Properties

Label 1305.2.a.k.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $2$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -3.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -3.00000 q^{7} -2.23607 q^{8} +1.61803 q^{10} -3.47214 q^{11} -1.76393 q^{13} -4.85410 q^{14} -4.85410 q^{16} +5.47214 q^{17} -5.70820 q^{19} +0.618034 q^{20} -5.61803 q^{22} +1.00000 q^{25} -2.85410 q^{26} -1.85410 q^{28} -1.00000 q^{29} -8.00000 q^{31} -3.38197 q^{32} +8.85410 q^{34} -3.00000 q^{35} -8.00000 q^{37} -9.23607 q^{38} -2.23607 q^{40} -4.47214 q^{41} -1.23607 q^{43} -2.14590 q^{44} +6.70820 q^{47} +2.00000 q^{49} +1.61803 q^{50} -1.09017 q^{52} +11.2361 q^{53} -3.47214 q^{55} +6.70820 q^{56} -1.61803 q^{58} -0.763932 q^{59} +7.70820 q^{61} -12.9443 q^{62} +4.23607 q^{64} -1.76393 q^{65} -2.52786 q^{67} +3.38197 q^{68} -4.85410 q^{70} -2.76393 q^{71} -8.00000 q^{73} -12.9443 q^{74} -3.52786 q^{76} +10.4164 q^{77} +16.1803 q^{79} -4.85410 q^{80} -7.23607 q^{82} -9.70820 q^{83} +5.47214 q^{85} -2.00000 q^{86} +7.76393 q^{88} +11.1803 q^{89} +5.29180 q^{91} +10.8541 q^{94} -5.70820 q^{95} +7.23607 q^{97} +3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 2 q^{5} - 6 q^{7} + q^{10} + 2 q^{11} - 8 q^{13} - 3 q^{14} - 3 q^{16} + 2 q^{17} + 2 q^{19} - q^{20} - 9 q^{22} + 2 q^{25} + q^{26} + 3 q^{28} - 2 q^{29} - 16 q^{31} - 9 q^{32} + 11 q^{34} - 6 q^{35} - 16 q^{37} - 14 q^{38} + 2 q^{43} - 11 q^{44} + 4 q^{49} + q^{50} + 9 q^{52} + 18 q^{53} + 2 q^{55} - q^{58} - 6 q^{59} + 2 q^{61} - 8 q^{62} + 4 q^{64} - 8 q^{65} - 14 q^{67} + 9 q^{68} - 3 q^{70} - 10 q^{71} - 16 q^{73} - 8 q^{74} - 16 q^{76} - 6 q^{77} + 10 q^{79} - 3 q^{80} - 10 q^{82} - 6 q^{83} + 2 q^{85} - 4 q^{86} + 20 q^{88} + 24 q^{91} + 15 q^{94} + 2 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 1.61803 0.511667
\(11\) −3.47214 −1.04689 −0.523444 0.852060i \(-0.675353\pi\)
−0.523444 + 0.852060i \(0.675353\pi\)
\(12\) 0 0
\(13\) −1.76393 −0.489227 −0.244613 0.969621i \(-0.578661\pi\)
−0.244613 + 0.969621i \(0.578661\pi\)
\(14\) −4.85410 −1.29731
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 5.47214 1.32719 0.663594 0.748093i \(-0.269030\pi\)
0.663594 + 0.748093i \(0.269030\pi\)
\(18\) 0 0
\(19\) −5.70820 −1.30955 −0.654776 0.755823i \(-0.727237\pi\)
−0.654776 + 0.755823i \(0.727237\pi\)
\(20\) 0.618034 0.138197
\(21\) 0 0
\(22\) −5.61803 −1.19777
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.85410 −0.559735
\(27\) 0 0
\(28\) −1.85410 −0.350392
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) 8.85410 1.51847
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −9.23607 −1.49829
\(39\) 0 0
\(40\) −2.23607 −0.353553
\(41\) −4.47214 −0.698430 −0.349215 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −1.23607 −0.188499 −0.0942493 0.995549i \(-0.530045\pi\)
−0.0942493 + 0.995549i \(0.530045\pi\)
\(44\) −2.14590 −0.323506
\(45\) 0 0
\(46\) 0 0
\(47\) 6.70820 0.978492 0.489246 0.872146i \(-0.337272\pi\)
0.489246 + 0.872146i \(0.337272\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.61803 0.228825
\(51\) 0 0
\(52\) −1.09017 −0.151179
\(53\) 11.2361 1.54339 0.771696 0.635991i \(-0.219409\pi\)
0.771696 + 0.635991i \(0.219409\pi\)
\(54\) 0 0
\(55\) −3.47214 −0.468183
\(56\) 6.70820 0.896421
\(57\) 0 0
\(58\) −1.61803 −0.212458
\(59\) −0.763932 −0.0994555 −0.0497277 0.998763i \(-0.515835\pi\)
−0.0497277 + 0.998763i \(0.515835\pi\)
\(60\) 0 0
\(61\) 7.70820 0.986934 0.493467 0.869764i \(-0.335729\pi\)
0.493467 + 0.869764i \(0.335729\pi\)
\(62\) −12.9443 −1.64392
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −1.76393 −0.218789
\(66\) 0 0
\(67\) −2.52786 −0.308828 −0.154414 0.988006i \(-0.549349\pi\)
−0.154414 + 0.988006i \(0.549349\pi\)
\(68\) 3.38197 0.410124
\(69\) 0 0
\(70\) −4.85410 −0.580176
\(71\) −2.76393 −0.328018 −0.164009 0.986459i \(-0.552443\pi\)
−0.164009 + 0.986459i \(0.552443\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −12.9443 −1.50474
\(75\) 0 0
\(76\) −3.52786 −0.404674
\(77\) 10.4164 1.18706
\(78\) 0 0
\(79\) 16.1803 1.82043 0.910215 0.414136i \(-0.135916\pi\)
0.910215 + 0.414136i \(0.135916\pi\)
\(80\) −4.85410 −0.542705
\(81\) 0 0
\(82\) −7.23607 −0.799090
\(83\) −9.70820 −1.06561 −0.532807 0.846237i \(-0.678863\pi\)
−0.532807 + 0.846237i \(0.678863\pi\)
