Properties

Label 3025.2.a.g.1.1
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
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] = [3025,2,Mod(1,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.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: \(24.1547466114\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +6.00000 q^{9} -3.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} +6.00000 q^{18} +4.00000 q^{19} +9.00000 q^{21} +8.00000 q^{23} -9.00000 q^{24} -4.00000 q^{26} +9.00000 q^{27} -3.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} +5.00000 q^{32} -6.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -12.0000 q^{39} -5.00000 q^{41} +9.00000 q^{42} -5.00000 q^{43} +8.00000 q^{46} +3.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +4.00000 q^{52} -4.00000 q^{53} +9.00000 q^{54} -9.00000 q^{56} +12.0000 q^{57} +6.00000 q^{58} -2.00000 q^{59} -11.0000 q^{61} -2.00000 q^{62} +18.0000 q^{63} +7.00000 q^{64} +13.0000 q^{67} +24.0000 q^{69} +2.00000 q^{71} -18.0000 q^{72} +8.00000 q^{73} +8.00000 q^{74} -4.00000 q^{76} -12.0000 q^{78} +10.0000 q^{79} +9.00000 q^{81} -5.00000 q^{82} -4.00000 q^{83} -9.00000 q^{84} -5.00000 q^{86} +18.0000 q^{87} +1.00000 q^{89} -12.0000 q^{91} -8.00000 q^{92} -6.00000 q^{93} +3.00000 q^{94} +15.0000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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 \(0.384973\pi\)
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −3.00000 −1.06066
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 6.00000 1.41421
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 9.00000 1.96396
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −9.00000 −1.83712
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 9.00000 1.73205
\(28\) −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
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 9.00000 1.38873
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −3.00000 −0.433013
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 12.0000 1.58944
\(58\) 6.00000 0.787839
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −2.00000 −0.254000
\(63\) 18.0000 2.26779
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 24.0000 2.88926
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −18.0000 −2.12132
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −12.0000 −1.35873
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −5.00000 −0.552158
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −9.00000 −0.981981
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) −8.00000 −0.834058
\(93\) −6.00000 −0.622171
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 15.0000 1.53093
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 0 0
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) −9.00000 −0.866025
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) −3.00000 −0.283473
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −24.0000 −2.21880
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) −11.0000 −0.995893
\(123\) −15.0000 −1.35250
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 18.0000 1.60357
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) −3.00000 −0.265165
\(129\) −15.0000 −1.32068
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 13.0000 1.12303
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 24.0000 2.04302
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 6.00000 0.494872
\(148\) −8.00000 −0.657596
\(149\) −17.0000 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 12.0000 0.960769
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 10.0000 0.795557
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 9.00000 0.707107
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 5.00000 0.390434
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 7.00000 0.541676 0.270838 0.962625i \(-0.412699\pi\)
0.270838 + 0.962625i \(0.412699\pi\)
\(168\) −27.0000 −2.08310
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 24.0000 1.83533
\(172\) 5.00000 0.381246
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 18.0000 1.36458
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 1.00000 0.0749532
\(179\) −26.0000 −1.94333 −0.971666 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 0 0
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) −12.0000 −0.889499
\(183\) −33.0000 −2.43943
\(184\) −24.0000 −1.76930
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) −3.00000 −0.218797
\(189\) 27.0000 1.96396
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 21.0000 1.51554
