920796i \(-0.372460\pi\)
0.390045 + 0.920796i \(0.372460\pi\)
\(480\) 0 0
\(481\) 19.0385 0.868081
\(482\) 0 0
\(483\) −7.37204 −0.335439
\(484\) 0 0
\(485\) −11.5116 −0.522713
\(486\) 0 0
\(487\) −12.8259 −0.581197 −0.290598 0.956845i \(-0.593854\pi\)
−0.290598 + 0.956845i \(0.593854\pi\)
\(488\) 0 0
\(489\) 3.22553 0.145863
\(490\) 0 0
\(491\) 10.4054 0.469591 0.234796 0.972045i \(-0.424558\pi\)
0.234796 + 0.972045i \(0.424558\pi\)
\(492\) 0 0
\(493\) 1.97773 0.0890723
\(494\) 0 0
\(495\) −16.1997 −0.728123
\(496\) 0 0
\(497\) 50.1317 2.24872
\(498\) 0 0
\(499\) −36.1612 −1.61880 −0.809399 0.587258i \(-0.800207\pi\)
−0.809399 + 0.587258i \(0.800207\pi\)
\(500\) 0 0
\(501\) 0.839995 0.0375282
\(502\) 0 0
\(503\) −33.2536 −1.48270 −0.741352 0.671117i \(-0.765815\pi\)
−0.741352 + 0.671117i \(0.765815\pi\)
\(504\) 0 0
\(505\) 11.8511 0.527368
\(506\) 0 0
\(507\) −4.37416 −0.194263
\(508\) 0 0
\(509\) 7.29084 0.323161 0.161580 0.986860i \(-0.448341\pi\)
0.161580 + 0.986860i \(0.448341\pi\)
\(510\) 0 0
\(511\) 23.4953 1.03937
\(512\) 0 0
\(513\) −1.75489 −0.0774805
\(514\) 0 0
\(515\) 1.35458 0.0596899
\(516\) 0 0
\(517\) −45.9555 −2.02112
\(518\) 0 0
\(519\) 2.75067 0.120741
\(520\) 0 0
\(521\) −9.87849 −0.432785 −0.216392 0.976306i \(-0.569429\pi\)
−0.216392 + 0.976306i \(0.569429\pi\)
\(522\) 0 0
\(523\) 11.1925 0.489414 0.244707 0.969597i \(-0.421308\pi\)
0.244707 + 0.969597i \(0.421308\pi\)
\(524\) 0 0
\(525\) −1.05774 −0.0461635
\(526\) 0 0
\(527\) −2.46794 −0.107505
\(528\) 0 0
\(529\) 25.5756 1.11198
\(530\) 0 0
\(531\) 27.5907 1.19733
\(532\) 0 0
\(533\) 22.9016 0.991980
\(534\) 0 0
\(535\) −7.06287 −0.305355
\(536\) 0 0
\(537\) −1.06586 −0.0459953
\(538\) 0 0
\(539\) −31.6950 −1.36520
\(540\) 0 0
\(541\) 2.22587 0.0956974 0.0478487 0.998855i \(-0.484763\pi\)
0.0478487 + 0.998855i \(0.484763\pi\)
\(542\) 0 0
\(543\) 5.86649 0.251755
\(544\) 0 0
\(545\) −10.1844 −0.436250
\(546\) 0 0
\(547\) 34.9675 1.49510 0.747552 0.664204i \(-0.231229\pi\)
0.747552 + 0.664204i \(0.231229\pi\)
\(548\) 0 0
\(549\) −26.8322 −1.14517
\(550\) 0 0
\(551\) −1.40632 −0.0599111
\(552\) 0 0
\(553\) −19.5098 −0.829641
\(554\) 0 0
\(555\) 1.07310 0.0455504
\(556\) 0 0
\(557\) −8.88539 −0.376486 −0.188243 0.982122i \(-0.560279\pi\)
−0.188243 + 0.982122i \(0.560279\pi\)
\(558\) 0 0
\(559\) 18.7660 0.793719
\(560\) 0 0
