Properties

Label 6525.2.a.bs.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $5$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) 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 \(1.71250\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} +O(q^{10})\) \(q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} -4.26814 q^{11} +0.779856 q^{13} +1.98063 q^{14} +1.21181 q^{16} +1.90354 q^{17} +6.72036 q^{19} +3.04103 q^{22} -2.17168 q^{23} -0.555643 q^{26} +4.14852 q^{28} -1.00000 q^{29} -8.82061 q^{31} -5.83998 q^{32} -1.35626 q^{34} +1.48402 q^{37} -4.78822 q^{38} +7.71389 q^{41} -8.19624 q^{43} +6.36956 q^{44} +1.54731 q^{46} +5.19381 q^{47} +0.727598 q^{49} -1.16382 q^{52} +11.7853 q^{53} -6.91707 q^{56} +0.712495 q^{58} +4.46028 q^{59} -5.24905 q^{61} +6.28464 q^{62} +1.73733 q^{64} -8.49375 q^{67} -2.84075 q^{68} -0.663102 q^{71} +16.5345 q^{73} -1.05736 q^{74} -10.0291 q^{76} +11.8648 q^{77} +9.54554 q^{79} -5.49611 q^{82} +0.0123998 q^{83} +5.83978 q^{86} -10.6203 q^{88} +5.46783 q^{89} -2.16789 q^{91} +3.24091 q^{92} -3.70056 q^{94} -0.952006 q^{97} -0.518410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} - 12 q^{11} - 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 6 q^{28} - 5 q^{29} + 2 q^{31} + q^{32} + 4 q^{34} - 16 q^{37} - 14 q^{38} + 14 q^{41} - 20 q^{44} - 6 q^{46} + 2 q^{47} - 7 q^{49} - 16 q^{52} + 26 q^{53} + 2 q^{56} - 3 q^{58} - 4 q^{59} - 12 q^{61} - 9 q^{64} - 12 q^{67} - 20 q^{68} - 30 q^{71} + 12 q^{73} - 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} - 10 q^{82} + 2 q^{83} - 30 q^{86} - 42 q^{88} + 22 q^{89} - 12 q^{91} + 20 q^{92} - 50 q^{94} - 20 q^{97} - 9 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.712495 −0.503810 −0.251905 0.967752i \(-0.581057\pi\)
−0.251905 + 0.967752i \(0.581057\pi\)
\(3\) 0 0
\(4\) −1.49235 −0.746175
\(5\) 0 0
\(6\) 0 0
\(7\) −2.77986 −1.05069 −0.525343 0.850890i \(-0.676063\pi\)
−0.525343 + 0.850890i \(0.676063\pi\)
\(8\) 2.48828 0.879741
\(9\) 0 0
\(10\) 0 0
\(11\) −4.26814 −1.28689 −0.643446 0.765491i \(-0.722496\pi\)
−0.643446 + 0.765491i \(0.722496\pi\)
\(12\) 0 0
\(13\) 0.779856 0.216293 0.108147 0.994135i \(-0.465508\pi\)
0.108147 + 0.994135i \(0.465508\pi\)
\(14\) 1.98063 0.529347
\(15\) 0 0
\(16\) 1.21181 0.302953
\(17\) 1.90354 0.461677 0.230838 0.972992i \(-0.425853\pi\)
0.230838 + 0.972992i \(0.425853\pi\)
\(18\) 0 0
\(19\) 6.72036 1.54176 0.770878 0.636983i \(-0.219818\pi\)
0.770878 + 0.636983i \(0.219818\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.04103 0.648349
\(23\) −2.17168 −0.452827 −0.226413 0.974031i \(-0.572700\pi\)
−0.226413 + 0.974031i \(0.572700\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.555643 −0.108971
\(27\) 0 0
\(28\) 4.14852 0.783997
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.82061 −1.58423 −0.792114 0.610373i \(-0.791020\pi\)
−0.792114 + 0.610373i \(0.791020\pi\)
\(32\) −5.83998 −1.03237
\(33\) 0 0
\(34\) −1.35626 −0.232597
\(35\) 0 0
\(36\) 0 0
\(37\) 1.48402 0.243971 0.121986 0.992532i \(-0.461074\pi\)
0.121986 + 0.992532i \(0.461074\pi\)
\(38\) −4.78822 −0.776752
\(39\) 0 0
\(40\) 0 0
\(41\) 7.71389 1.20471 0.602354 0.798229i \(-0.294230\pi\)
0.602354 + 0.798229i \(0.294230\pi\)
\(42\) 0 0
\(43\) −8.19624 −1.24991 −0.624957 0.780659i \(-0.714884\pi\)
−0.624957 + 0.780659i \(0.714884\pi\)
\(44\) 6.36956 0.960247
\(45\) 0 0
\(46\) 1.54731 0.228139
\(47\) 5.19381 0.757595 0.378798 0.925480i \(-0.376338\pi\)
0.378798 + 0.925480i \(0.376338\pi\)
\(48\) 0 0
\(49\) 0.727598 0.103943
\(50\) 0 0
\(51\) 0 0
\(52\) −1.16382 −0.161393
\(53\) 11.7853 1.61884 0.809419 0.587231i \(-0.199782\pi\)
0.809419 + 0.587231i \(0.199782\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.91707 −0.924332
\(57\) 0 0
\(58\) 0.712495 0.0935552
\(59\) 4.46028 0.580679 0.290339 0.956924i \(-0.406232\pi\)
0.290339 + 0.956924i \(0.406232\pi\)
\(60\) 0 0
\(61\) −5.24905 −0.672071 −0.336036 0.941849i \(-0.609086\pi\)
−0.336036 + 0.941849i \(0.609086\pi\)
\(62\) 6.28464 0.798150
\(63\) 0 0
\(64\) 1.73733 0.217166
\(65\) 0 0
\(66\) 0 0
\(67\) −8.49375 −1.03768 −0.518838 0.854872i \(-0.673635\pi\)
−0.518838 + 0.854872i \(0.673635\pi\)
\(68\) −2.84075 −0.344492
\(69\) 0 0
\(70\) 0 0
\(71\) −0.663102 −0.0786957 −0.0393479 0.999226i \(-0.512528\pi\)
−0.0393479 + 0.999226i \(0.512528\pi\)