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) 39.0000 2.75085
\(202\) −5.00000 −0.351799
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 48.0000 3.33623
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 4.00000 0.274721
\(213\) 6.00000 0.411113
\(214\) −9.00000 −0.615227
\(215\) 0 0
\(216\) −27.0000 −1.83712
\(217\) −6.00000 −0.407307
\(218\) −9.00000 −0.609557
\(219\) 24.0000 1.62177
\(220\) 0 0
\(221\) 0 0
\(222\) 24.0000 1.61077
\(223\) −5.00000 −0.334825 −0.167412 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 15.0000 1.00223
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 1.00000 0.0663723 0.0331862 0.999449i \(-0.489435\pi\)
0.0331862 + 0.999449i \(0.489435\pi\)
\(228\) −12.0000 −0.794719
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −24.0000 −1.56893
\(235\) 0 0
\(236\) 2.00000 0.130189
\(237\) 30.0000 1.94871
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 11.0000 0.704203
\(245\) 0 0
\(246\) −15.0000 −0.956365
\(247\) −16.0000 −1.01806
\(248\) 6.00000 0.381000
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −18.0000 −1.13389
\(253\) 0 0
\(254\) −11.0000 −0.690201
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −15.0000 −0.933859
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 36.0000 2.22834
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 3.00000 0.183597
\(268\) −13.0000 −0.794101
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) −36.0000 −2.17882
\(274\) 0 0
\(275\) 0 0
\(276\) −24.0000 −1.44463
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −18.0000 −1.07957
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 9.00000 0.535942
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −15.0000 −0.885422
\(288\) 30.0000 1.76777
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 24.0000 1.40690
\(292\) −8.00000 −0.468165
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −24.0000 −1.39497
\(297\) 0 0
\(298\) −17.0000 −0.984784
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) −15.0000 −0.864586
\(302\) −14.0000 −0.805609
\(303\) −15.0000 −0.861727
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) −24.0000 −1.36531
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 36.0000 2.03810
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 0 0
\(321\) −27.0000 −1.50699
\(322\) 24.0000 1.33747
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 17.0000 0.941543
\(327\) −27.0000 −1.49310
\(328\) 15.0000 0.828236
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) 48.0000 2.63038
\(334\) 7.00000 0.383023
\(335\) 0 0
\(336\) −9.00000 −0.490990
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 3.00000 0.163178
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 24.0000 1.29777
\(343\) −15.0000 −0.809924
\(344\) 15.0000 0.808746
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −7.00000 −0.375780 −0.187890 0.982190i \(-0.560165\pi\)
−0.187890 + 0.982190i \(0.560165\pi\)
\(348\) −18.0000 −0.964901
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) −26.0000 −1.37414
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −19.0000 −0.998618
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) −33.0000 −1.72494
\(367\) −1.00000 −0.0521996 −0.0260998 0.999659i \(-0.508309\pi\)
−0.0260998 + 0.999659i \(0.508309\pi\)
\(368\) −8.00000 −0.417029
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 6.00000 0.311086
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −24.0000 −1.23606
\(378\) 27.0000 1.38873
\(379\) −22.0000 −1.13006 −0.565032 0.825069i \(-0.691136\pi\)
−0.565032 + 0.825069i \(0.691136\pi\)
\(380\) 0 0
\(381\) −33.0000 −1.69064
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −9.00000 −0.459279
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) −30.0000 −1.52499
\(388\) −8.00000 −0.406138
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) −6.00000 −0.300753
\(399\) 36.0000 1.80225
\(400\) 0 0
\(401\) −37.0000 −1.84769 −0.923846 0.382765i \(-0.874972\pi\)
−0.923846 + 0.382765i \(0.874972\pi\)
\(402\) 39.0000 1.94514
\(403\) 8.00000 0.398508
\(404\) 5.00000 0.248759
\(405\) 0 0
\(406\) 18.0000 0.893325
\(407\) 0 0
\(408\) 0 0
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −6.00000 −0.295241
\(414\) 48.0000 2.35907
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) −54.0000 −2.64439
\(418\) 0 0
\(419\) 32.0000 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) −22.0000 −1.07094
\(423\) 18.0000 0.875190
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) −33.0000 −1.59698
\(428\) 9.00000 0.435031
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −9.00000 −0.433013
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 9.00000 0.431022