\(561\) 2.32242 0.0980527
\(562\) 0 0
\(563\) 29.6767 1.25072 0.625362 0.780335i \(-0.284952\pi\)
0.625362 + 0.780335i \(0.284952\pi\)
\(564\) 0 0
\(565\) −1.86015 −0.0782570
\(566\) 0 0
\(567\) 29.2716 1.22929
\(568\) 0 0
\(569\) −22.1152 −0.927116 −0.463558 0.886067i \(-0.653427\pi\)
−0.463558 + 0.886067i \(0.653427\pi\)
\(570\) 0 0
\(571\) 33.2656 1.39212 0.696060 0.717983i \(-0.254935\pi\)
0.696060 + 0.717983i \(0.254935\pi\)
\(572\) 0 0
\(573\) 2.46794 0.103100
\(574\) 0 0
\(575\) 6.96962 0.290653
\(576\) 0 0
\(577\) 0.934140 0.0388887 0.0194444 0.999811i \(-0.493810\pi\)
0.0194444 + 0.999811i \(0.493810\pi\)
\(578\) 0 0
\(579\) 6.60516 0.274501
\(580\) 0 0
\(581\) 14.8126 0.614532
\(582\) 0 0
\(583\) 42.3648 1.75457
\(584\) 0 0
\(585\) 15.3353 0.634038
\(586\) 0 0
\(587\) −1.57531 −0.0650200 −0.0325100 0.999471i \(-0.510350\pi\)
−0.0325100 + 0.999471i \(0.510350\pi\)
\(588\) 0 0
\(589\) 1.75489 0.0723092
\(590\) 0 0
\(591\) 2.61596 0.107606
\(592\) 0 0
\(593\) 26.0162 1.06836 0.534180 0.845371i \(-0.320621\pi\)
0.534180 + 0.845371i \(0.320621\pi\)
\(594\) 0 0
\(595\) −5.01114 −0.205437
\(596\) 0 0
\(597\) 6.25323 0.255928
\(598\) 0 0
\(599\) −19.5359 −0.798217 −0.399109 0.916904i \(-0.630680\pi\)
−0.399109 + 0.916904i \(0.630680\pi\)
\(600\) 0 0
\(601\) 43.5299 1.77562 0.887811 0.460208i \(-0.152225\pi\)
0.887811 + 0.460208i \(0.152225\pi\)
\(602\) 0 0
\(603\) −13.8632 −0.564552
\(604\) 0 0
\(605\) −19.9504 −0.811098
\(606\) 0 0
\(607\) 12.5847 0.510796 0.255398 0.966836i \(-0.417794\pi\)
0.255398 + 0.966836i \(0.417794\pi\)
\(608\) 0 0
\(609\) −1.48751 −0.0602771
\(610\) 0 0
\(611\) 43.5034 1.75996
\(612\) 0 0
\(613\) −18.2412 −0.736756 −0.368378 0.929676i \(-0.620087\pi\)
−0.368378 + 0.929676i \(0.620087\pi\)
\(614\) 0 0
\(615\) 1.29084 0.0520517
\(616\) 0 0
\(617\) 26.7314 1.07617 0.538084 0.842892i \(-0.319148\pi\)
0.538084 + 0.842892i \(0.319148\pi\)
\(618\) 0 0
\(619\) −2.92690 −0.117642 −0.0588211 0.998269i \(-0.518734\pi\)
−0.0588211 + 0.998269i \(0.518734\pi\)
\(620\) 0 0
\(621\) 12.2310 0.490811
\(622\) 0 0
\(623\) −32.8896 −1.31770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.65142 −0.0659514
\(628\) 0 0
\(629\) 5.08389 0.202708
\(630\) 0 0
\(631\) 13.2493 0.527447 0.263724 0.964598i \(-0.415049\pi\)
0.263724 + 0.964598i \(0.415049\pi\)
\(632\) 0 0
\(633\) 1.58679 0.0630691
