Properties

Label 2475.2.a.k.1.1
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} -3.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} +1.00000 q^{22} +1.00000 q^{23} -2.00000 q^{26} -3.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} +3.00000 q^{34} -1.00000 q^{37} -1.00000 q^{38} -5.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} +3.00000 q^{47} +2.00000 q^{49} +2.00000 q^{52} +10.0000 q^{53} -9.00000 q^{56} +6.00000 q^{58} +11.0000 q^{59} +14.0000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -2.00000 q^{67} -3.00000 q^{68} -5.00000 q^{71} -2.00000 q^{73} -1.00000 q^{74} +1.00000 q^{76} +3.00000 q^{77} +5.00000 q^{79} -5.00000 q^{82} +8.00000 q^{83} -4.00000 q^{86} -3.00000 q^{88} -10.0000 q^{89} -6.00000 q^{91} -1.00000 q^{92} +3.00000 q^{94} +17.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.00000 −0.552158
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 0 0
\(101\) −11.0000 −1.09454 −0.547270 0.836956i \(-0.684333\pi\)
−0.547270 + 0.836956i \(0.684333\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 11.0000 1.01263
\(119\) 9.00000 0.825029
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −9.00000 −0.771744
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.00000 −0.419591
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 7.00000 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 3.00000 0.243332
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 5.00000 0.397779
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 5.00000 0.390434
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −1.00000 −0.0747435 −0.0373718 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 17.0000 1.22053
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 23.0000 1.63868 0.819341 0.573306i \(-0.194340\pi\)
0.819341 + 0.573306i \(0.194340\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −11.0000 −0.773957
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) −12.0000 −0.812743
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 15.0000 1.00223
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 11.0000 0.726900 0.363450 0.931614i \(-0.381599\pi\)
0.363450 + 0.931614i \(0.381599\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −11.0000 −0.716039
\(237\) 0 0
\(238\) 9.00000 0.583383
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 0 0
\(283\) 23.0000 1.36721 0.683604 0.729853i \(-0.260412\pi\)
0.683604 + 0.729853i \(0.260412\pi\)
\(284\) 5.00000 0.296695
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −15.0000 −0.885422
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 13.0000 0.759468 0.379734 0.925096i \(-0.376015\pi\)
0.379734 + 0.925096i \(0.376015\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) 7.00000 0.405499
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) 15.0000 0.828236
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 10.0000 0.547176
\(335\) 0 0
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −1.00000 −0.0528516
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −25.0000 −1.31397
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 30.0000 1.55752
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 3.00000 0.155126
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 19.0000 0.972125
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) −17.0000 −0.863044
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 23.0000 1.15872
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −18.0000 −0.902258
\(399\) 0 0
\(400\) 0 0
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 11.0000 0.547270
\(405\) 0 0
\(406\) 18.0000 0.893325
\(407\) −1.00000 −0.0495682
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) 33.0000 1.62382
\(414\) 0 0
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) −1.00000 −0.0489116
\(419\) −23.0000 −1.12362 −0.561812 0.827265i \(-0.689895\pi\)
−0.561812 + 0.827265i \(0.689895\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −30.0000 −1.45693
\(425\) 0 0
\(426\) 0 0
\(427\) 42.0000 2.03252
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −31.0000 −1.47285 −0.736427 0.676517i \(-0.763489\pi\)
−0.736427 + 0.676517i \(0.763489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) 0 0
\(448\) 21.0000 0.992157
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 11.0000 0.513996
\(459\) 0 0
\(460\) 0 0
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 9.00000 0.416917
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) 0 0
\(472\) −33.0000 −1.51895
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) −9.00000 −0.412514
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −42.0000 −1.90125
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −15.0000 −0.672842
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.00000 0.0444554
\(507\) 0 0
\(508\) −5.00000 −0.221839
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 3.00000 0.131940
\(518\) −3.00000 −0.131812
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 3.00000 0.130066
\(533\) 10.0000 0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) −3.00000 −0.128861
\(543\) 0 0
\(544\) 15.0000 0.643120
\(545\) 0 0
\(546\) 0 0
\(547\) 33.0000 1.41098 0.705489 0.708721i \(-0.250727\pi\)
0.705489 + 0.708721i \(0.250727\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 15.0000 0.637865
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −21.0000 −0.885832
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 23.0000 0.966762
\(567\) 0 0
\(568\) 15.0000 0.629386
\(569\) 37.0000 1.55112 0.775560 0.631273i \(-0.217467\pi\)
0.775560 + 0.631273i \(0.217467\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −15.0000 −0.626088
\(575\) 0 0
\(576\) 0 0
\(577\) 43.0000 1.79011 0.895057 0.445952i \(-0.147135\pi\)
0.895057 + 0.445952i \(0.147135\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 13.0000 0.537025
\(587\) 9.00000 0.371470 0.185735 0.982600i \(-0.440533\pi\)