\(84\) 0 0
\(85\) 5.47214 0.593536
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 7.76393 0.827638
\(89\) 11.1803 1.18511 0.592557 0.805529i \(-0.298119\pi\)
0.592557 + 0.805529i \(0.298119\pi\)
\(90\) 0 0
\(91\) 5.29180 0.554731
\(92\) 0 0
\(93\) 0 0
\(94\) 10.8541 1.11952
\(95\) −5.70820 −0.585649
\(96\) 0 0
\(97\) 7.23607 0.734711 0.367356 0.930081i \(-0.380263\pi\)
0.367356 + 0.930081i \(0.380263\pi\)
\(98\) 3.23607 0.326892
\(99\) 0 0
\(100\) 0.618034 0.0618034
\(101\) 0.236068 0.0234896 0.0117448 0.999931i \(-0.496261\pi\)
0.0117448 + 0.999931i \(0.496261\pi\)
\(102\) 0 0
\(103\) −19.4164 −1.91316 −0.956578 0.291477i \(-0.905853\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(104\) 3.94427 0.386768
\(105\) 0 0
\(106\) 18.1803 1.76583
\(107\) −16.4721 −1.59242 −0.796211 0.605019i \(-0.793165\pi\)
−0.796211 + 0.605019i \(0.793165\pi\)
\(108\) 0 0
\(109\) 10.4164 0.997711 0.498855 0.866685i \(-0.333754\pi\)
0.498855 + 0.866685i \(0.333754\pi\)
\(110\) −5.61803 −0.535659
\(111\) 0 0
\(112\) 14.5623 1.37601
\(113\) −9.94427 −0.935478 −0.467739 0.883867i \(-0.654931\pi\)
−0.467739 + 0.883867i \(0.654931\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.618034 −0.0573830
\(117\) 0 0
\(118\) −1.23607 −0.113789
\(119\) −16.4164 −1.50489
\(120\) 0 0
\(121\) 1.05573 0.0959753
\(122\) 12.4721 1.12917
\(123\) 0 0
\(124\) −4.94427 −0.444009
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) −2.85410 −0.250321
\(131\) 5.47214 0.478103 0.239051 0.971007i \(-0.423164\pi\)
0.239051 + 0.971007i \(0.423164\pi\)
\(132\) 0 0
\(133\) 17.1246 1.48489
\(134\) −4.09017 −0.353337
\(135\) 0 0
\(136\) −12.2361 −1.04923
\(137\) −6.94427 −0.593289 −0.296645 0.954988i \(-0.595868\pi\)
−0.296645 + 0.954988i \(0.595868\pi\)
\(138\) 0 0
\(139\) 12.7082 1.07790 0.538948 0.842339i \(-0.318822\pi\)
0.538948 + 0.842339i \(0.318822\pi\)
\(140\) −1.85410 −0.156700
\(141\) 0 0
\(142\) −4.47214 −0.375293
\(143\) 6.12461 0.512166
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) −12.9443 −1.07128
\(147\) 0 0
\(148\) −4.94427 −0.406417
\(149\) 2.18034 0.178620 0.0893102 0.996004i \(-0.471534\pi\)
0.0893102 + 0.996004i \(0.471534\pi\)
\(150\) 0 0
\(151\) −6.47214 −0.526695 −0.263347 0.964701i \(-0.584827\pi\)
−0.263347 + 0.964701i \(0.584827\pi\)
\(152\) 12.7639 1.03529
\(153\) 0 0
\(154\) 16.8541 1.35814
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 1.70820 0.136330 0.0681648 0.997674i \(-0.478286\pi\)
0.0681648 + 0.997674i \(0.478286\pi\)
\(158\) 26.1803 2.08280
\(159\) 0 0
\(160\) −3.38197 −0.267368
\(161\) 0 0
\(162\) 0 0
\(163\) 25.1246 1.96791 0.983956 0.178413i \(-0.0570962\pi\)
0.983956 + 0.178413i \(0.0570962\pi\)
\(164\) −2.76393 −0.215827
\(165\) 0 0
\(166\) −15.7082 −1.21919
\(167\) −17.8885 −1.38426 −0.692129 0.721774i \(-0.743327\pi\)
−0.692129 + 0.721774i \(0.743327\pi\)
\(168\) 0 0
\(169\) −9.88854 −0.760657
\(170\) 8.85410 0.679079
\(171\) 0 0
\(172\) −0.763932 −0.0582493
\(173\) −7.41641 −0.563859 −0.281930 0.959435i \(-0.590974\pi\)
−0.281930 + 0.959435i \(0.590974\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 16.8541 1.27043
\(177\) 0 0
\(178\) 18.0902 1.35592
\(179\) 18.1803 1.35886 0.679431 0.733739i \(-0.262227\pi\)
0.679431 + 0.733739i \(0.262227\pi\)
\(180\) 0 0
\(181\) −18.4164 −1.36888 −0.684440 0.729069i \(-0.739953\pi\)
−0.684440 + 0.729069i \(0.739953\pi\)
\(182\) 8.56231 0.634680
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −19.0000 −1.38942
\(188\) 4.14590 0.302371
\(189\) 0 0
\(190\) −9.23607 −0.670055
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 11.7082 0.840600
\(195\) 0 0
\(196\) 1.23607 0.0882906
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) 0 0
\(199\) 7.29180 0.516902 0.258451 0.966024i \(-0.416788\pi\)
0.258451 + 0.966024i \(0.416788\pi\)
\(200\) −2.23607 −0.158114
\(201\) 0 0
\(202\) 0.381966 0.0268750
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) −4.47214 −0.312348
\(206\) −31.4164 −2.18888
\(207\) 0 0
\(208\) 8.56231 0.593689
\(209\) 19.8197 1.37095
\(210\) 0 0
\(211\) 7.88854 0.543070 0.271535 0.962429i \(-0.412469\pi\)
0.271535 + 0.962429i \(0.412469\pi\)
\(212\) 6.94427 0.476935
\(213\) 0 0
\(214\) −26.6525 −1.82193
\(215\) −1.23607 −0.0842991