\(634\) 0 0
\(635\) −4.89053 −0.194075
\(636\) 0 0
\(637\) 30.0039 1.18880
\(638\) 0 0
\(639\) −40.9669 −1.62063
\(640\) 0 0
\(641\) 6.93026 0.273729 0.136864 0.990590i \(-0.456298\pi\)
0.136864 + 0.990590i \(0.456298\pi\)
\(642\) 0 0
\(643\) −14.4794 −0.571012 −0.285506 0.958377i \(-0.592162\pi\)
−0.285506 + 0.958377i \(0.592162\pi\)
\(644\) 0 0
\(645\) 1.05774 0.0416484
\(646\) 0 0
\(647\) −6.35549 −0.249860 −0.124930 0.992166i \(-0.539871\pi\)
−0.124930 + 0.992166i \(0.539871\pi\)
\(648\) 0 0
\(649\) 52.7134 2.06918
\(650\) 0 0
\(651\) 1.85622 0.0727510
\(652\) 0 0
\(653\) −34.4030 −1.34629 −0.673146 0.739509i \(-0.735058\pi\)
−0.673146 + 0.739509i \(0.735058\pi\)
\(654\) 0 0
\(655\) 2.81263 0.109899
\(656\) 0 0
\(657\) −19.2000 −0.749065
\(658\) 0 0
\(659\) 19.6214 0.764341 0.382170 0.924092i \(-0.375177\pi\)
0.382170 + 0.924092i \(0.375177\pi\)
\(660\) 0 0
\(661\) 39.8054 1.54825 0.774126 0.633032i \(-0.218190\pi\)
0.774126 + 0.633032i \(0.218190\pi\)
\(662\) 0 0
\(663\) −2.19850 −0.0853828
\(664\) 0 0
\(665\) 3.56331 0.138179
\(666\) 0 0
\(667\) 9.80150 0.379515
\(668\) 0 0
\(669\) −0.923022 −0.0356861
\(670\) 0 0
\(671\) −51.2644 −1.97904
\(672\) 0 0
\(673\) −8.90374 −0.343214 −0.171607 0.985166i \(-0.554896\pi\)
−0.171607 + 0.985166i \(0.554896\pi\)
\(674\) 0 0
\(675\) 1.75489 0.0675459
\(676\) 0 0
\(677\) 22.2695 0.855886 0.427943 0.903806i \(-0.359238\pi\)
0.427943 + 0.903806i \(0.359238\pi\)
\(678\) 0 0
\(679\) 41.0193 1.57417
\(680\) 0 0
\(681\) 4.29506 0.164587
\(682\) 0 0
\(683\) −15.4054 −0.589472 −0.294736 0.955579i \(-0.595232\pi\)
−0.294736 + 0.955579i \(0.595232\pi\)
\(684\) 0 0
\(685\) −9.23009 −0.352663
\(686\) 0 0
\(687\) 1.54795 0.0590580
\(688\) 0 0
\(689\) −40.1044 −1.52785
\(690\) 0 0
\(691\) 17.8773 0.680084 0.340042 0.940410i \(-0.389559\pi\)
0.340042 + 0.940410i \(0.389559\pi\)
\(692\) 0 0
\(693\) 57.7245 2.19277
\(694\) 0 0
\(695\) −3.67878 −0.139544
\(696\) 0 0
\(697\) 6.11548 0.231640
\(698\) 0 0
\(699\) 0.946144 0.0357864
\(700\) 0 0
\(701\) 35.3609 1.33556 0.667782 0.744357i \(-0.267244\pi\)
0.667782 + 0.744357i \(0.267244\pi\)
\(702\) 0 0
\(703\) −3.61504 −0.136344
\(704\) 0 0
\(705\) 2.45205 0.0923496
\(706\) 0 0
\(707\) −42.2292 −1.58819
\(708\) 0 0
\(709\) 6.90410 0.259289 0.129644 0.991561i \(-0.458616\pi\)
0.129644 + 0.991561i \(0.458616\pi\)
\(710\) 0 0