\(72\) 0 0
\(73\) 16.5345 1.93522 0.967609 0.252455i \(-0.0812380\pi\)
0.967609 + 0.252455i \(0.0812380\pi\)
\(74\) −1.05736 −0.122915
\(75\) 0 0
\(76\) −10.0291 −1.15042
\(77\) 11.8648 1.35212
\(78\) 0 0
\(79\) 9.54554 1.07396 0.536978 0.843596i \(-0.319566\pi\)
0.536978 + 0.843596i \(0.319566\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.49611 −0.606944
\(83\) 0.0123998 0.00136106 0.000680528 1.00000i \(-0.499783\pi\)
0.000680528 1.00000i \(0.499783\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.83978 0.629720
\(87\) 0 0
\(88\) −10.6203 −1.13213
\(89\) 5.46783 0.579588 0.289794 0.957089i \(-0.406413\pi\)
0.289794 + 0.957089i \(0.406413\pi\)
\(90\) 0 0
\(91\) −2.16789 −0.227256
\(92\) 3.24091 0.337888
\(93\) 0 0
\(94\) −3.70056 −0.381684
\(95\) 0 0
\(96\) 0 0
\(97\) −0.952006 −0.0966615 −0.0483308 0.998831i \(-0.515390\pi\)
−0.0483308 + 0.998831i \(0.515390\pi\)
\(98\) −0.518410 −0.0523673
\(99\) 0 0
\(100\) 0 0
\(101\) −8.31781 −0.827653 −0.413826 0.910356i \(-0.635808\pi\)
−0.413826 + 0.910356i \(0.635808\pi\)
\(102\) 0 0
\(103\) 12.4091 1.22271 0.611355 0.791357i \(-0.290625\pi\)
0.611355 + 0.791357i \(0.290625\pi\)
\(104\) 1.94050 0.190282
\(105\) 0 0
\(106\) −8.39698 −0.815587
\(107\) 17.5579 1.69739 0.848695 0.528883i \(-0.177389\pi\)
0.848695 + 0.528883i \(0.177389\pi\)
\(108\) 0 0
\(109\) 1.00973 0.0967146 0.0483573 0.998830i \(-0.484601\pi\)
0.0483573 + 0.998830i \(0.484601\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.36866 −0.318309
\(113\) 15.5787 1.46552 0.732761 0.680487i \(-0.238232\pi\)
0.732761 + 0.680487i \(0.238232\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.49235 0.138561
\(117\) 0 0
\(118\) −3.17792 −0.292552
\(119\) −5.29157 −0.485078
\(120\) 0 0
\(121\) 7.21701 0.656091
\(122\) 3.73992 0.338596
\(123\) 0 0
\(124\) 13.1634 1.18211
\(125\) 0 0
\(126\) 0 0
\(127\) −16.9536 −1.50438 −0.752192 0.658943i \(-0.771004\pi\)
−0.752192 + 0.658943i \(0.771004\pi\)
\(128\) 10.4421 0.922961
\(129\) 0 0
\(130\) 0 0
\(131\) −11.9324 −1.04254 −0.521268 0.853393i \(-0.674541\pi\)
−0.521268 + 0.853393i \(0.674541\pi\)
\(132\) 0 0
\(133\) −18.6816 −1.61990
\(134\) 6.05175 0.522792
\(135\) 0 0
\(136\) 4.73655 0.406156
\(137\) 0.239785 0.0204862 0.0102431 0.999948i \(-0.496739\pi\)
0.0102431 + 0.999948i \(0.496739\pi\)
\(138\) 0 0
\(139\) −17.0255 −1.44408 −0.722040 0.691851i \(-0.756795\pi\)
−0.722040 + 0.691851i \(0.756795\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.472457 0.0396477
\(143\) −3.32853 −0.278346
\(144\) 0 0
\(145\) 0 0
\(146\) −11.7808 −0.974982
\(147\) 0 0
\(148\) −2.21468 −0.182045
\(149\) 4.13046 0.338380 0.169190 0.985583i \(-0.445885\pi\)
0.169190 + 0.985583i \(0.445885\pi\)
\(150\) 0 0
\(151\) −3.21580 −0.261698 −0.130849 0.991402i \(-0.541770\pi\)
−0.130849 + 0.991402i \(0.541770\pi\)
\(152\) 16.7221 1.35635
\(153\) 0 0
\(154\) −8.45362 −0.681212
\(155\) 0 0
\(156\) 0 0
\(157\) 6.84237 0.546081 0.273040 0.962003i \(-0.411971\pi\)
0.273040 + 0.962003i \(0.411971\pi\)
\(158\) −6.80115 −0.541070
\(159\) 0 0
\(160\) 0 0
\(161\) 6.03696 0.475779
\(162\) 0 0
\(163\) −14.0502 −1.10050 −0.550248 0.835001i \(-0.685467\pi\)
−0.550248 + 0.835001i \(0.685467\pi\)
\(164\) −11.5118 −0.898923
\(165\) 0 0
\(166\) −0.00883480 −0.000685713 0
\(167\) −17.8725 −1.38301 −0.691507 0.722370i \(-0.743053\pi\)
−0.691507 + 0.722370i \(0.743053\pi\)
\(168\) 0 0
\(169\) −12.3918 −0.953217
\(170\) 0 0
\(171\) 0 0
\(172\) 12.2317 0.932656
\(173\) 2.82518 0.214795 0.107397 0.994216i \(-0.465748\pi\)
0.107397 + 0.994216i \(0.465748\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.17218 −0.389868
\(177\) 0 0
\(178\) −3.89580 −0.292002
\(179\) −20.2829 −1.51601 −0.758006 0.652247i \(-0.773826\pi\)
−0.758006 + 0.652247i \(0.773826\pi\)
\(180\) 0 0
\(181\) −8.63783 −0.642045 −0.321023 0.947072i \(-0.604027\pi\)
−0.321023 + 0.947072i \(0.604027\pi\)
\(182\) 1.54461 0.114494
\(183\) 0 0
\(184\) −5.40376 −0.398370
\(185\) 0 0
\(186\) 0 0
\(187\) −8.12458 −0.594128
\(188\) −7.75099 −0.565299
\(189\) 0 0
\(190\) 0 0
\(191\) −0.896850 −0.0648938 −0.0324469 0.999473i \(-0.510330\pi\)
−0.0324469 + 0.999473i \(0.510330\pi\)
\(192\) 0 0
\(193\) −10.4506 −0.752251 −0.376126 0.926569i \(-0.622744\pi\)
−0.376126 + 0.926569i \(0.622744\pi\)
\(194\) 0.678299 0.0486991
\(195\) 0 0
\(196\) −1.08583 −0.0775594