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 16.8541 1.14150
\(219\) 0 0
\(220\) −2.14590 −0.144676
\(221\) −9.65248 −0.649296
\(222\) 0 0
\(223\) −13.0000 −0.870544 −0.435272 0.900299i \(-0.643348\pi\)
−0.435272 + 0.900299i \(0.643348\pi\)
\(224\) 10.1459 0.677901
\(225\) 0 0
\(226\) −16.0902 −1.07030
\(227\) 28.1803 1.87039 0.935197 0.354127i \(-0.115222\pi\)
0.935197 + 0.354127i \(0.115222\pi\)
\(228\) 0 0
\(229\) −17.4164 −1.15091 −0.575454 0.817834i \(-0.695175\pi\)
−0.575454 + 0.817834i \(0.695175\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.23607 0.146805
\(233\) 25.4164 1.66508 0.832542 0.553962i \(-0.186885\pi\)
0.832542 + 0.553962i \(0.186885\pi\)
\(234\) 0 0
\(235\) 6.70820 0.437595
\(236\) −0.472136 −0.0307334
\(237\) 0 0
\(238\) −26.5623 −1.72178
\(239\) 23.8885 1.54522 0.772611 0.634880i \(-0.218950\pi\)
0.772611 + 0.634880i \(0.218950\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 1.70820 0.109808
\(243\) 0 0
\(244\) 4.76393 0.304979
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 10.0689 0.640668
\(248\) 17.8885 1.13592
\(249\) 0 0
\(250\) 1.61803 0.102333
\(251\) −10.8885 −0.687279 −0.343639 0.939102i \(-0.611660\pi\)
−0.343639 + 0.939102i \(0.611660\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −9.70820 −0.609147
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −3.70820 −0.231311 −0.115656 0.993289i \(-0.536897\pi\)
−0.115656 + 0.993289i \(0.536897\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) −1.09017 −0.0676095
\(261\) 0 0
\(262\) 8.85410 0.547008
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 11.2361 0.690226
\(266\) 27.7082 1.69890
\(267\) 0 0
\(268\) −1.56231 −0.0954330
\(269\) 25.7639 1.57085 0.785427 0.618954i \(-0.212443\pi\)
0.785427 + 0.618954i \(0.212443\pi\)
\(270\) 0 0
\(271\) −30.3607 −1.84428 −0.922140 0.386856i \(-0.873561\pi\)
−0.922140 + 0.386856i \(0.873561\pi\)
\(272\) −26.5623 −1.61058
\(273\) 0 0
\(274\) −11.2361 −0.678796
\(275\) −3.47214 −0.209378
\(276\) 0 0
\(277\) −4.70820 −0.282889 −0.141444 0.989946i \(-0.545175\pi\)
−0.141444 + 0.989946i \(0.545175\pi\)
\(278\) 20.5623 1.23325
\(279\) 0 0
\(280\) 6.70820 0.400892
\(281\) −17.1246 −1.02157 −0.510784 0.859709i \(-0.670645\pi\)
−0.510784 + 0.859709i \(0.670645\pi\)
\(282\) 0 0
\(283\) 9.52786 0.566373 0.283186 0.959065i \(-0.408609\pi\)
0.283186 + 0.959065i \(0.408609\pi\)
\(284\) −1.70820 −0.101363
\(285\) 0 0
\(286\) 9.90983 0.585981
\(287\) 13.4164 0.791946
\(288\) 0 0
\(289\) 12.9443 0.761428
\(290\) −1.61803 −0.0950142
\(291\) 0 0
\(292\) −4.94427 −0.289342
\(293\) 0.0557281 0.00325567 0.00162783 0.999999i \(-0.499482\pi\)
0.00162783 + 0.999999i \(0.499482\pi\)
\(294\) 0 0
\(295\) −0.763932 −0.0444778
\(296\) 17.8885 1.03975
\(297\) 0 0
\(298\) 3.52786 0.204364
\(299\) 0 0
\(300\) 0 0
\(301\) 3.70820 0.213737
\(302\) −10.4721 −0.602604
\(303\) 0 0
\(304\) 27.7082 1.58917
\(305\) 7.70820 0.441370
\(306\) 0 0
\(307\) 7.05573 0.402692 0.201346 0.979520i \(-0.435468\pi\)
0.201346 + 0.979520i \(0.435468\pi\)
\(308\) 6.43769 0.366822
\(309\) 0 0
\(310\) −12.9443 −0.735185
\(311\) −14.5279 −0.823800 −0.411900 0.911229i \(-0.635135\pi\)
−0.411900 + 0.911229i \(0.635135\pi\)
\(312\) 0 0
\(313\) −21.1803 −1.19718 −0.598592 0.801054i \(-0.704273\pi\)
−0.598592 + 0.801054i \(0.704273\pi\)
\(314\) 2.76393 0.155978
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 1.58359 0.0889434 0.0444717 0.999011i \(-0.485840\pi\)
0.0444717 + 0.999011i \(0.485840\pi\)
\(318\) 0 0
\(319\) 3.47214 0.194402
\(320\) 4.23607 0.236803
\(321\) 0 0
\(322\) 0 0
\(323\) −31.2361 −1.73802
\(324\) 0 0
\(325\) −1.76393 −0.0978453
\(326\) 40.6525 2.25153
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −20.1246 −1.10951
\(330\) 0 0
\(331\) 19.8885 1.09317 0.546587 0.837403i \(-0.315927\pi\)
0.546587 + 0.837403i \(0.315927\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) −28.9443 −1.58376
\(335\) −2.52786 −0.138112
\(336\) 0 0
\(337\) −11.5279 −0.627963 −0.313981 0.949429i \(-0.601663\pi\)
−0.313981 + 0.949429i \(0.601663\pi\)
\(338\) −16.0000 −0.870285
\(339\) 0 0
\(340\) 3.38197 0.183413
\(341\) 27.7771 1.50421
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 2.76393 0.149021
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −26.4721 −1.42110 −0.710549 0.703647i \(-0.751554\pi\)