\(711\) 15.9431 0.597914
\(712\) 0 0
\(713\) −12.2310 −0.458053
\(714\) 0 0
\(715\) 29.2990 1.09572
\(716\) 0 0
\(717\) 4.90410 0.183147
\(718\) 0 0
\(719\) 38.9431 1.45233 0.726167 0.687518i \(-0.241300\pi\)
0.726167 + 0.687518i \(0.241300\pi\)
\(720\) 0 0
\(721\) −4.82678 −0.179759
\(722\) 0 0
\(723\) 3.62951 0.134983
\(724\) 0 0
\(725\) 1.40632 0.0522293
\(726\) 0 0
\(727\) 6.18347 0.229332 0.114666 0.993404i \(-0.463420\pi\)
0.114666 + 0.993404i \(0.463420\pi\)
\(728\) 0 0
\(729\) −22.3576 −0.828058
\(730\) 0 0
\(731\) 5.01114 0.185344
\(732\) 0 0
\(733\) −0.688715 −0.0254383 −0.0127191 0.999919i \(-0.504049\pi\)
−0.0127191 + 0.999919i \(0.504049\pi\)
\(734\) 0 0
\(735\) 1.69115 0.0623792
\(736\) 0 0
\(737\) −26.4863 −0.975636
\(738\) 0 0
\(739\) 26.2532 0.965741 0.482870 0.875692i \(-0.339594\pi\)
0.482870 + 0.875692i \(0.339594\pi\)
\(740\) 0 0
\(741\) 1.56331 0.0574295
\(742\) 0 0
\(743\) 32.5688 1.19483 0.597416 0.801931i \(-0.296194\pi\)
0.597416 + 0.801931i \(0.296194\pi\)
\(744\) 0 0
\(745\) −7.09925 −0.260096
\(746\) 0 0
\(747\) −12.1047 −0.442887
\(748\) 0 0
\(749\) 25.1672 0.919589
\(750\) 0 0
\(751\) −45.4833 −1.65971 −0.829855 0.557979i \(-0.811577\pi\)
−0.829855 + 0.557979i \(0.811577\pi\)
\(752\) 0 0
\(753\) 1.34200 0.0489052
\(754\) 0 0
\(755\) −18.3567 −0.668069
\(756\) 0 0
\(757\) 2.74947 0.0999311 0.0499656 0.998751i \(-0.484089\pi\)
0.0499656 + 0.998751i \(0.484089\pi\)
\(758\) 0 0
\(759\) 11.5098 0.417779
\(760\) 0 0
\(761\) −33.2978 −1.20704 −0.603521 0.797347i \(-0.706236\pi\)
−0.603521 + 0.797347i \(0.706236\pi\)
\(762\) 0 0
\(763\) 36.2900 1.31379
\(764\) 0 0
\(765\) 4.09503 0.148056
\(766\) 0 0
\(767\) −49.9008 −1.80181
\(768\) 0 0
\(769\) −19.1540 −0.690710 −0.345355 0.938472i \(-0.612241\pi\)
−0.345355 + 0.938472i \(0.612241\pi\)
\(770\) 0 0
\(771\) −4.72840 −0.170289
\(772\) 0 0
\(773\) 0.569309 0.0204766 0.0102383 0.999948i \(-0.496741\pi\)
0.0102383 + 0.999948i \(0.496741\pi\)
\(774\) 0 0
\(775\) −1.75489 −0.0630377
\(776\) 0 0
\(777\) −3.82377 −0.137177
\(778\) 0 0
\(779\) −4.34858 −0.155804
\(780\) 0 0
\(781\) −78.2695 −2.80070
\(782\) 0 0
\(783\) 2.46794 0.0881969
\(784\) 0 0
\(785\) −17.2301 −0.614968
\(786\) 0 0
\(787\) −15.9991 −0.570307 −0.285153 0.958482i \(-0.592044\pi\)
−0.285153 + 0.958482i \(0.592044\pi\)
\(788\) 0 0
\(789\) 0.253546 0.00902649
\(790\) 0 0