\(197\) −9.57740 −0.682361 −0.341181 0.939998i \(-0.610827\pi\)
−0.341181 + 0.939998i \(0.610827\pi\)
\(198\) 0 0
\(199\) 5.61768 0.398227 0.199113 0.979976i \(-0.436194\pi\)
0.199113 + 0.979976i \(0.436194\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 5.92640 0.416980
\(203\) 2.77986 0.195108
\(204\) 0 0
\(205\) 0 0
\(206\) −8.84145 −0.616013
\(207\) 0 0
\(208\) 0.945039 0.0655267
\(209\) −28.6834 −1.98407
\(210\) 0 0
\(211\) −0.605216 −0.0416648 −0.0208324 0.999783i \(-0.506632\pi\)
−0.0208324 + 0.999783i \(0.506632\pi\)
\(212\) −17.5878 −1.20794
\(213\) 0 0
\(214\) −12.5099 −0.855162
\(215\) 0 0
\(216\) 0 0
\(217\) 24.5200 1.66453
\(218\) −0.719428 −0.0487258
\(219\) 0 0
\(220\) 0 0
\(221\) 1.48449 0.0998575
\(222\) 0 0
\(223\) 16.5537 1.10852 0.554260 0.832344i \(-0.313001\pi\)
0.554260 + 0.832344i \(0.313001\pi\)
\(224\) 16.2343 1.08470
\(225\) 0 0
\(226\) −11.0997 −0.738344
\(227\) 22.0061 1.46060 0.730299 0.683128i \(-0.239381\pi\)
0.730299 + 0.683128i \(0.239381\pi\)
\(228\) 0 0
\(229\) −24.5647 −1.62328 −0.811641 0.584157i \(-0.801425\pi\)
−0.811641 + 0.584157i \(0.801425\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.48828 −0.163364
\(233\) 5.56824 0.364787 0.182394 0.983226i \(-0.441615\pi\)
0.182394 + 0.983226i \(0.441615\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.65630 −0.433288
\(237\) 0 0
\(238\) 3.77022 0.244387
\(239\) −6.90640 −0.446738 −0.223369 0.974734i \(-0.571705\pi\)
−0.223369 + 0.974734i \(0.571705\pi\)
\(240\) 0 0
\(241\) −25.1989 −1.62320 −0.811602 0.584210i \(-0.801404\pi\)
−0.811602 + 0.584210i \(0.801404\pi\)
\(242\) −5.14208 −0.330545
\(243\) 0 0
\(244\) 7.83342 0.501483
\(245\) 0 0
\(246\) 0 0
\(247\) 5.24091 0.333471
\(248\) −21.9482 −1.39371
\(249\) 0 0
\(250\) 0 0
\(251\) −28.3687 −1.79062 −0.895309 0.445446i \(-0.853045\pi\)
−0.895309 + 0.445446i \(0.853045\pi\)
\(252\) 0 0
\(253\) 9.26903 0.582739
\(254\) 12.0793 0.757924
\(255\) 0 0
\(256\) −10.9146 −0.682163
\(257\) −8.71100 −0.543377 −0.271688 0.962385i \(-0.587582\pi\)
−0.271688 + 0.962385i \(0.587582\pi\)
\(258\) 0 0
\(259\) −4.12536 −0.256337
\(260\) 0 0
\(261\) 0 0
\(262\) 8.50175 0.525240
\(263\) 13.2128 0.814736 0.407368 0.913264i \(-0.366447\pi\)
0.407368 + 0.913264i \(0.366447\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 13.3106 0.816123
\(267\) 0 0
\(268\) 12.6757 0.774289
\(269\) 10.3447 0.630725 0.315363 0.948971i \(-0.397874\pi\)
0.315363 + 0.948971i \(0.397874\pi\)
\(270\) 0 0
\(271\) 9.76022 0.592891 0.296445 0.955050i \(-0.404199\pi\)
0.296445 + 0.955050i \(0.404199\pi\)
\(272\) 2.30674 0.139866
\(273\) 0 0
\(274\) −0.170845 −0.0103212
\(275\) 0 0
\(276\) 0 0
\(277\) 1.87503 0.112659 0.0563297 0.998412i \(-0.482060\pi\)
0.0563297 + 0.998412i \(0.482060\pi\)
\(278\) 12.1306 0.727542
\(279\) 0 0
\(280\) 0 0
\(281\) 21.2930 1.27024 0.635118 0.772415i \(-0.280951\pi\)
0.635118 + 0.772415i \(0.280951\pi\)
\(282\) 0 0
\(283\) −15.8729 −0.943544 −0.471772 0.881721i \(-0.656385\pi\)
−0.471772 + 0.881721i \(0.656385\pi\)
\(284\) 0.989581 0.0587208
\(285\) 0 0
\(286\) 2.37156 0.140233
\(287\) −21.4435 −1.26577
\(288\) 0 0
\(289\) −13.3765 −0.786855
\(290\) 0 0
\(291\) 0 0
\(292\) −24.6753 −1.44401
\(293\) 33.1369 1.93588 0.967938 0.251189i \(-0.0808216\pi\)
0.967938 + 0.251189i \(0.0808216\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.69266 0.214631
\(297\) 0 0
\(298\) −2.94293 −0.170479
\(299\) −1.69360 −0.0979433
\(300\) 0 0
\(301\) 22.7844 1.31327
\(302\) 2.29124 0.131846
\(303\) 0 0
\(304\) 8.14381 0.467080
\(305\) 0 0
\(306\) 0 0
\(307\) −17.2605 −0.985109 −0.492555 0.870282i \(-0.663937\pi\)
−0.492555 + 0.870282i \(0.663937\pi\)
\(308\) −17.7065 −1.00892
\(309\) 0 0
\(310\) 0 0
\(311\) 32.5116 1.84356 0.921781 0.387711i \(-0.126734\pi\)
0.921781 + 0.387711i \(0.126734\pi\)
\(312\) 0 0
\(313\) −6.26304 −0.354008 −0.177004 0.984210i \(-0.556641\pi\)
−0.177004 + 0.984210i \(0.556641\pi\)
\(314\) −4.87516 −0.275121
\(315\) 0 0
\(316\) −14.2453 −0.801360
\(317\) −18.8523 −1.05885 −0.529426 0.848356i \(-0.677593\pi\)
−0.529426 + 0.848356i \(0.677593\pi\)
\(318\) 0 0
\(319\) 4.26814 0.238970
\(320\) 0 0
\(321\) 0 0
\(322\) −4.30130 −0.239702
\(323\) 12.7925 0.711793
\(324\) 0 0
\(325\) 0 0
\(326\) 10.0107 0.554441
\(327\) 0 0
\(328\) 19.1943 1.05983
\(329\) −14.4380 −0.795995
