Properties

Label 2925.2.a.z.1.1
Level $2925$
Weight $2$
Character 2925.1
Self dual yes
Analytic conductor $23.356$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2925,2,Mod(1,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, 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("2925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.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: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +4.73205 q^{11} -1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} -6.19615 q^{19} -8.19615 q^{22} +1.26795 q^{23} +1.73205 q^{26} -2.00000 q^{28} +2.53590 q^{29} +10.1962 q^{31} +5.19615 q^{32} +6.00000 q^{34} +4.00000 q^{37} +10.7321 q^{38} -3.46410 q^{41} +0.196152 q^{43} +4.73205 q^{44} -2.19615 q^{46} +6.00000 q^{47} -3.00000 q^{49} -1.00000 q^{52} +10.3923 q^{53} -3.46410 q^{56} -4.39230 q^{58} -9.12436 q^{59} -8.39230 q^{61} -17.6603 q^{62} +1.00000 q^{64} -6.39230 q^{67} -3.46410 q^{68} -4.73205 q^{71} +4.00000 q^{73} -6.92820 q^{74} -6.19615 q^{76} -9.46410 q^{77} -8.39230 q^{79} +6.00000 q^{82} -6.00000 q^{83} -0.339746 q^{86} +8.19615 q^{88} +12.9282 q^{89} +2.00000 q^{91} +1.26795 q^{92} -10.3923 q^{94} -2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{7} + 6 q^{11} - 2 q^{13} - 10 q^{16} - 2 q^{19} - 6 q^{22} + 6 q^{23} - 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 8 q^{37} + 18 q^{38} - 10 q^{43} + 6 q^{44} + 6 q^{46} + 12 q^{47} - 6 q^{49} - 2 q^{52} + 12 q^{58} + 6 q^{59} + 4 q^{61} - 18 q^{62} + 2 q^{64} + 8 q^{67} - 6 q^{71} + 8 q^{73} - 2 q^{76} - 12 q^{77} + 4 q^{79} + 12 q^{82} - 12 q^{83} - 18 q^{86} + 6 q^{88} + 12 q^{89} + 4 q^{91} + 6 q^{92} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −6.19615 −1.42149 −0.710747 0.703447i \(-0.751643\pi\)
−0.710747 + 0.703447i \(0.751643\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.19615 −1.74743
\(23\) 1.26795 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.73205 0.339683
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 2.53590 0.470905 0.235452 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) 0 0
\(31\) 10.1962 1.83128 0.915642 0.401996i \(-0.131683\pi\)
0.915642 + 0.401996i \(0.131683\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 10.7321 1.74097
\(39\) 0 0
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) 0.196152 0.0299130 0.0149565 0.999888i \(-0.495239\pi\)
0.0149565 + 0.999888i \(0.495239\pi\)
\(44\) 4.73205 0.713384
\(45\) 0 0
\(46\) −2.19615 −0.323805
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) −4.39230 −0.576738
\(59\) −9.12436 −1.18789 −0.593945 0.804506i \(-0.702430\pi\)
−0.593945 + 0.804506i \(0.702430\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) −17.6603 −2.24285
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.39230 −0.780944 −0.390472 0.920615i \(-0.627688\pi\)
−0.390472 + 0.920615i \(0.627688\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) 0 0
\(71\) −4.73205 −0.561591 −0.280796 0.959768i \(-0.590598\pi\)
−0.280796 + 0.959768i \(0.590598\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0 0
\(76\) −6.19615 −0.710747
\(77\) −9.46410 −1.07853
\(78\) 0 0
\(79\) −8.39230 −0.944208 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.339746 −0.0366357
\(87\) 0 0
\(88\) 8.19615 0.873713
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 1.26795 0.132193
\(93\) 0 0
\(94\) −10.3923 −1.07188
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) 0 0
\(101\) 0.928203 0.0923597 0.0461798 0.998933i \(-0.485295\pi\)
0.0461798 + 0.998933i \(0.485295\pi\)
\(102\) 0 0
\(103\) 0.196152 0.0193275 0.00966374 0.999953i \(-0.496924\pi\)
0.00966374 + 0.999953i \(0.496924\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) 17.6603 1.70728 0.853641 0.520862i \(-0.174390\pi\)
0.853641 + 0.520862i \(0.174390\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0000 0.944911