\(791\) 6.62828 0.235675
\(792\) 0 0
\(793\) 48.5290 1.72332
\(794\) 0 0
\(795\) −2.26046 −0.0801704
\(796\) 0 0
\(797\) −35.7528 −1.26643 −0.633214 0.773976i \(-0.718265\pi\)
−0.633214 + 0.773976i \(0.718265\pi\)
\(798\) 0 0
\(799\) 11.6168 0.410974
\(800\) 0 0
\(801\) 26.8769 0.949650
\(802\) 0 0
\(803\) −36.6827 −1.29450
\(804\) 0 0
\(805\) −24.8349 −0.875315
\(806\) 0 0
\(807\) −3.08457 −0.108582
\(808\) 0 0
\(809\) 23.2036 0.815796 0.407898 0.913028i \(-0.366262\pi\)
0.407898 + 0.913028i \(0.366262\pi\)
\(810\) 0 0
\(811\) 21.7549 0.763918 0.381959 0.924179i \(-0.375250\pi\)
0.381959 + 0.924179i \(0.375250\pi\)
\(812\) 0 0
\(813\) −2.43402 −0.0853647
\(814\) 0 0
\(815\) 10.8662 0.380625
\(816\) 0 0
\(817\) −3.56331 −0.124664
\(818\) 0 0
\(819\) −54.6445 −1.90943
\(820\) 0 0
\(821\) 52.2532 1.82365 0.911825 0.410579i \(-0.134673\pi\)
0.911825 + 0.410579i \(0.134673\pi\)
\(822\) 0 0
\(823\) 9.44783 0.329331 0.164665 0.986349i \(-0.447346\pi\)
0.164665 + 0.986349i \(0.447346\pi\)
\(824\) 0 0
\(825\) 1.65142 0.0574951
\(826\) 0 0
\(827\) 50.2216 1.74638 0.873188 0.487383i \(-0.162048\pi\)
0.873188 + 0.487383i \(0.162048\pi\)
\(828\) 0 0
\(829\) 44.2066 1.53536 0.767680 0.640834i \(-0.221411\pi\)
0.767680 + 0.640834i \(0.221411\pi\)
\(830\) 0 0
\(831\) −4.34826 −0.150839
\(832\) 0 0
\(833\) 8.01200 0.277599
\(834\) 0 0
\(835\) 2.82977 0.0979283
\(836\) 0 0
\(837\) −3.07965 −0.106448
\(838\) 0 0
\(839\) 30.4033 1.04964 0.524819 0.851214i \(-0.324133\pi\)
0.524819 + 0.851214i \(0.324133\pi\)
\(840\) 0 0
\(841\) −27.0223 −0.931803
\(842\) 0 0
\(843\) 9.38404 0.323204
\(844\) 0 0
\(845\) −14.7357 −0.506922
\(846\) 0 0
\(847\) 71.0893 2.44266
\(848\) 0 0
\(849\) 7.34168 0.251966
\(850\) 0 0
\(851\) 25.1955 0.863690
\(852\) 0 0
\(853\) 3.93925 0.134877 0.0674386 0.997723i \(-0.478517\pi\)
0.0674386 + 0.997723i \(0.478517\pi\)
\(854\) 0 0
\(855\) −2.91188 −0.0995844
\(856\) 0 0
\(857\) 27.4388 0.937292 0.468646 0.883386i \(-0.344742\pi\)
0.468646 + 0.883386i \(0.344742\pi\)
\(858\) 0 0
\(859\) −32.4517 −1.10724 −0.553619 0.832770i \(-0.686754\pi\)
−0.553619 + 0.832770i \(0.686754\pi\)
\(860\) 0 0
\(861\) −4.59966 −0.156756
\(862\) 0 0
\(863\) 2.10861 0.0717778 0.0358889 0.999356i \(-0.488574\pi\)
0.0358889 + 0.999356i \(0.488574\pi\)
\(864\) 0 0
\(865\) 9.26647 0.315069
\(866\) 0 0
\(867\) 4.45924 0.151444
\(868\) 0 0