\(330\) 0 0
\(331\) −0.596245 −0.0327726 −0.0163863 0.999866i \(-0.505216\pi\)
−0.0163863 + 0.999866i \(0.505216\pi\)
\(332\) −0.0185049 −0.00101559
\(333\) 0 0
\(334\) 12.7341 0.696776
\(335\) 0 0
\(336\) 0 0
\(337\) −32.6414 −1.77809 −0.889046 0.457818i \(-0.848631\pi\)
−0.889046 + 0.457818i \(0.848631\pi\)
\(338\) 8.82911 0.480240
\(339\) 0 0
\(340\) 0 0
\(341\) 37.6476 2.03873
\(342\) 0 0
\(343\) 17.4364 0.941476
\(344\) −20.3946 −1.09960
\(345\) 0 0
\(346\) −2.01293 −0.108216
\(347\) −13.1985 −0.708531 −0.354265 0.935145i \(-0.615269\pi\)
−0.354265 + 0.935145i \(0.615269\pi\)
\(348\) 0 0
\(349\) 15.6412 0.837255 0.418628 0.908158i \(-0.362511\pi\)
0.418628 + 0.908158i \(0.362511\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 24.9258 1.32855
\(353\) −5.65486 −0.300978 −0.150489 0.988612i \(-0.548085\pi\)
−0.150489 + 0.988612i \(0.548085\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.15991 −0.432475
\(357\) 0 0
\(358\) 14.4514 0.763782
\(359\) −25.2693 −1.33366 −0.666832 0.745208i \(-0.732350\pi\)
−0.666832 + 0.745208i \(0.732350\pi\)
\(360\) 0 0
\(361\) 26.1632 1.37701
\(362\) 6.15441 0.323469
\(363\) 0 0
\(364\) 3.23525 0.169573
\(365\) 0 0
\(366\) 0 0
\(367\) 26.2788 1.37174 0.685872 0.727722i \(-0.259421\pi\)
0.685872 + 0.727722i \(0.259421\pi\)
\(368\) −2.63167 −0.137185
\(369\) 0 0
\(370\) 0 0
\(371\) −32.7615 −1.70089
\(372\) 0 0
\(373\) 1.43033 0.0740596 0.0370298 0.999314i \(-0.488210\pi\)
0.0370298 + 0.999314i \(0.488210\pi\)
\(374\) 5.78872 0.299328
\(375\) 0 0
\(376\) 12.9237 0.666487
\(377\) −0.779856 −0.0401646
\(378\) 0 0
\(379\) −35.2336 −1.80983 −0.904915 0.425591i \(-0.860066\pi\)
−0.904915 + 0.425591i \(0.860066\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.639001 0.0326941
\(383\) −32.0701 −1.63870 −0.819352 0.573291i \(-0.805666\pi\)
−0.819352 + 0.573291i \(0.805666\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.44601 0.378992
\(387\) 0 0
\(388\) 1.42073 0.0721265
\(389\) −1.63922 −0.0831117 −0.0415558 0.999136i \(-0.513231\pi\)
−0.0415558 + 0.999136i \(0.513231\pi\)
\(390\) 0 0
\(391\) −4.13389 −0.209060
\(392\) 1.81047 0.0914425
\(393\) 0 0
\(394\) 6.82385 0.343781
\(395\) 0 0
\(396\) 0 0
\(397\) −2.04098 −0.102434 −0.0512170 0.998688i \(-0.516310\pi\)
−0.0512170 + 0.998688i \(0.516310\pi\)
\(398\) −4.00257 −0.200631
\(399\) 0 0
\(400\) 0 0
\(401\) −17.4108 −0.869455 −0.434727 0.900562i \(-0.643155\pi\)
−0.434727 + 0.900562i \(0.643155\pi\)
\(402\) 0 0
\(403\) −6.87880 −0.342658
\(404\) 12.4131 0.617574
\(405\) 0 0
\(406\) −1.98063 −0.0982972
\(407\) −6.33400 −0.313965
\(408\) 0 0
\(409\) −37.3010 −1.84441 −0.922207 0.386696i \(-0.873616\pi\)
−0.922207 + 0.386696i \(0.873616\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −18.5188 −0.912356
\(413\) −12.3989 −0.610111
\(414\) 0 0
\(415\) 0 0
\(416\) −4.55434 −0.223295
\(417\) 0 0
\(418\) 20.4368 0.999596
\(419\) −4.32941 −0.211506 −0.105753 0.994392i \(-0.533725\pi\)
−0.105753 + 0.994392i \(0.533725\pi\)
\(420\) 0 0
\(421\) −34.2505 −1.66927 −0.834634 0.550806i \(-0.814321\pi\)
−0.834634 + 0.550806i \(0.814321\pi\)
\(422\) 0.431214 0.0209911
\(423\) 0 0
\(424\) 29.3252 1.42416
\(425\) 0 0
\(426\) 0 0
\(427\) 14.5916 0.706137
\(428\) −26.2026 −1.26655
\(429\) 0 0
\(430\) 0 0
\(431\) −34.4885 −1.66125 −0.830626 0.556831i \(-0.812017\pi\)
−0.830626 + 0.556831i \(0.812017\pi\)
\(432\) 0 0
\(433\) −3.02258 −0.145256 −0.0726279 0.997359i \(-0.523139\pi\)
−0.0726279 + 0.997359i \(0.523139\pi\)
\(434\) −17.4704 −0.838606
\(435\) 0 0
\(436\) −1.50687 −0.0721661
\(437\) −14.5945 −0.698148
\(438\) 0 0
\(439\) 8.37392 0.399665 0.199833 0.979830i \(-0.435960\pi\)
0.199833 + 0.979830i \(0.435960\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.05769 −0.0503092
\(443\) −29.8770 −1.41950 −0.709749 0.704454i \(-0.751192\pi\)
−0.709749 + 0.704454i \(0.751192\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.7945 −0.558484
\(447\) 0 0
\(448\) −4.82952 −0.228174
\(449\) −12.0780 −0.569994 −0.284997 0.958528i \(-0.591993\pi\)
−0.284997 + 0.958528i \(0.591993\pi\)
\(450\) 0 0
\(451\) −32.9240 −1.55033
\(452\) −23.2489 −1.09354
\(453\) 0 0
\(454\) −15.6793 −0.735864
\(455\) 0 0
\(456\) 0 0
\(457\) −3.03357 −0.141905 −0.0709523 0.997480i \(-0.522604\pi\)
−0.0709523 + 0.997480i \(0.522604\pi\)
\(458\) 17.5022 0.817826
\(459\) 0 0