\(437\) 32.0000 1.53077
\(438\) 24.0000 1.14676
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) −24.0000 −1.13899
\(445\) 0 0
\(446\) −5.00000 −0.236757
\(447\) −51.0000 −2.41222
\(448\) 21.0000 0.992157
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −42.0000 −1.97333
\(454\) 1.00000 0.0469323
\(455\) 0 0
\(456\) −36.0000 −1.68585
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −1.00000 −0.0467269
\(459\) 0 0
\(460\) 0 0
\(461\) 7.00000 0.326023 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(462\) 0 0
\(463\) 15.0000 0.697109 0.348555 0.937288i \(-0.386673\pi\)
0.348555 + 0.937288i \(0.386673\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 24.0000 1.10940
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 30.0000 1.37795
\(475\) 0 0
\(476\) 0 0
\(477\) −24.0000 −1.09888
\(478\) 4.00000 0.182956
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 23.0000 1.04762
\(483\) 72.0000 3.27611
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 33.0000 1.49384
\(489\) 51.0000 2.30630
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 15.0000 0.676252
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 6.00000 0.269137
\(498\) −12.0000 −0.537733
\(499\) −42.0000 −1.88018 −0.940089 0.340929i \(-0.889258\pi\)
−0.940089 + 0.340929i \(0.889258\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) 18.0000 0.803379
\(503\) −13.0000 −0.579641 −0.289821 0.957081i \(-0.593596\pi\)
−0.289821 + 0.957081i \(0.593596\pi\)
\(504\) −54.0000 −2.40535
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 11.0000 0.488046
\(509\) 23.0000 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) −11.0000 −0.486136
\(513\) 36.0000 1.58944
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 15.0000 0.660338
\(517\) 0 0
\(518\) 24.0000 1.05450
\(519\) −72.0000 −3.16045
\(520\) 0 0
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 36.0000 1.57568
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −12.0000 −0.520266
\(533\) 20.0000 0.866296
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) −39.0000 −1.68454
\(537\) −78.0000 −3.36595
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 31.0000 1.33279 0.666397 0.745597i \(-0.267836\pi\)
0.666397 + 0.745597i \(0.267836\pi\)
\(542\) 0 0
\(543\) −57.0000 −2.44610
\(544\) 0 0
\(545\) 0 0
\(546\) −36.0000 −1.54066
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) −66.0000 −2.81681
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) −72.0000 −3.06452
\(553\) 30.0000 1.27573
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 18.0000 0.763370
\(557\) 28.0000 1.18640 0.593199 0.805056i \(-0.297865\pi\)
0.593199 + 0.805056i \(0.297865\pi\)
\(558\) −12.0000 −0.508001
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) −9.00000 −0.378968
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 27.0000 1.13389
\(568\) −6.00000 −0.251754
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) −18.0000 −0.753277 −0.376638 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) −15.0000 −0.626088
\(575\) 0 0
\(576\) 42.0000 1.75000
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −17.0000 −0.707107
\(579\) 30.0000 1.24676
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 24.0000 0.994832
\(583\) 0 0
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) 34.0000 1.40453
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) −6.00000 −0.247436
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 42.0000 1.72765
\(592\) −8.00000 −0.328798
\(593\) −44.0000 −1.80686 −0.903432 0.428732i \(-0.858960\pi\)
−0.903432 + 0.428732i \(0.858960\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) −18.0000 −0.736691
\(598\) −32.0000 −1.30858
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −15.0000 −0.611354
\(603\) 78.0000 3.17641
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) −15.0000 −0.609333
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 20.0000 0.811107
\(609\) 54.0000 2.18819
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) −24.0000 −0.965422
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 72.0000 2.88926
\(622\) 24.0000 0.962312
\(623\) 3.00000 0.120192
\(624\) 12.0000 0.480384
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 0 0
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) −30.0000 −1.19334
\(633\) −66.0000 −2.62326
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) −8.00000 −0.316972
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) −27.0000 −1.06561
\(643\) −35.0000 −1.38027 −0.690133 0.723683i \(-0.742448\pi\)