\(869\) 30.4602 1.03329
\(870\) 0 0
\(871\) 25.0731 0.849569
\(872\) 0 0
\(873\) −33.5204 −1.13449
\(874\) 0 0
\(875\) −3.56331 −0.120462
\(876\) 0 0
\(877\) −37.6613 −1.27173 −0.635866 0.771799i \(-0.719357\pi\)
−0.635866 + 0.771799i \(0.719357\pi\)
\(878\) 0 0
\(879\) −8.99007 −0.303227
\(880\) 0 0
\(881\) −39.8818 −1.34365 −0.671827 0.740708i \(-0.734490\pi\)
−0.671827 + 0.740708i \(0.734490\pi\)
\(882\) 0 0
\(883\) −36.0458 −1.21304 −0.606518 0.795070i \(-0.707434\pi\)
−0.606518 + 0.795070i \(0.707434\pi\)
\(884\) 0 0
\(885\) −2.81263 −0.0945456
\(886\) 0 0
\(887\) 26.2057 0.879902 0.439951 0.898022i \(-0.354996\pi\)
0.439951 + 0.898022i \(0.354996\pi\)
\(888\) 0 0
\(889\) 17.4264 0.584464
\(890\) 0 0
\(891\) −45.7011 −1.53104
\(892\) 0 0
\(893\) −8.26046 −0.276426
\(894\) 0 0
\(895\) −3.59067 −0.120023
\(896\) 0 0
\(897\) −10.8957 −0.363796
\(898\) 0 0
\(899\) −2.46794 −0.0823103
\(900\) 0 0
\(901\) −10.7092 −0.356774
\(902\) 0 0
\(903\) −3.76905 −0.125426
\(904\) 0 0
\(905\) 19.7630 0.656945
\(906\) 0 0
\(907\) 22.8036 0.757182 0.378591 0.925564i \(-0.376409\pi\)
0.378591 + 0.925564i \(0.376409\pi\)
\(908\) 0 0
\(909\) 34.5091 1.14460
\(910\) 0 0
\(911\) −4.44146 −0.147152 −0.0735761 0.997290i \(-0.523441\pi\)
−0.0735761 + 0.997290i \(0.523441\pi\)
\(912\) 0 0
\(913\) −23.1266 −0.765379
\(914\) 0 0
\(915\) 2.73532 0.0904268
\(916\) 0 0
\(917\) −10.0223 −0.330965
\(918\) 0 0
\(919\) −37.0111 −1.22088 −0.610442 0.792061i \(-0.709008\pi\)
−0.610442 + 0.792061i \(0.709008\pi\)
\(920\) 0 0
\(921\) −6.84602 −0.225584
\(922\) 0 0
\(923\) 74.0932 2.43881
\(924\) 0 0
\(925\) 3.61504 0.118862
\(926\) 0 0
\(927\) 3.94438 0.129550
\(928\) 0 0
\(929\) −3.04185 −0.0997999 −0.0499000 0.998754i \(-0.515890\pi\)
−0.0499000 + 0.998754i \(0.515890\pi\)
\(930\) 0 0
\(931\) −5.69716 −0.186717
\(932\) 0 0
\(933\) −3.06198 −0.100245
\(934\) 0 0
\(935\) 7.82377 0.255865
\(936\) 0 0
\(937\) 38.6099 1.26133 0.630666 0.776055i \(-0.282782\pi\)
0.630666 + 0.776055i \(0.282782\pi\)
\(938\) 0 0
\(939\) 1.46948 0.0479547
\(940\) 0 0
\(941\) 2.69716 0.0879248 0.0439624 0.999033i \(-0.486002\pi\)
0.0439624 + 0.999033i \(0.486002\pi\)
\(942\) 0 0
\(943\) 30.3080 0.986963
\(944\) 0 0
\(945\) −6.25323 −0.203418
\(946\) 0 0
\(947\) 20.1877 0.656012 0.328006 0.944676i \(-0.393623\pi\)
0.328006 + 0.944676i \(0.393623\pi\)