\(460\) 0 0
\(461\) 4.44508 0.207028 0.103514 0.994628i \(-0.466991\pi\)
0.103514 + 0.994628i \(0.466991\pi\)
\(462\) 0 0
\(463\) 9.04875 0.420531 0.210266 0.977644i \(-0.432567\pi\)
0.210266 + 0.977644i \(0.432567\pi\)
\(464\) −1.21181 −0.0562570
\(465\) 0 0
\(466\) −3.96734 −0.183784
\(467\) 8.11327 0.375437 0.187719 0.982223i \(-0.439891\pi\)
0.187719 + 0.982223i \(0.439891\pi\)
\(468\) 0 0
\(469\) 23.6114 1.09027
\(470\) 0 0
\(471\) 0 0
\(472\) 11.0984 0.510847
\(473\) 34.9827 1.60851
\(474\) 0 0
\(475\) 0 0
\(476\) 7.89688 0.361953
\(477\) 0 0
\(478\) 4.92078 0.225071
\(479\) −6.62947 −0.302908 −0.151454 0.988464i \(-0.548396\pi\)
−0.151454 + 0.988464i \(0.548396\pi\)
\(480\) 0 0
\(481\) 1.15732 0.0527693
\(482\) 17.9541 0.817787
\(483\) 0 0
\(484\) −10.7703 −0.489559
\(485\) 0 0
\(486\) 0 0
\(487\) 29.2047 1.32339 0.661695 0.749773i \(-0.269837\pi\)
0.661695 + 0.749773i \(0.269837\pi\)
\(488\) −13.0611 −0.591249
\(489\) 0 0
\(490\) 0 0
\(491\) 4.32464 0.195168 0.0975841 0.995227i \(-0.468889\pi\)
0.0975841 + 0.995227i \(0.468889\pi\)
\(492\) 0 0
\(493\) −1.90354 −0.0857312
\(494\) −3.73412 −0.168006
\(495\) 0 0
\(496\) −10.6889 −0.479947
\(497\) 1.84333 0.0826846
\(498\) 0 0
\(499\) 4.29688 0.192355 0.0961774 0.995364i \(-0.469338\pi\)
0.0961774 + 0.995364i \(0.469338\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.2126 0.902131
\(503\) −14.2512 −0.635429 −0.317715 0.948186i \(-0.602915\pi\)
−0.317715 + 0.948186i \(0.602915\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.60414 −0.293590
\(507\) 0 0
\(508\) 25.3007 1.12254
\(509\) 19.4974 0.864206 0.432103 0.901824i \(-0.357772\pi\)
0.432103 + 0.901824i \(0.357772\pi\)
\(510\) 0 0
\(511\) −45.9635 −2.03331
\(512\) −13.1076 −0.579280
\(513\) 0 0
\(514\) 6.20654 0.273759
\(515\) 0 0
\(516\) 0 0
\(517\) −22.1679 −0.974943
\(518\) 2.93930 0.129145
\(519\) 0 0
\(520\) 0 0
\(521\) −24.5229 −1.07437 −0.537185 0.843465i \(-0.680512\pi\)
−0.537185 + 0.843465i \(0.680512\pi\)
\(522\) 0 0
\(523\) 10.8566 0.474725 0.237362 0.971421i \(-0.423717\pi\)
0.237362 + 0.971421i \(0.423717\pi\)
\(524\) 17.8073 0.777914
\(525\) 0 0
\(526\) −9.41406 −0.410472
\(527\) −16.7904 −0.731401
\(528\) 0 0
\(529\) −18.2838 −0.794948
\(530\) 0 0
\(531\) 0 0
\(532\) 27.8795 1.20873
\(533\) 6.01572 0.260570
\(534\) 0 0
\(535\) 0 0
\(536\) −21.1348 −0.912886
\(537\) 0 0
\(538\) −7.37052 −0.317766
\(539\) −3.10549 −0.133763
\(540\) 0 0
\(541\) −22.7032 −0.976085 −0.488043 0.872820i \(-0.662289\pi\)
−0.488043 + 0.872820i \(0.662289\pi\)
\(542\) −6.95410 −0.298704
\(543\) 0 0
\(544\) −11.1166 −0.476622
\(545\) 0 0
\(546\) 0 0
\(547\) 32.8229 1.40340 0.701702 0.712471i \(-0.252424\pi\)
0.701702 + 0.712471i \(0.252424\pi\)
\(548\) −0.357843 −0.0152863
\(549\) 0 0
\(550\) 0 0
\(551\) −6.72036 −0.286297
\(552\) 0 0
\(553\) −26.5352 −1.12839
\(554\) −1.33595 −0.0567590
\(555\) 0 0
\(556\) 25.4080 1.07754
\(557\) 26.3234 1.11536 0.557679 0.830057i \(-0.311692\pi\)
0.557679 + 0.830057i \(0.311692\pi\)
\(558\) 0 0
\(559\) −6.39189 −0.270348
\(560\) 0 0
\(561\) 0 0
\(562\) −15.1712 −0.639958
\(563\) 28.4750 1.20008 0.600039 0.799971i \(-0.295152\pi\)
0.600039 + 0.799971i \(0.295152\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 11.3093 0.475367
\(567\) 0 0
\(568\) −1.64999 −0.0692318
\(569\) 14.7555 0.618582 0.309291 0.950967i \(-0.399908\pi\)
0.309291 + 0.950967i \(0.399908\pi\)
\(570\) 0 0
\(571\) −0.204397 −0.00855376 −0.00427688 0.999991i \(-0.501361\pi\)
−0.00427688 + 0.999991i \(0.501361\pi\)
\(572\) 4.96734 0.207695
\(573\) 0 0
\(574\) 15.2784 0.637708
\(575\) 0 0
\(576\) 0 0
\(577\) 37.3300 1.55407 0.777035 0.629457i \(-0.216723\pi\)
0.777035 + 0.629457i \(0.216723\pi\)
\(578\) 9.53071 0.396425
\(579\) 0 0
\(580\) 0 0
\(581\) −0.0344697 −0.00143004
\(582\) 0 0
\(583\) −50.3014 −2.08327
\(584\) 41.1425 1.70249
\(585\) 0 0
\(586\) −23.6098 −0.975314
\(587\) −10.8497 −0.447813 −0.223907 0.974611i \(-0.571881\pi\)
−0.223907 + 0.974611i \(0.571881\pi\)
\(588\) 0 0
\(589\) −59.2776 −2.44249
\(590\) 0 0
\(591\) 0 0
\(592\) 1.79835 0.0739119
\(593\) 14.0575 0.577272 0.288636 0.957439i \(-0.406798\pi\)
0.288636 + 0.957439i \(0.406798\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.16409 −0.252491
\(597\) 0 0
\(598\) 1.20668 0.0493448
\(599\) 5.52446 0.225724 0.112862 0.993611i \(-0.463998\pi\)