−0.710549 + 0.703647i \(0.751554\pi\)
\(348\) 0 0
\(349\) −1.05573 −0.0565118 −0.0282559 0.999601i \(-0.508995\pi\)
−0.0282559 + 0.999601i \(0.508995\pi\)
\(350\) −4.85410 −0.259463
\(351\) 0 0
\(352\) 11.7426 0.625885
\(353\) −12.4721 −0.663825 −0.331912 0.943310i \(-0.607694\pi\)
−0.331912 + 0.943310i \(0.607694\pi\)
\(354\) 0 0
\(355\) −2.76393 −0.146694
\(356\) 6.90983 0.366220
\(357\) 0 0
\(358\) 29.4164 1.55471
\(359\) 31.4164 1.65809 0.829047 0.559178i \(-0.188883\pi\)
0.829047 + 0.559178i \(0.188883\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) −29.7984 −1.56617
\(363\) 0 0
\(364\) 3.27051 0.171421
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) −11.4164 −0.595932 −0.297966 0.954577i \(-0.596308\pi\)
−0.297966 + 0.954577i \(0.596308\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −12.9443 −0.672941
\(371\) −33.7082 −1.75004
\(372\) 0 0
\(373\) −19.8885 −1.02979 −0.514895 0.857253i \(-0.672169\pi\)
−0.514895 + 0.857253i \(0.672169\pi\)
\(374\) −30.7426 −1.58966
\(375\) 0 0
\(376\) −15.0000 −0.773566
\(377\) 1.76393 0.0908471
\(378\) 0 0
\(379\) −10.9443 −0.562169 −0.281085 0.959683i \(-0.590694\pi\)
−0.281085 + 0.959683i \(0.590694\pi\)
\(380\) −3.52786 −0.180976
\(381\) 0 0
\(382\) −19.4164 −0.993430
\(383\) −38.0689 −1.94523 −0.972615 0.232424i \(-0.925334\pi\)
−0.972615 + 0.232424i \(0.925334\pi\)
\(384\) 0 0
\(385\) 10.4164 0.530869
\(386\) −9.70820 −0.494135
\(387\) 0 0
\(388\) 4.47214 0.227038
\(389\) −17.7639 −0.900667 −0.450334 0.892860i \(-0.648695\pi\)
−0.450334 + 0.892860i \(0.648695\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −4.47214 −0.225877
\(393\) 0 0
\(394\) 4.76393 0.240003
\(395\) 16.1803 0.814121
\(396\) 0 0
\(397\) −33.4164 −1.67712 −0.838561 0.544808i \(-0.816602\pi\)
−0.838561 + 0.544808i \(0.816602\pi\)
\(398\) 11.7984 0.591399
\(399\) 0 0
\(400\) −4.85410 −0.242705
\(401\) 5.34752 0.267043 0.133521 0.991046i \(-0.457372\pi\)
0.133521 + 0.991046i \(0.457372\pi\)
\(402\) 0 0
\(403\) 14.1115 0.702942
\(404\) 0.145898 0.00725870
\(405\) 0 0
\(406\) 4.85410 0.240905
\(407\) 27.7771 1.37686
\(408\) 0 0
\(409\) −22.6525 −1.12009 −0.560046 0.828461i \(-0.689217\pi\)
−0.560046 + 0.828461i \(0.689217\pi\)
\(410\) −7.23607 −0.357364
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) 2.29180 0.112772
\(414\) 0 0
\(415\) −9.70820 −0.476557
\(416\) 5.96556 0.292486
\(417\) 0 0
\(418\) 32.0689 1.56854
\(419\) 10.3607 0.506152 0.253076 0.967446i \(-0.418558\pi\)
0.253076 + 0.967446i \(0.418558\pi\)
\(420\) 0 0
\(421\) −24.1803 −1.17848 −0.589239 0.807959i \(-0.700572\pi\)
−0.589239 + 0.807959i \(0.700572\pi\)
\(422\) 12.7639 0.621338
\(423\) 0 0
\(424\) −25.1246 −1.22016
\(425\) 5.47214 0.265438
\(426\) 0 0
\(427\) −23.1246 −1.11908
\(428\) −10.1803 −0.492085
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) −25.5279 −1.22963 −0.614817 0.788670i \(-0.710770\pi\)
−0.614817 + 0.788670i \(0.710770\pi\)
\(432\) 0 0
\(433\) 26.6525 1.28084 0.640418 0.768026i \(-0.278761\pi\)
0.640418 + 0.768026i \(0.278761\pi\)
\(434\) 38.8328 1.86403
\(435\) 0 0
\(436\) 6.43769 0.308310
\(437\) 0 0
\(438\) 0 0
\(439\) 30.1246 1.43777 0.718885 0.695129i \(-0.244653\pi\)
0.718885 + 0.695129i \(0.244653\pi\)
\(440\) 7.76393 0.370131
\(441\) 0 0
\(442\) −15.6180 −0.742874
\(443\) −28.2361 −1.34154 −0.670768 0.741667i \(-0.734035\pi\)
−0.670768 + 0.741667i \(0.734035\pi\)
\(444\) 0 0
\(445\) 11.1803 0.529999
\(446\) −21.0344 −0.996010
\(447\) 0 0
\(448\) −12.7082 −0.600406
\(449\) −6.23607 −0.294298 −0.147149 0.989114i \(-0.547010\pi\)
−0.147149 + 0.989114i \(0.547010\pi\)
\(450\) 0 0
\(451\) 15.5279 0.731179
\(452\) −6.14590 −0.289079
\(453\) 0 0
\(454\) 45.5967 2.13996
\(455\) 5.29180 0.248083
\(456\) 0 0
\(457\) −34.1246 −1.59628 −0.798141 0.602471i \(-0.794183\pi\)
−0.798141 + 0.602471i \(0.794183\pi\)
\(458\) −28.1803 −1.31678
\(459\) 0 0
\(460\) 0 0
\(461\) 42.3607 1.97293 0.986467 0.163961i \(-0.0524272\pi\)
0.986467 + 0.163961i \(0.0524272\pi\)
\(462\) 0 0
\(463\) −11.4721 −0.533155 −0.266578 0.963813i \(-0.585893\pi\)
−0.266578 + 0.963813i \(0.585893\pi\)
\(464\) 4.85410 0.225346
\(465\) 0 0
\(466\) 41.1246 1.90506
\(467\) −4.94427 −0.228794 −0.114397 0.993435i \(-0.536494\pi\)
−0.114397 + 0.993435i \(0.536494\pi\)