−0.690133 + 0.723683i \(0.742448\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) 25.0000 0.982851 0.491426 0.870919i \(-0.336476\pi\)
0.491426 + 0.870919i \(0.336476\pi\)
\(648\) −27.0000 −1.06066
\(649\) 0 0
\(650\) 0 0
\(651\) −18.0000 −0.705476
\(652\) −17.0000 −0.665771
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −27.0000 −1.05578
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 48.0000 1.87266
\(658\) 9.00000 0.350857
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −37.0000 −1.43913 −0.719567 0.694423i \(-0.755660\pi\)
−0.719567 + 0.694423i \(0.755660\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 48.0000 1.85996
\(667\) 48.0000 1.85857
\(668\) −7.00000 −0.270838
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) 0 0
\(672\) 45.0000 1.73591
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 28.0000 1.07613 0.538064 0.842904i \(-0.319156\pi\)
0.538064 + 0.842904i \(0.319156\pi\)
\(678\) −18.0000 −0.691286
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) −47.0000 −1.79841 −0.899203 0.437533i \(-0.855852\pi\)
−0.899203 + 0.437533i \(0.855852\pi\)
\(684\) −24.0000 −0.917663
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) −3.00000 −0.114457
\(688\) 5.00000 0.190623
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) −7.00000 −0.265716
\(695\) 0 0
\(696\) −54.0000 −2.04686
\(697\) 0 0
\(698\) 22.0000 0.832712
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) −36.0000 −1.35873
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −15.0000 −0.564133
\(708\) 6.00000 0.225494
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 60.0000 2.25018
\(712\) −3.00000 −0.112430
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 26.0000 0.971666
\(717\) 12.0000 0.448148
\(718\) −28.0000 −1.04495
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −3.00000 −0.111648
\(723\) 69.0000 2.56614
\(724\) 19.0000 0.706129
\(725\) 0 0
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 36.0000 1.33425
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 33.0000 1.21972
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) −1.00000 −0.0369107
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) −30.0000 −1.10432
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) −48.0000 −1.76332
\(742\) −12.0000 −0.440534
\(743\) 7.00000 0.256805 0.128403 0.991722i \(-0.459015\pi\)
0.128403 + 0.991722i \(0.459015\pi\)
\(744\) 18.0000 0.659912
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) −27.0000 −0.986559
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) −3.00000 −0.109399
\(753\) 54.0000 1.96787
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) −27.0000 −0.981981
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) −22.0000 −0.799076
\(759\) 0 0
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) −33.0000 −1.19546
\(763\) −27.0000 −0.977466
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) −51.0000 −1.84030
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −10.0000 −0.359908
\(773\) 22.0000 0.791285 0.395643 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) −30.0000 −1.07833
\(775\) 0 0
\(776\) −24.0000 −0.861550
\(777\) 72.0000 2.58299
\(778\) −3.00000 −0.107555
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 54.0000 1.92980
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 0 0
\(787\) 43.0000 1.53278 0.766392 0.642373i \(-0.222050\pi\)
0.766392 + 0.642373i \(0.222050\pi\)
\(788\) −14.0000 −0.498729
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 44.0000 1.56249
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 36.0000 1.27439
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −37.0000 −1.30652
\(803\) 0 0
\(804\) −39.0000 −1.37542
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 63.0000 2.21771
\(808\) 15.0000 0.527698
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) 42.0000 1.47482 0.737410 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) −18.0000 −0.631676
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 21.0000 0.734248
\(819\) −72.0000 −2.51588
\(820\) 0 0
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) −51.0000 −1.77775 −0.888874 0.458151i \(-0.848512\pi\)
−0.888874 + 0.458151i \(0.848512\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −45.0000 −1.56480 −0.782402 0.622774i \(-0.786006\pi\)
−0.782402 + 0.622774i \(0.786006\pi\)
\(828\) −48.0000 −1.66812
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 0 0
\(831\) −42.0000 −1.45696
\(832\) −28.0000 −0.970725
\(833\) 0 0
\(834\) −54.0000 −1.86987
\(835\) 0 0
\(836\) 0 0
\(837\) −18.0000 −0.622171
\(838\) 32.0000 1.10542
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 3.00000 0.103387