0.112862 + 0.993611i \(0.463998\pi\)
\(600\) 0 0
\(601\) −6.85394 −0.279578 −0.139789 0.990181i \(-0.544642\pi\)
−0.139789 + 0.990181i \(0.544642\pi\)
\(602\) −16.2337 −0.661638
\(603\) 0 0
\(604\) 4.79910 0.195273
\(605\) 0 0
\(606\) 0 0
\(607\) −30.4979 −1.23787 −0.618935 0.785442i \(-0.712436\pi\)
−0.618935 + 0.785442i \(0.712436\pi\)
\(608\) −39.2467 −1.59166
\(609\) 0 0
\(610\) 0 0
\(611\) 4.05042 0.163863
\(612\) 0 0
\(613\) −29.9211 −1.20850 −0.604250 0.796794i \(-0.706527\pi\)
−0.604250 + 0.796794i \(0.706527\pi\)
\(614\) 12.2980 0.496308
\(615\) 0 0
\(616\) 29.5230 1.18952
\(617\) −1.04305 −0.0419917 −0.0209959 0.999780i \(-0.506684\pi\)
−0.0209959 + 0.999780i \(0.506684\pi\)
\(618\) 0 0
\(619\) 22.1661 0.890930 0.445465 0.895299i \(-0.353038\pi\)
0.445465 + 0.895299i \(0.353038\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −23.1643 −0.928805
\(623\) −15.1998 −0.608966
\(624\) 0 0
\(625\) 0 0
\(626\) 4.46238 0.178353
\(627\) 0 0
\(628\) −10.2112 −0.407472
\(629\) 2.82489 0.112636
\(630\) 0 0
\(631\) 20.1089 0.800524 0.400262 0.916401i \(-0.368919\pi\)
0.400262 + 0.916401i \(0.368919\pi\)
\(632\) 23.7520 0.944804
\(633\) 0 0
\(634\) 13.4322 0.533460
\(635\) 0 0
\(636\) 0 0
\(637\) 0.567422 0.0224821
\(638\) −3.04103 −0.120395
\(639\) 0 0
\(640\) 0 0
\(641\) −5.93422 −0.234388 −0.117194 0.993109i \(-0.537390\pi\)
−0.117194 + 0.993109i \(0.537390\pi\)
\(642\) 0 0
\(643\) −0.951317 −0.0375163 −0.0187581 0.999824i \(-0.505971\pi\)
−0.0187581 + 0.999824i \(0.505971\pi\)
\(644\) −9.00926 −0.355015
\(645\) 0 0
\(646\) −9.11458 −0.358608
\(647\) −35.1384 −1.38143 −0.690716 0.723126i \(-0.742705\pi\)
−0.690716 + 0.723126i \(0.742705\pi\)
\(648\) 0 0
\(649\) −19.0371 −0.747271
\(650\) 0 0
\(651\) 0 0
\(652\) 20.9678 0.821163
\(653\) −45.9930 −1.79984 −0.899922 0.436050i \(-0.856377\pi\)
−0.899922 + 0.436050i \(0.856377\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.34779 0.364970
\(657\) 0 0
\(658\) 10.2870 0.401030
\(659\) 19.3162 0.752451 0.376226 0.926528i \(-0.377222\pi\)
0.376226 + 0.926528i \(0.377222\pi\)
\(660\) 0 0
\(661\) 44.3224 1.72394 0.861970 0.506959i \(-0.169230\pi\)
0.861970 + 0.506959i \(0.169230\pi\)
\(662\) 0.424821 0.0165111
\(663\) 0 0
\(664\) 0.0308542 0.00119738
\(665\) 0 0
\(666\) 0 0
\(667\) 2.17168 0.0840878
\(668\) 26.6720 1.03197
\(669\) 0 0
\(670\) 0 0
\(671\) 22.4037 0.864883
\(672\) 0 0
\(673\) −12.7421 −0.491170 −0.245585 0.969375i \(-0.578980\pi\)
−0.245585 + 0.969375i \(0.578980\pi\)
\(674\) 23.2569 0.895821
\(675\) 0 0
\(676\) 18.4930 0.711267
\(677\) −9.41654 −0.361907 −0.180954 0.983492i \(-0.557918\pi\)
−0.180954 + 0.983492i \(0.557918\pi\)
\(678\) 0 0
\(679\) 2.64644 0.101561
\(680\) 0 0
\(681\) 0 0
\(682\) −26.8237 −1.02713
\(683\) 6.92316 0.264907 0.132454 0.991189i \(-0.457714\pi\)
0.132454 + 0.991189i \(0.457714\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12.4233 −0.474325
\(687\) 0 0
\(688\) −9.93231 −0.378666
\(689\) 9.19085 0.350144
\(690\) 0 0
\(691\) −48.7272 −1.85367 −0.926835 0.375469i \(-0.877482\pi\)
−0.926835 + 0.375469i \(0.877482\pi\)
\(692\) −4.21616 −0.160274
\(693\) 0 0
\(694\) 9.40384 0.356965
\(695\) 0 0
\(696\) 0 0
\(697\) 14.6837 0.556186
\(698\) −11.1443 −0.421818
\(699\) 0 0
\(700\) 0 0
\(701\) −2.88580 −0.108995 −0.0544975 0.998514i \(-0.517356\pi\)
−0.0544975 + 0.998514i \(0.517356\pi\)
\(702\) 0 0
\(703\) 9.97314 0.376144
\(704\) −7.41516 −0.279469
\(705\) 0 0
\(706\) 4.02906 0.151636
\(707\) 23.1223 0.869604
\(708\) 0 0
\(709\) −1.73056 −0.0649924 −0.0324962 0.999472i \(-0.510346\pi\)
−0.0324962 + 0.999472i \(0.510346\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 13.6055 0.509887
\(713\) 19.1555 0.717381
\(714\) 0 0
\(715\) 0 0
\(716\) 30.2692 1.13121
\(717\) 0 0
\(718\) 18.0043 0.671913
\(719\) 12.9089 0.481420 0.240710 0.970597i \(-0.422620\pi\)
0.240710 + 0.970597i \(0.422620\pi\)
\(720\) 0 0
\(721\) −34.4956 −1.28468
\(722\) −18.6412 −0.693752
\(723\) 0 0
\(724\) 12.8907 0.479078
\(725\) 0 0
\(726\) 0 0
\(727\) 38.3405 1.42197 0.710985 0.703207i \(-0.248249\pi\)
0.710985 + 0.703207i \(0.248249\pi\)
\(728\) −5.39431 −0.199927
\(729\) 0 0
\(730\) 0 0
\(731\) −15.6019 −0.577057
\(732\) 0 0
\(733\) 34.8966 1.28894 0.644468 0.764631i \(-0.277079\pi\)
0.644468 + 0.764631i \(0.277079\pi\)
\(734\) −18.7235 −0.691098