\(468\) 0 0
\(469\) 7.58359 0.350178
\(470\) 10.8541 0.500662
\(471\) 0 0
\(472\) 1.70820 0.0786265
\(473\) 4.29180 0.197337
\(474\) 0 0
\(475\) −5.70820 −0.261910
\(476\) −10.1459 −0.465036
\(477\) 0 0
\(478\) 38.6525 1.76792
\(479\) −2.47214 −0.112955 −0.0564774 0.998404i \(-0.517987\pi\)
−0.0564774 + 0.998404i \(0.517987\pi\)
\(480\) 0 0
\(481\) 14.1115 0.643427
\(482\) 11.3262 0.515896
\(483\) 0 0
\(484\) 0.652476 0.0296580
\(485\) 7.23607 0.328573
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −17.2361 −0.780240
\(489\) 0 0
\(490\) 3.23607 0.146191
\(491\) 9.88854 0.446264 0.223132 0.974788i \(-0.428372\pi\)
0.223132 + 0.974788i \(0.428372\pi\)
\(492\) 0 0
\(493\) −5.47214 −0.246453
\(494\) 16.2918 0.733003
\(495\) 0 0
\(496\) 38.8328 1.74364
\(497\) 8.29180 0.371938
\(498\) 0 0
\(499\) −28.2361 −1.26402 −0.632010 0.774960i \(-0.717770\pi\)
−0.632010 + 0.774960i \(0.717770\pi\)
\(500\) 0.618034 0.0276393
\(501\) 0 0
\(502\) −17.6180 −0.786331
\(503\) −18.5967 −0.829188 −0.414594 0.910006i \(-0.636076\pi\)
−0.414594 + 0.910006i \(0.636076\pi\)
\(504\) 0 0
\(505\) 0.236068 0.0105049
\(506\) 0 0
\(507\) 0 0
\(508\) −3.70820 −0.164525
\(509\) 16.1803 0.717181 0.358590 0.933495i \(-0.383257\pi\)
0.358590 + 0.933495i \(0.383257\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −19.4164 −0.855589
\(516\) 0 0
\(517\) −23.2918 −1.02437
\(518\) 38.8328 1.70622
\(519\) 0 0
\(520\) 3.94427 0.172968
\(521\) 25.2361 1.10561 0.552806 0.833310i \(-0.313557\pi\)
0.552806 + 0.833310i \(0.313557\pi\)
\(522\) 0 0
\(523\) 23.8328 1.04214 0.521068 0.853515i \(-0.325534\pi\)
0.521068 + 0.853515i \(0.325534\pi\)
\(524\) 3.38197 0.147742
\(525\) 0 0
\(526\) −38.8328 −1.69319
\(527\) −43.7771 −1.90696
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 18.1803 0.789704
\(531\) 0 0
\(532\) 10.5836 0.458857
\(533\) 7.88854 0.341691
\(534\) 0 0
\(535\) −16.4721 −0.712153
\(536\) 5.65248 0.244150
\(537\) 0 0
\(538\) 41.6869 1.79725
\(539\) −6.94427 −0.299111
\(540\) 0 0
\(541\) −6.36068 −0.273467 −0.136733 0.990608i \(-0.543660\pi\)
−0.136733 + 0.990608i \(0.543660\pi\)
\(542\) −49.1246 −2.11008
\(543\) 0 0
\(544\) −18.5066 −0.793463
\(545\) 10.4164 0.446190
\(546\) 0 0
\(547\) 18.5279 0.792194 0.396097 0.918209i \(-0.370364\pi\)
0.396097 + 0.918209i \(0.370364\pi\)
\(548\) −4.29180 −0.183336
\(549\) 0 0
\(550\) −5.61803 −0.239554
\(551\) 5.70820 0.243178
\(552\) 0 0
\(553\) −48.5410 −2.06417
\(554\) −7.61803 −0.323659
\(555\) 0 0
\(556\) 7.85410 0.333088
\(557\) −7.23607 −0.306602 −0.153301 0.988180i \(-0.548990\pi\)
−0.153301 + 0.988180i \(0.548990\pi\)
\(558\) 0 0
\(559\) 2.18034 0.0922186
\(560\) 14.5623 0.615370
\(561\) 0 0
\(562\) −27.7082 −1.16880
\(563\) −3.87539 −0.163328 −0.0816641 0.996660i \(-0.526023\pi\)
−0.0816641 + 0.996660i \(0.526023\pi\)
\(564\) 0 0
\(565\) −9.94427 −0.418359
\(566\) 15.4164 0.648000
\(567\) 0 0
\(568\) 6.18034 0.259321
\(569\) −17.1803 −0.720237 −0.360119 0.932907i \(-0.617264\pi\)
−0.360119 + 0.932907i \(0.617264\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 3.78522 0.158268
\(573\) 0 0
\(574\) 21.7082 0.906083
\(575\) 0 0
\(576\) 0 0
\(577\) 33.3050 1.38650 0.693252 0.720696i \(-0.256177\pi\)
0.693252 + 0.720696i \(0.256177\pi\)
\(578\) 20.9443 0.871167
\(579\) 0 0
\(580\) −0.618034 −0.0256625
\(581\) 29.1246 1.20829
\(582\) 0 0
\(583\) −39.0132 −1.61576
\(584\) 17.8885 0.740233
\(585\) 0 0
\(586\) 0.0901699 0.00372489
\(587\) 24.1803 0.998029 0.499015 0.866594i \(-0.333695\pi\)
0.499015 + 0.866594i \(0.333695\pi\)
\(588\) 0 0
\(589\) 45.6656 1.88162
\(590\) −1.23607 −0.0508881
\(591\) 0 0
\(592\) 38.8328 1.59602
\(593\) −6.65248 −0.273184 −0.136592 0.990627i \(-0.543615\pi\)
−0.136592 + 0.990627i \(0.543615\pi\)
\(594\) 0 0
\(595\) −16.4164 −0.673007
\(596\) 1.34752 0.0551967
\(597\) 0 0
\(598\) 0 0
\(599\) −26.8885 −1.09864 −0.549318 0.835613i \(-0.685112\pi\)
−0.549318 + 0.835613i \(0.685112\pi\)
\(600\) 0 0
\(601\) 43.8885 1.79025 0.895126 0.445814i \(-0.147086\pi\)
0.895126 + 0.445814i \(0.147086\pi\)
\(602\) 6.00000 0.244542
\(603\) 0 0
\(604\) −4.00000 −0.162758
\(605\) 1.05573 0.0429215
\(606\) 0 0
\(607\) 9.34752 0.379404 0.189702 0.981842i \(-0.439248\pi\)
0.189702 + 0.981842i \(0.439248\pi\)
\(608\) 19.3050 0.782919
\(609\) 0 0