\(948\) 0 0
\(949\) 34.7254 1.12723
\(950\) 0 0
\(951\) 4.22801 0.137103
\(952\) 0 0
\(953\) 5.18559 0.167978 0.0839888 0.996467i \(-0.473234\pi\)
0.0839888 + 0.996467i \(0.473234\pi\)
\(954\) 0 0
\(955\) 8.31398 0.269034
\(956\) 0 0
\(957\) 2.32242 0.0750732
\(958\) 0 0
\(959\) 32.8896 1.06206
\(960\) 0 0
\(961\) −27.9203 −0.900656
\(962\) 0 0
\(963\) −20.5663 −0.662739
\(964\) 0 0
\(965\) 22.2514 0.716299
\(966\) 0 0
\(967\) −58.0054 −1.86533 −0.932665 0.360744i \(-0.882523\pi\)
−0.932665 + 0.360744i \(0.882523\pi\)
\(968\) 0 0
\(969\) 0.417453 0.0134105
\(970\) 0 0
\(971\) 24.3221 0.780533 0.390267 0.920702i \(-0.372383\pi\)
0.390267 + 0.920702i \(0.372383\pi\)
\(972\) 0 0
\(973\) 13.1086 0.420244
\(974\) 0 0
\(975\) −1.56331 −0.0500659
\(976\) 0 0
\(977\) −36.1134 −1.15537 −0.577685 0.816260i \(-0.696044\pi\)
−0.577685 + 0.816260i \(0.696044\pi\)
\(978\) 0 0
\(979\) 51.3498 1.64115
\(980\) 0 0
\(981\) −29.6557 −0.946832
\(982\) 0 0
\(983\) −21.1603 −0.674909 −0.337455 0.941342i \(-0.609566\pi\)
−0.337455 + 0.941342i \(0.609566\pi\)
\(984\) 0 0
\(985\) 8.81263 0.280794
\(986\) 0 0
\(987\) −8.73741 −0.278115
\(988\) 0 0
\(989\) 24.8349 0.789704
\(990\) 0 0
\(991\) −20.2652 −0.643746 −0.321873 0.946783i \(-0.604312\pi\)
−0.321873 + 0.946783i \(0.604312\pi\)
\(992\) 0 0
\(993\) 0.606364 0.0192424
\(994\) 0 0
\(995\) 21.0659 0.667833
\(996\) 0 0
\(997\) 24.2875 0.769193 0.384597 0.923085i \(-0.374341\pi\)
0.384597 + 0.923085i \(0.374341\pi\)
\(998\) 0 0
\(999\) 6.34402 0.200716
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 1520.2.a.t.1.3 4
4.3 odd 2 95.2.a.b.1.4 4
5.4 even 2 7600.2.a.cf.1.2 4
8.3 odd 2 6080.2.a.cc.1.3 4
8.5 even 2 6080.2.a.ch.1.2 4
12.11 even 2 855.2.a.m.1.1 4
20.3 even 4 475.2.b.e.324.3 8
20.7 even 4 475.2.b.e.324.6 8
20.19 odd 2 475.2.a.i.1.1 4
28.27 even 2 4655.2.a.y.1.4 4
60.59 even 2 4275.2.a.bo.1.4 4
76.75 even 2 1805.2.a.p.1.1 4
380.379 even 2 9025.2.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.4 4 4.3 odd 2
475.2.a.i.1.1 4 20.19 odd 2
475.2.b.e.324.3 8 20.3 even 4
475.2.b.e.324.6 8 20.7 even 4
855.2.a.m.1.1 4 12.11 even 2
1520.2.a.t.1.3 4 1.1 even 1 trivial
1805.2.a.p.1.1 4 76.75 even 2
4275.2.a.bo.1.4 4 60.59 even 2
4655.2.a.y.1.4 4 28.27 even 2
6080.2.a.cc.1.3 4 8.3 odd 2
6080.2.a.ch.1.2 4 8.5 even 2
7600.2.a.cf.1.2 4 5.4 even 2
9025.2.a.bf.1.4 4 380.379 even 2