\(735\) 0 0
\(736\) 12.6826 0.467485
\(737\) 36.2525 1.33538
\(738\) 0 0
\(739\) 14.2969 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 23.3424 0.856927
\(743\) −44.1357 −1.61918 −0.809591 0.586995i \(-0.800311\pi\)
−0.809591 + 0.586995i \(0.800311\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.01910 −0.0373120
\(747\) 0 0
\(748\) 12.1247 0.443324
\(749\) −48.8085 −1.78343
\(750\) 0 0
\(751\) 9.56375 0.348986 0.174493 0.984658i \(-0.444171\pi\)
0.174493 + 0.984658i \(0.444171\pi\)
\(752\) 6.29393 0.229516
\(753\) 0 0
\(754\) 0.555643 0.0202353
\(755\) 0 0
\(756\) 0 0
\(757\) −30.1910 −1.09731 −0.548656 0.836048i \(-0.684860\pi\)
−0.548656 + 0.836048i \(0.684860\pi\)
\(758\) 25.1038 0.911811
\(759\) 0 0
\(760\) 0 0
\(761\) −8.27038 −0.299801 −0.149900 0.988701i \(-0.547895\pi\)
−0.149900 + 0.988701i \(0.547895\pi\)
\(762\) 0 0
\(763\) −2.80690 −0.101617
\(764\) 1.33841 0.0484221
\(765\) 0 0
\(766\) 22.8498 0.825595
\(767\) 3.47837 0.125597
\(768\) 0 0
\(769\) −22.4592 −0.809901 −0.404950 0.914339i \(-0.632711\pi\)
−0.404950 + 0.914339i \(0.632711\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 15.5960 0.561311
\(773\) −26.9520 −0.969398 −0.484699 0.874681i \(-0.661071\pi\)
−0.484699 + 0.874681i \(0.661071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2.36886 −0.0850371
\(777\) 0 0
\(778\) 1.16794 0.0418725
\(779\) 51.8401 1.85737
\(780\) 0 0
\(781\) 2.83021 0.101273
\(782\) 2.94537 0.105326
\(783\) 0 0
\(784\) 0.881713 0.0314897
\(785\) 0 0
\(786\) 0 0
\(787\) −1.16423 −0.0415004 −0.0207502 0.999785i \(-0.506605\pi\)
−0.0207502 + 0.999785i \(0.506605\pi\)
\(788\) 14.2928 0.509161
\(789\) 0 0
\(790\) 0 0
\(791\) −43.3065 −1.53980
\(792\) 0 0
\(793\) −4.09350 −0.145364
\(794\) 1.45419 0.0516073
\(795\) 0 0
\(796\) −8.38354 −0.297147
\(797\) 15.9624 0.565418 0.282709 0.959206i \(-0.408767\pi\)
0.282709 + 0.959206i \(0.408767\pi\)
\(798\) 0 0
\(799\) 9.88664 0.349764
\(800\) 0 0
\(801\) 0 0
\(802\) 12.4051 0.438040
\(803\) −70.5715 −2.49042
\(804\) 0 0
\(805\) 0 0
\(806\) 4.90111 0.172634
\(807\) 0 0
\(808\) −20.6971 −0.728120
\(809\) 7.83605 0.275501 0.137750 0.990467i \(-0.456013\pi\)
0.137750 + 0.990467i \(0.456013\pi\)
\(810\) 0 0
\(811\) −23.9935 −0.842526 −0.421263 0.906939i \(-0.638413\pi\)
−0.421263 + 0.906939i \(0.638413\pi\)
\(812\) −4.14852 −0.145585
\(813\) 0 0
\(814\) 4.51294 0.158179
\(815\) 0 0
\(816\) 0 0
\(817\) −55.0817 −1.92706
\(818\) 26.5768 0.929235
\(819\) 0 0
\(820\) 0 0
\(821\) 12.8804 0.449528 0.224764 0.974413i \(-0.427839\pi\)
0.224764 + 0.974413i \(0.427839\pi\)
\(822\) 0 0
\(823\) −47.8165 −1.66678 −0.833389 0.552687i \(-0.813603\pi\)
−0.833389 + 0.552687i \(0.813603\pi\)
\(824\) 30.8775 1.07567
\(825\) 0 0
\(826\) 8.83417 0.307380
\(827\) −14.6873 −0.510726 −0.255363 0.966845i \(-0.582195\pi\)
−0.255363 + 0.966845i \(0.582195\pi\)
\(828\) 0 0
\(829\) 41.9876 1.45829 0.729145 0.684359i \(-0.239918\pi\)
0.729145 + 0.684359i \(0.239918\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.35487 0.0469715
\(833\) 1.38501 0.0479879
\(834\) 0 0
\(835\) 0 0
\(836\) 42.8057 1.48047
\(837\) 0 0
\(838\) 3.08468 0.106559
\(839\) 4.04408 0.139617 0.0698085 0.997560i \(-0.477761\pi\)
0.0698085 + 0.997560i \(0.477761\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 24.4033 0.840994
\(843\) 0 0
\(844\) 0.903195 0.0310892
\(845\) 0 0
\(846\) 0 0
\(847\) −20.0622 −0.689347
\(848\) 14.2816 0.490432
\(849\) 0 0
\(850\) 0 0
\(851\) −3.22282 −0.110477
\(852\) 0 0
\(853\) 43.8299 1.50071 0.750353 0.661037i \(-0.229883\pi\)
0.750353 + 0.661037i \(0.229883\pi\)
\(854\) −10.3964 −0.355759
\(855\) 0 0
\(856\) 43.6891 1.49326
\(857\) 2.09275 0.0714868 0.0357434 0.999361i \(-0.488620\pi\)
0.0357434 + 0.999361i \(0.488620\pi\)
\(858\) 0 0
\(859\) 46.5885 1.58958 0.794790 0.606885i \(-0.207581\pi\)
0.794790 + 0.606885i \(0.207581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.5729 0.836955
\(863\) 32.1466 1.09428 0.547141 0.837040i \(-0.315716\pi\)
0.547141 + 0.837040i \(0.315716\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.15357 0.0731814
\(867\) 0 0
\(868\) −36.5925 −1.24203
\(869\) −40.7417 −1.38207
\(870\) 0 0
\(871\) −6.62390 −0.224442
\(872\) 2.51249 0.0850838
\(873\) 0 0
\(874\) 10.3985 0.351734
\(875\) 0 0
\(876\) 0 0
\(877\) −48.0544 −1.62268 −0.811342 0.584572i \(-0.801262\pi\)