\(610\) 12.4721 0.504982
\(611\) −11.8328 −0.478704
\(612\) 0 0
\(613\) 20.7082 0.836396 0.418198 0.908356i \(-0.362662\pi\)
0.418198 + 0.908356i \(0.362662\pi\)
\(614\) 11.4164 0.460729
\(615\) 0 0
\(616\) −23.2918 −0.938453
\(617\) 19.3050 0.777188 0.388594 0.921409i \(-0.372961\pi\)
0.388594 + 0.921409i \(0.372961\pi\)
\(618\) 0 0
\(619\) −12.4721 −0.501297 −0.250649 0.968078i \(-0.580644\pi\)
−0.250649 + 0.968078i \(0.580644\pi\)
\(620\) −4.94427 −0.198567
\(621\) 0 0
\(622\) −23.5066 −0.942528
\(623\) −33.5410 −1.34379
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.2705 −1.36973
\(627\) 0 0
\(628\) 1.05573 0.0421281
\(629\) −43.7771 −1.74551
\(630\) 0 0
\(631\) 46.4853 1.85055 0.925275 0.379297i \(-0.123834\pi\)
0.925275 + 0.379297i \(0.123834\pi\)
\(632\) −36.1803 −1.43918
\(633\) 0 0
\(634\) 2.56231 0.101762
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −3.52786 −0.139779
\(638\) 5.61803 0.222420
\(639\) 0 0
\(640\) 13.6180 0.538300
\(641\) −41.7639 −1.64958 −0.824788 0.565442i \(-0.808706\pi\)
−0.824788 + 0.565442i \(0.808706\pi\)
\(642\) 0 0
\(643\) 17.8328 0.703258 0.351629 0.936140i \(-0.385628\pi\)
0.351629 + 0.936140i \(0.385628\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −50.5410 −1.98851
\(647\) −6.76393 −0.265918 −0.132959 0.991122i \(-0.542448\pi\)
−0.132959 + 0.991122i \(0.542448\pi\)
\(648\) 0 0
\(649\) 2.65248 0.104119
\(650\) −2.85410 −0.111947
\(651\) 0 0
\(652\) 15.5279 0.608118
\(653\) 26.8885 1.05223 0.526115 0.850413i \(-0.323648\pi\)
0.526115 + 0.850413i \(0.323648\pi\)
\(654\) 0 0
\(655\) 5.47214 0.213814
\(656\) 21.7082 0.847563
\(657\) 0 0
\(658\) −32.5623 −1.26941
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) 15.8328 0.615825 0.307913 0.951415i \(-0.400370\pi\)
0.307913 + 0.951415i \(0.400370\pi\)
\(662\) 32.1803 1.25072
\(663\) 0 0
\(664\) 21.7082 0.842442
\(665\) 17.1246 0.664064
\(666\) 0 0
\(667\) 0 0
\(668\) −11.0557 −0.427759
\(669\) 0 0
\(670\) −4.09017 −0.158017
\(671\) −26.7639 −1.03321
\(672\) 0 0
\(673\) −46.7082 −1.80047 −0.900234 0.435405i \(-0.856605\pi\)
−0.900234 + 0.435405i \(0.856605\pi\)
\(674\) −18.6525 −0.718467
\(675\) 0 0
\(676\) −6.11146 −0.235056
\(677\) −14.8885 −0.572213 −0.286107 0.958198i \(-0.592361\pi\)
−0.286107 + 0.958198i \(0.592361\pi\)
\(678\) 0 0
\(679\) −21.7082 −0.833084
\(680\) −12.2361 −0.469232
\(681\) 0 0
\(682\) 44.9443 1.72101
\(683\) 19.4164 0.742948 0.371474 0.928443i \(-0.378852\pi\)
0.371474 + 0.928443i \(0.378852\pi\)
\(684\) 0 0
\(685\) −6.94427 −0.265327
\(686\) 24.2705 0.926652
\(687\) 0 0
\(688\) 6.00000 0.228748
\(689\) −19.8197 −0.755069
\(690\) 0 0
\(691\) −21.2918 −0.809978 −0.404989 0.914322i \(-0.632725\pi\)
−0.404989 + 0.914322i \(0.632725\pi\)
\(692\) −4.58359 −0.174242
\(693\) 0 0
\(694\) −42.8328 −1.62591
\(695\) 12.7082 0.482050
\(696\) 0 0
\(697\) −24.4721 −0.926948
\(698\) −1.70820 −0.0646565
\(699\) 0 0
\(700\) −1.85410 −0.0700785
\(701\) −35.1246 −1.32664 −0.663319 0.748337i \(-0.730853\pi\)
−0.663319 + 0.748337i \(0.730853\pi\)
\(702\) 0 0
\(703\) 45.6656 1.72231
\(704\) −14.7082 −0.554336
\(705\) 0 0
\(706\) −20.1803 −0.759497
\(707\) −0.708204 −0.0266348
\(708\) 0 0
\(709\) −15.8885 −0.596707 −0.298353 0.954455i \(-0.596437\pi\)
−0.298353 + 0.954455i \(0.596437\pi\)
\(710\) −4.47214 −0.167836
\(711\) 0 0
\(712\) −25.0000 −0.936915
\(713\) 0 0
\(714\) 0 0
\(715\) 6.12461 0.229047
\(716\) 11.2361 0.419912
\(717\) 0 0
\(718\) 50.8328 1.89706
\(719\) 6.76393 0.252252 0.126126 0.992014i \(-0.459746\pi\)
0.126126 + 0.992014i \(0.459746\pi\)
\(720\) 0 0
\(721\) 58.2492 2.16931
\(722\) 21.9787 0.817963
\(723\) 0 0
\(724\) −11.3820 −0.423007
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 37.8885 1.40521 0.702604 0.711581i \(-0.252020\pi\)
0.702604 + 0.711581i \(0.252020\pi\)
\(728\) −11.8328 −0.438553
\(729\) 0 0
\(730\) −12.9443 −0.479089
\(731\) −6.76393 −0.250173
\(732\) 0 0
\(733\) −27.7082 −1.02343 −0.511713 0.859156i \(-0.670989\pi\)
−0.511713 + 0.859156i \(0.670989\pi\)
\(734\) −18.4721 −0.681819
\(735\) 0 0
\(736\) 0 0
\(737\) 8.77709 0.323308
\(738\) 0 0
\(739\) −43.4853 −1.59963 −0.799816 0.600245i \(-0.795070\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(740\) −4.94427 −0.181755
\(741\) 0 0
\(742\) −54.5410 −2.00226