0.185735 + 0.982600i \(0.440533\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.00000 −0.286731
\(597\) 0 0
\(598\) −2.00000 −0.0817861
\(599\) −27.0000 −1.10319 −0.551595 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 0 0
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) −15.0000 −0.596668
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) −13.0000 −0.511083 −0.255541 0.966798i \(-0.582254\pi\)
−0.255541 + 0.966798i \(0.582254\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) 9.00000 0.350857
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −24.0000 −0.931381
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) −10.0000 −0.386912
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −14.0000 −0.538064 −0.269032 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) 51.0000 1.95720
\(680\) 0 0
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 47.0000 1.79841 0.899203 0.437533i \(-0.144148\pi\)
0.899203 + 0.437533i \(0.144148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) −10.0000 −0.379595
\(695\) 0 0
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) 8.00000 0.302804
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) 1.00000 0.0377157
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −33.0000 −1.24109
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 30.0000 1.12430
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) 1.00000 0.0373718
\(717\) 0 0
\(718\) −28.0000 −1.04495
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) −18.0000 −0.669891
\(723\) 0 0
\(724\) 25.0000 0.929118
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 18.0000 0.667124
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) 1.00000 0.0367856 0.0183928 0.999831i \(-0.494145\pi\)
0.0183928 + 0.999831i \(0.494145\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 30.0000 1.10133
\(743\) −14.0000 −0.513610 −0.256805 0.966463i \(-0.582670\pi\)
−0.256805 + 0.966463i \(0.582670\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 0 0
\(748\) −3.00000 −0.109691
\(749\) 54.0000 1.97312
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −3.00000 −0.109399
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) −19.0000 −0.687396
\(765\) 0 0
\(766\) 0 0
\(767\) −22.0000 −0.794374
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 44.0000 1.58257 0.791285 0.611448i \(-0.209412\pi\)
0.791285 + 0.611448i \(0.209412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −51.0000 −1.83079
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 5.00000 0.179144
\(780\) 0 0
\(781\) −5.00000 −0.178914
\(782\) 3.00000 0.107280
\(783\) 0 0
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 0 0
\(787\) −43.0000 −1.53278 −0.766392 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) −23.0000 −0.819341
\(789\) 0 0
\(790\) 0 0
\(791\) 54.0000 1.92002
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) −38.0000 −1.34183
\(803\) −2.00000 −0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 33.0000 1.16094
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) −18.0000 −0.631676
\(813\) 0 0
\(814\) −1.00000 −0.0350500
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) 33.0000 1.14822
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) −23.0000 −0.794522
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 9.00000 0.310160
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 0 0
\(851\) −1.00000 −0.0342796
\(852\) 0 0
\(853\) 40.0000 1.36957 0.684787 0.728743i \(-0.259895\pi\)
0.684787 + 0.728743i \(0.259895\pi\)
\(854\) 42.0000 1.43721
\(855\) 0 0
\(856\) −54.0000 −1.84568
\(857\) 39.0000 1.33221 0.666107 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 36.0000 1.21911
\(873\) 0 0
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) 0 0
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) −35.0000 −1.18119
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −31.0000 −1.04147
\(887\) 14.0000 0.470074 0.235037 0.971986i \(-0.424479\pi\)
0.235037 + 0.971986i \(0.424479\pi\)
\(888\) 0 0
\(889\) 15.0000 0.503084
\(890\) 0 0
\(891\) 0 0
\(892\) −22.0000 −0.736614
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 0 0
\(896\) −9.00000 −0.300669
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) −5.00000 −0.166482
\(903\) 0 0
\(904\) −54.0000 −1.79601
\(905\) 0 0
\(906\) 0 0
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) −53.0000 −1.75597 −0.877984 0.478690i \(-0.841112\pi\)
−0.877984 + 0.478690i \(0.841112\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) −28.0000 −0.926158
\(915\) 0 0
\(916\) −11.0000 −0.363450
\(917\) 36.0000 1.18882
\(918\) 0 0
\(919\) −39.0000 −1.28649 −0.643246 0.765660i \(-0.722413\pi\)
−0.643246 + 0.765660i \(0.722413\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 34.0000 1.11973
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 30.0000 0.984798
\(929\) 32.0000 1.04989 0.524943 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) −6.00000 −0.195907
\(939\) 0 0
\(940\) 0 0
\(941\) 5.00000 0.162995 0.0814977 0.996674i \(-0.474030\pi\)
0.0814977 + 0.996674i \(0.474030\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) −11.0000 −0.358020
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) −27.0000 −0.875075
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) −15.0000 −0.475771
\(995\) 0 0
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 24.0000 0.759707
\(999\) 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 2475.2.a.k.1.1 yes 1
3.2 odd 2 2475.2.a.d.1.1 yes 1
5.2 odd 4 2475.2.c.e.199.2 2
5.3 odd 4 2475.2.c.e.199.1 2
5.4 even 2 2475.2.a.b.1.1 1
15.2 even 4 2475.2.c.c.199.1 2
15.8 even 4 2475.2.c.c.199.2 2
15.14 odd 2 2475.2.a.h.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.b.1.1 1 5.4 even 2
2475.2.a.d.1.1 yes 1 3.2 odd 2
2475.2.a.h.1.1 yes 1 15.14 odd 2
2475.2.a.k.1.1 yes 1 1.1 even 1 trivial
2475.2.c.c.199.1 2 15.2 even 4
2475.2.c.c.199.2 2 15.8 even 4
2475.2.c.e.199.1 2 5.3 odd 4
2475.2.c.e.199.2 2 5.2 odd 4