−0.811342 + 0.584572i \(0.801262\pi\)
\(878\) −5.96637 −0.201355
\(879\) 0 0
\(880\) 0 0
\(881\) 30.9846 1.04390 0.521950 0.852976i \(-0.325205\pi\)
0.521950 + 0.852976i \(0.325205\pi\)
\(882\) 0 0
\(883\) −9.60378 −0.323193 −0.161597 0.986857i \(-0.551664\pi\)
−0.161597 + 0.986857i \(0.551664\pi\)
\(884\) −2.21538 −0.0745112
\(885\) 0 0
\(886\) 21.2872 0.715158
\(887\) −29.7279 −0.998165 −0.499083 0.866554i \(-0.666330\pi\)
−0.499083 + 0.866554i \(0.666330\pi\)
\(888\) 0 0
\(889\) 47.1284 1.58064
\(890\) 0 0
\(891\) 0 0
\(892\) −24.7040 −0.827150
\(893\) 34.9043 1.16803
\(894\) 0 0
\(895\) 0 0
\(896\) −29.0276 −0.969743
\(897\) 0 0
\(898\) 8.60548 0.287169
\(899\) 8.82061 0.294184
\(900\) 0 0
\(901\) 22.4339 0.747380
\(902\) 23.4582 0.781071
\(903\) 0 0
\(904\) 38.7642 1.28928
\(905\) 0 0
\(906\) 0 0
\(907\) −4.35876 −0.144730 −0.0723651 0.997378i \(-0.523055\pi\)
−0.0723651 + 0.997378i \(0.523055\pi\)
\(908\) −32.8409 −1.08986
\(909\) 0 0
\(910\) 0 0
\(911\) 0.389317 0.0128986 0.00644932 0.999979i \(-0.497947\pi\)
0.00644932 + 0.999979i \(0.497947\pi\)
\(912\) 0 0
\(913\) −0.0529241 −0.00175153
\(914\) 2.16141 0.0714930
\(915\) 0 0
\(916\) 36.6592 1.21125
\(917\) 33.1703 1.09538
\(918\) 0 0
\(919\) −54.9915 −1.81400 −0.907001 0.421128i \(-0.861634\pi\)
−0.907001 + 0.421128i \(0.861634\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.16710 −0.104303
\(923\) −0.517124 −0.0170213
\(924\) 0 0
\(925\) 0 0
\(926\) −6.44719 −0.211868
\(927\) 0 0
\(928\) 5.83998 0.191707
\(929\) −17.4617 −0.572901 −0.286450 0.958095i \(-0.592475\pi\)
−0.286450 + 0.958095i \(0.592475\pi\)
\(930\) 0 0
\(931\) 4.88972 0.160254
\(932\) −8.30977 −0.272195
\(933\) 0 0
\(934\) −5.78066 −0.189149
\(935\) 0 0
\(936\) 0 0
\(937\) −27.4261 −0.895970 −0.447985 0.894041i \(-0.647858\pi\)
−0.447985 + 0.894041i \(0.647858\pi\)
\(938\) −16.8230 −0.549291
\(939\) 0 0
\(940\) 0 0
\(941\) 35.1587 1.14614 0.573070 0.819506i \(-0.305752\pi\)
0.573070 + 0.819506i \(0.305752\pi\)
\(942\) 0 0
\(943\) −16.7521 −0.545524
\(944\) 5.40502 0.175918
\(945\) 0 0
\(946\) −24.9250 −0.810381
\(947\) −44.1438 −1.43448 −0.717240 0.696827i \(-0.754595\pi\)
−0.717240 + 0.696827i \(0.754595\pi\)
\(948\) 0 0
\(949\) 12.8945 0.418574
\(950\) 0 0
\(951\) 0 0
\(952\) −13.1669 −0.426743
\(953\) −27.3169 −0.884881 −0.442441 0.896798i \(-0.645887\pi\)
−0.442441 + 0.896798i \(0.645887\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3068 0.333345
\(957\) 0 0
\(958\) 4.72346 0.152608
\(959\) −0.666567 −0.0215246
\(960\) 0 0
\(961\) 46.8031 1.50978
\(962\) −0.824585 −0.0265857
\(963\) 0 0
\(964\) 37.6056 1.21120
\(965\) 0 0
\(966\) 0 0
\(967\) −54.3910 −1.74910 −0.874548 0.484939i \(-0.838842\pi\)
−0.874548 + 0.484939i \(0.838842\pi\)
\(968\) 17.9579 0.577190
\(969\) 0 0
\(970\) 0 0
\(971\) 14.2906 0.458606 0.229303 0.973355i \(-0.426355\pi\)
0.229303 + 0.973355i \(0.426355\pi\)
\(972\) 0 0
\(973\) 47.3283 1.51728
\(974\) −20.8082 −0.666737
\(975\) 0 0
\(976\) −6.36086 −0.203606
\(977\) −8.94905 −0.286306 −0.143153 0.989701i \(-0.545724\pi\)
−0.143153 + 0.989701i \(0.545724\pi\)
\(978\) 0 0
\(979\) −23.3374 −0.745868
\(980\) 0 0
\(981\) 0 0
\(982\) −3.08128 −0.0983277
\(983\) −22.1981 −0.708010 −0.354005 0.935244i \(-0.615180\pi\)
−0.354005 + 0.935244i \(0.615180\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.35626 0.0431923
\(987\) 0 0
\(988\) −7.82128 −0.248828
\(989\) 17.7996 0.565995
\(990\) 0 0
\(991\) −13.2764 −0.421740 −0.210870 0.977514i \(-0.567630\pi\)
−0.210870 + 0.977514i \(0.567630\pi\)
\(992\) 51.5121 1.63551
\(993\) 0 0
\(994\) −1.31336 −0.0416573
\(995\) 0 0
\(996\) 0 0
\(997\) 19.6929 0.623682 0.311841 0.950134i \(-0.399054\pi\)
0.311841 + 0.950134i \(0.399054\pi\)
\(998\) −3.06150 −0.0969102
\(999\) 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 6525.2.a.bs.1.2 5
3.2 odd 2 2175.2.a.w.1.4 5
5.2 odd 4 1305.2.c.j.784.5 10
5.3 odd 4 1305.2.c.j.784.6 10
5.4 even 2 6525.2.a.bl.1.4 5
15.2 even 4 435.2.c.e.349.6 yes 10
15.8 even 4 435.2.c.e.349.5 10
15.14 odd 2 2175.2.a.z.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.5 10 15.8 even 4
435.2.c.e.349.6 yes 10 15.2 even 4
1305.2.c.j.784.5 10 5.2 odd 4
1305.2.c.j.784.6 10 5.3 odd 4
2175.2.a.w.1.4 5 3.2 odd 2
2175.2.a.z.1.2 5 15.14 odd 2
6525.2.a.bl.1.4 5 5.4 even 2
6525.2.a.bs.1.2 5 1.1 even 1 trivial