\(743\) 33.1803 1.21727 0.608634 0.793451i \(-0.291718\pi\)
0.608634 + 0.793451i \(0.291718\pi\)
\(744\) 0 0
\(745\) 2.18034 0.0798815
\(746\) −32.1803 −1.17821
\(747\) 0 0
\(748\) −11.7426 −0.429354
\(749\) 49.4164 1.80564
\(750\) 0 0
\(751\) 16.5836 0.605144 0.302572 0.953127i \(-0.402155\pi\)
0.302572 + 0.953127i \(0.402155\pi\)
\(752\) −32.5623 −1.18743
\(753\) 0 0
\(754\) 2.85410 0.103940
\(755\) −6.47214 −0.235545
\(756\) 0 0
\(757\) 18.8328 0.684490 0.342245 0.939611i \(-0.388813\pi\)
0.342245 + 0.939611i \(0.388813\pi\)
\(758\) −17.7082 −0.643191
\(759\) 0 0
\(760\) 12.7639 0.462996
\(761\) 43.0132 1.55923 0.779613 0.626262i \(-0.215416\pi\)
0.779613 + 0.626262i \(0.215416\pi\)
\(762\) 0 0
\(763\) −31.2492 −1.13130
\(764\) −7.41641 −0.268316
\(765\) 0 0
\(766\) −61.5967 −2.22558
\(767\) 1.34752 0.0486563
\(768\) 0 0
\(769\) 3.34752 0.120715 0.0603574 0.998177i \(-0.480776\pi\)
0.0603574 + 0.998177i \(0.480776\pi\)
\(770\) 16.8541 0.607380
\(771\) 0 0
\(772\) −3.70820 −0.133461
\(773\) −28.2492 −1.01605 −0.508027 0.861341i \(-0.669625\pi\)
−0.508027 + 0.861341i \(0.669625\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −16.1803 −0.580840
\(777\) 0 0
\(778\) −28.7426 −1.03047
\(779\) 25.5279 0.914631
\(780\) 0 0
\(781\) 9.59675 0.343399
\(782\) 0 0
\(783\) 0 0
\(784\) −9.70820 −0.346722
\(785\) 1.70820 0.0609684
\(786\) 0 0
\(787\) −33.8885 −1.20800 −0.603998 0.796986i \(-0.706427\pi\)
−0.603998 + 0.796986i \(0.706427\pi\)
\(788\) 1.81966 0.0648227
\(789\) 0 0
\(790\) 26.1803 0.931455
\(791\) 29.8328 1.06073
\(792\) 0 0
\(793\) −13.5967 −0.482835
\(794\) −54.0689 −1.91883
\(795\) 0 0
\(796\) 4.50658 0.159731
\(797\) 10.9443 0.387666 0.193833 0.981035i \(-0.437908\pi\)
0.193833 + 0.981035i \(0.437908\pi\)
\(798\) 0 0
\(799\) 36.7082 1.29864
\(800\) −3.38197 −0.119571
\(801\) 0 0
\(802\) 8.65248 0.305530
\(803\) 27.7771 0.980232
\(804\) 0 0
\(805\) 0 0
\(806\) 22.8328 0.804252
\(807\) 0 0
\(808\) −0.527864 −0.0185702
\(809\) −39.7639 −1.39803 −0.699013 0.715109i \(-0.746377\pi\)
−0.699013 + 0.715109i \(0.746377\pi\)
\(810\) 0 0
\(811\) −9.29180 −0.326279 −0.163140 0.986603i \(-0.552162\pi\)
−0.163140 + 0.986603i \(0.552162\pi\)
\(812\) 1.85410 0.0650662
\(813\) 0 0
\(814\) 44.9443 1.57530
\(815\) 25.1246 0.880077
\(816\) 0 0
\(817\) 7.05573 0.246849
\(818\) −36.6525 −1.28152
\(819\) 0 0
\(820\) −2.76393 −0.0965207
\(821\) 37.4164 1.30584 0.652921 0.757426i \(-0.273543\pi\)
0.652921 + 0.757426i \(0.273543\pi\)
\(822\) 0 0
\(823\) −33.2361 −1.15854 −0.579268 0.815137i \(-0.696662\pi\)
−0.579268 + 0.815137i \(0.696662\pi\)
\(824\) 43.4164 1.51248
\(825\) 0 0
\(826\) 3.70820 0.129025
\(827\) −19.4164 −0.675175 −0.337587 0.941294i \(-0.609611\pi\)
−0.337587 + 0.941294i \(0.609611\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) −15.7082 −0.545240
\(831\) 0 0
\(832\) −7.47214 −0.259050
\(833\) 10.9443 0.379197
\(834\) 0 0
\(835\) −17.8885 −0.619059
\(836\) 12.2492 0.423648
\(837\) 0 0
\(838\) 16.7639 0.579100
\(839\) 20.8885 0.721153 0.360576 0.932730i \(-0.382580\pi\)
0.360576 + 0.932730i \(0.382580\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −39.1246 −1.34832
\(843\) 0 0
\(844\) 4.87539 0.167818
\(845\) −9.88854 −0.340176
\(846\) 0 0
\(847\) −3.16718 −0.108826
\(848\) −54.5410 −1.87295
\(849\) 0 0
\(850\) 8.85410 0.303693
\(851\) 0 0
\(852\) 0 0
\(853\) −33.2361 −1.13798 −0.568991 0.822344i \(-0.692666\pi\)
−0.568991 + 0.822344i \(0.692666\pi\)
\(854\) −37.4164 −1.28036
\(855\) 0 0
\(856\) 36.8328 1.25892
\(857\) −11.8885 −0.406105 −0.203052 0.979168i \(-0.565086\pi\)
−0.203052 + 0.979168i \(0.565086\pi\)
\(858\) 0 0
\(859\) −2.29180 −0.0781951 −0.0390975 0.999235i \(-0.512448\pi\)
−0.0390975 + 0.999235i \(0.512448\pi\)
\(860\) −0.763932 −0.0260499
\(861\) 0 0
\(862\) −41.3050 −1.40685
\(863\) 33.5967 1.14365 0.571823 0.820377i \(-0.306236\pi\)
0.571823 + 0.820377i \(0.306236\pi\)
\(864\) 0 0
\(865\) −7.41641 −0.252165
\(866\) 43.1246 1.46543
\(867\) 0 0
\(868\) 14.8328 0.503459
\(869\) −56.1803 −1.90579
\(870\) 0 0
\(871\) 4.45898 0.151087
\(872\) −23.2918 −0.788760
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −4.83282 −0.163193 −0.0815963 0.996665i \(-0.526002\pi\)
−0.0815963 + 0.996665i \(0.526002\pi\)
\(878\) 48.7426 1.64498
\(879\) 0 0