\(843\) −18.0000 −0.619953
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) 18.0000 0.618853
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) 39.0000 1.33848
\(850\) 0 0
\(851\) 64.0000 2.19389
\(852\) −6.00000 −0.205557
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) −33.0000 −1.12924
\(855\) 0 0
\(856\) 27.0000 0.922841
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −45.0000 −1.53360
\(862\) 18.0000 0.613082
\(863\) −31.0000 −1.05525 −0.527626 0.849477i \(-0.676918\pi\)
−0.527626 + 0.849477i \(0.676918\pi\)
\(864\) 45.0000 1.53093
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −51.0000 −1.73205
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) −52.0000 −1.76195
\(872\) 27.0000 0.914335
\(873\) 48.0000 1.62455
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) −24.0000 −0.810885
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 8.00000 0.269987
\(879\) 102.000 3.44037
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 12.0000 0.404061
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) −1.00000 −0.0335767 −0.0167884 0.999859i \(-0.505344\pi\)
−0.0167884 + 0.999859i \(0.505344\pi\)
\(888\) −72.0000 −2.41616
\(889\) −33.0000 −1.10678
\(890\) 0 0
\(891\) 0 0
\(892\) 5.00000 0.167412
\(893\) 12.0000 0.401565
\(894\) −51.0000 −1.70570
\(895\) 0 0
\(896\) −9.00000 −0.300669
\(897\) −96.0000 −3.20535
\(898\) −13.0000 −0.433816
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −45.0000 −1.49751
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −42.0000 −1.39536
\(907\) 43.0000 1.42779 0.713896 0.700252i \(-0.246929\pi\)
0.713896 + 0.700252i \(0.246929\pi\)
\(908\) −1.00000 −0.0331862
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) −34.0000 −1.12647 −0.563235 0.826297i \(-0.690443\pi\)
−0.563235 + 0.826297i \(0.690443\pi\)
\(912\) −12.0000 −0.397360
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) 1.00000 0.0330409
\(917\) 0 0
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 7.00000 0.230533
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) 15.0000 0.492931
\(927\) −48.0000 −1.57653
\(928\) 30.0000 0.984798
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) −24.0000 −0.786146
\(933\) 72.0000 2.35717
\(934\) −27.0000 −0.883467
\(935\) 0 0
\(936\) 72.0000 2.35339
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 39.0000 1.27340
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) −13.0000 −0.423788 −0.211894 0.977293i \(-0.567963\pi\)
−0.211894 + 0.977293i \(0.567963\pi\)
\(942\) 24.0000 0.781962
\(943\) −40.0000 −1.30258
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −30.0000 −0.974355
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) −54.0000 −1.75107
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −24.0000 −0.777029
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −32.0000 −1.03172
\(963\) −54.0000 −1.74013
\(964\) −23.0000 −0.740780
\(965\) 0 0
\(966\) 72.0000 2.31656
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −54.0000 −1.73116
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 51.0000 1.63080
\(979\) 0 0
\(980\) 0 0
\(981\) −54.0000 −1.72409
\(982\) 22.0000 0.702048
\(983\) 61.0000 1.94560 0.972799 0.231651i \(-0.0744128\pi\)
0.972799 + 0.231651i \(0.0744128\pi\)
\(984\) 45.0000 1.43455
\(985\) 0 0
\(986\) 0 0
\(987\) 27.0000 0.859419
\(988\) 16.0000 0.509028
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −10.0000 −0.317500
\(993\) −60.0000 −1.90404
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) −42.0000 −1.32949
\(999\) 72.0000 2.27798
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 3025.2.a.g.1.1 1
5.4 even 2 605.2.a.a.1.1 1
11.10 odd 2 3025.2.a.c.1.1 1
15.14 odd 2 5445.2.a.h.1.1 1
20.19 odd 2 9680.2.a.bf.1.1 1
55.4 even 10 605.2.g.d.511.1 4
55.9 even 10 605.2.g.d.81.1 4
55.14 even 10 605.2.g.d.251.1 4
55.19 odd 10 605.2.g.b.251.1 4
55.24 odd 10 605.2.g.b.81.1 4
55.29 odd 10 605.2.g.b.511.1 4
55.39 odd 10 605.2.g.b.366.1 4
55.49 even 10 605.2.g.d.366.1 4
55.54 odd 2 605.2.a.c.1.1 yes 1
165.164 even 2 5445.2.a.d.1.1 1
220.219 even 2 9680.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.a.1.1 1 5.4 even 2
605.2.a.c.1.1 yes 1 55.54 odd 2
605.2.g.b.81.1 4 55.24 odd 10
605.2.g.b.251.1 4 55.19 odd 10
605.2.g.b.366.1 4 55.39 odd 10
605.2.g.b.511.1 4 55.29 odd 10
605.2.g.d.81.1 4 55.9 even 10
605.2.g.d.251.1 4 55.14 even 10
605.2.g.d.366.1 4 55.49 even 10
605.2.g.d.511.1 4 55.4 even 10
3025.2.a.c.1.1 1 11.10 odd 2
3025.2.a.g.1.1 1 1.1 even 1 trivial
5445.2.a.d.1.1 1 165.164 even 2
5445.2.a.h.1.1 1 15.14 odd 2
9680.2.a.be.1.1 1 220.219 even 2
9680.2.a.bf.1.1 1 20.19 odd 2