\(880\) 16.8541 0.568152
\(881\) 0.708204 0.0238600 0.0119300 0.999929i \(-0.496202\pi\)
0.0119300 + 0.999929i \(0.496202\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) −5.96556 −0.200643
\(885\) 0 0
\(886\) −45.6869 −1.53488
\(887\) −4.59675 −0.154344 −0.0771718 0.997018i \(-0.524589\pi\)
−0.0771718 + 0.997018i \(0.524589\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 18.0902 0.606384
\(891\) 0 0
\(892\) −8.03444 −0.269013
\(893\) −38.2918 −1.28139
\(894\) 0 0
\(895\) 18.1803 0.607702
\(896\) −40.8541 −1.36484
\(897\) 0 0
\(898\) −10.0902 −0.336713
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 61.4853 2.04837
\(902\) 25.1246 0.836558
\(903\) 0 0
\(904\) 22.2361 0.739561
\(905\) −18.4164 −0.612182
\(906\) 0 0
\(907\) 31.2361 1.03718 0.518588 0.855024i \(-0.326458\pi\)
0.518588 + 0.855024i \(0.326458\pi\)
\(908\) 17.4164 0.577984
\(909\) 0 0
\(910\) 8.56231 0.283838
\(911\) 35.9443 1.19089 0.595443 0.803397i \(-0.296976\pi\)
0.595443 + 0.803397i \(0.296976\pi\)
\(912\) 0 0
\(913\) 33.7082 1.11558
\(914\) −55.2148 −1.82634
\(915\) 0 0
\(916\) −10.7639 −0.355650
\(917\) −16.4164 −0.542118
\(918\) 0 0
\(919\) −20.1246 −0.663850 −0.331925 0.943306i \(-0.607698\pi\)
−0.331925 + 0.943306i \(0.607698\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 68.5410 2.25728
\(923\) 4.87539 0.160475
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −18.5623 −0.609995
\(927\) 0 0
\(928\) 3.38197 0.111018
\(929\) 3.16718 0.103912 0.0519560 0.998649i \(-0.483454\pi\)
0.0519560 + 0.998649i \(0.483454\pi\)
\(930\) 0 0
\(931\) −11.4164 −0.374158
\(932\) 15.7082 0.514539
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) −19.0000 −0.621366
\(936\) 0 0
\(937\) 24.2361 0.791758 0.395879 0.918303i \(-0.370440\pi\)
0.395879 + 0.918303i \(0.370440\pi\)
\(938\) 12.2705 0.400646
\(939\) 0 0
\(940\) 4.14590 0.135224
\(941\) −12.0689 −0.393434 −0.196717 0.980460i \(-0.563028\pi\)
−0.196717 + 0.980460i \(0.563028\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 3.70820 0.120692
\(945\) 0 0
\(946\) 6.94427 0.225778
\(947\) −0.708204 −0.0230135 −0.0115068 0.999934i \(-0.503663\pi\)
−0.0115068 + 0.999934i \(0.503663\pi\)
\(948\) 0 0
\(949\) 14.1115 0.458077
\(950\) −9.23607 −0.299658
\(951\) 0 0
\(952\) 36.7082 1.18972
\(953\) 57.0132 1.84684 0.923419 0.383794i \(-0.125383\pi\)
0.923419 + 0.383794i \(0.125383\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 14.7639 0.477500
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) 20.8328 0.672727
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 22.8328 0.736160
\(963\) 0 0
\(964\) 4.32624 0.139339
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 5.12461 0.164796 0.0823982 0.996599i \(-0.473742\pi\)
0.0823982 + 0.996599i \(0.473742\pi\)
\(968\) −2.36068 −0.0758751
\(969\) 0 0
\(970\) 11.7082 0.375928
\(971\) 40.9443 1.31396 0.656982 0.753906i \(-0.271833\pi\)
0.656982 + 0.753906i \(0.271833\pi\)
\(972\) 0 0
\(973\) −38.1246 −1.22222
\(974\) −19.4164 −0.622142
\(975\) 0 0
\(976\) −37.4164 −1.19767
\(977\) 13.5279 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(978\) 0 0
\(979\) −38.8197 −1.24068
\(980\) 1.23607 0.0394847
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) −46.4721 −1.48223 −0.741115 0.671378i \(-0.765703\pi\)
−0.741115 + 0.671378i \(0.765703\pi\)
\(984\) 0 0
\(985\) 2.94427 0.0938123
\(986\) −8.85410 −0.281972
\(987\) 0 0
\(988\) 6.22291 0.197977
\(989\) 0 0
\(990\) 0 0
\(991\) −62.4853 −1.98491 −0.992455 0.122607i \(-0.960875\pi\)
−0.992455 + 0.122607i \(0.960875\pi\)
\(992\) 27.0557 0.859020
\(993\) 0 0
\(994\) 13.4164 0.425543
\(995\) 7.29180 0.231165
\(996\) 0 0
\(997\) −19.4164 −0.614924 −0.307462 0.951560i \(-0.599480\pi\)
−0.307462 + 0.951560i \(0.599480\pi\)
\(998\) −45.6869 −1.44619
\(999\) 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 1305.2.a.k.1.2 2
3.2 odd 2 435.2.a.e.1.1 2
5.4 even 2 6525.2.a.s.1.1 2
12.11 even 2 6960.2.a.bu.1.1 2
15.2 even 4 2175.2.c.j.349.1 4
15.8 even 4 2175.2.c.j.349.4 4
15.14 odd 2 2175.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.e.1.1 2 3.2 odd 2
1305.2.a.k.1.2 2 1.1 even 1 trivial
2175.2.a.q.1.2 2 15.14 odd 2
2175.2.c.j.349.1 4 15.2 even 4
2175.2.c.j.349.4 4 15.8 even 4
6525.2.a.s.1.1 2 5.4 even 2
6960.2.a.bu.1.1 2 12.11 even 2