\(113\) 8.53590 0.802990 0.401495 0.915861i \(-0.368491\pi\)
0.401495 + 0.915861i \(0.368491\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.53590 0.235452
\(117\) 0 0
\(118\) 15.8038 1.45486
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 14.5359 1.31602
\(123\) 0 0
\(124\) 10.1962 0.915642
\(125\) 0 0
\(126\) 0 0
\(127\) −16.1962 −1.43718 −0.718588 0.695436i \(-0.755211\pi\)
−0.718588 + 0.695436i \(0.755211\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 12.3923 1.07455
\(134\) 11.0718 0.956458
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 0.928203 0.0793018 0.0396509 0.999214i \(-0.487375\pi\)
0.0396509 + 0.999214i \(0.487375\pi\)
\(138\) 0 0
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.19615 0.687806
\(143\) −4.73205 −0.395714
\(144\) 0 0
\(145\) 0 0
\(146\) −6.92820 −0.573382
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 7.85641 0.643622 0.321811 0.946804i \(-0.395708\pi\)
0.321811 + 0.946804i \(0.395708\pi\)
\(150\) 0 0
\(151\) −1.80385 −0.146795 −0.0733975 0.997303i \(-0.523384\pi\)
−0.0733975 + 0.997303i \(0.523384\pi\)
\(152\) −10.7321 −0.870484
\(153\) 0 0
\(154\) 16.3923 1.32093
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 14.5359 1.15641
\(159\) 0 0
\(160\) 0 0
\(161\) −2.53590 −0.199857
\(162\) 0 0
\(163\) 14.3923 1.12729 0.563646 0.826016i \(-0.309398\pi\)
0.563646 + 0.826016i \(0.309398\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) 10.3923 0.806599
\(167\) −0.928203 −0.0718265 −0.0359133 0.999355i \(-0.511434\pi\)
−0.0359133 + 0.999355i \(0.511434\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0.196152 0.0149565
\(173\) 8.53590 0.648972 0.324486 0.945890i \(-0.394809\pi\)
0.324486 + 0.945890i \(0.394809\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −23.6603 −1.78346
\(177\) 0 0
\(178\) −22.3923 −1.67837
\(179\) 18.9282 1.41476 0.707380 0.706833i \(-0.249877\pi\)
0.707380 + 0.706833i \(0.249877\pi\)
\(180\) 0 0
\(181\) 0.392305 0.0291598 0.0145799 0.999894i \(-0.495359\pi\)
0.0145799 + 0.999894i \(0.495359\pi\)
\(182\) −3.46410 −0.256776
\(183\) 0 0
\(184\) 2.19615 0.161903
\(185\) 0 0
\(186\) 0 0
\(187\) −16.3923 −1.19872
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 5.07180 0.366982 0.183491 0.983021i \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 3.46410 0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −12.9282 −0.921096 −0.460548 0.887635i \(-0.652347\pi\)
−0.460548 + 0.887635i \(0.652347\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.60770 −0.113117
\(203\) −5.07180 −0.355970
\(204\) 0 0
\(205\) 0 0
\(206\) −0.339746 −0.0236712
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) −29.3205 −2.02814
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.3923 0.713746
\(213\) 0 0
\(214\) −30.5885 −2.09098
\(215\) 0 0
\(216\) 0 0
\(217\) −20.3923 −1.38432
\(218\) −3.46410 −0.234619
\(219\) 0 0
\(220\) 0 0
\(221\) 3.46410 0.233021
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) −14.7846 −0.983458
\(227\) −3.46410 −0.229920 −0.114960 0.993370i \(-0.536674\pi\)
−0.114960 + 0.993370i \(0.536674\pi\)
\(228\) 0 0
\(229\) 6.39230 0.422415 0.211208 0.977441i \(-0.432260\pi\)
0.211208 + 0.977441i \(0.432260\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.39230 0.288369
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.12436 −0.593945
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) 14.1962 0.918273 0.459136 0.888366i \(-0.348159\pi\)
0.459136 + 0.888366i \(0.348159\pi\)
\(240\) 0 0
\(241\) −2.39230 −0.154102 −0.0770510 0.997027i \(-0.524550\pi\)
−0.0770510 + 0.997027i \(0.524550\pi\)
\(242\) −19.7321 −1.26842
\(243\) 0 0
\(244\) −8.39230 −0.537262
\(245\) 0 0
\(246\) 0 0
\(247\) 6.19615 0.394252
\(248\) 17.6603 1.12143
\(249\) 0 0
\(250\) 0 0
\(251\) 21.4641 1.35480 0.677401 0.735614i \(-0.263106\pi\)
0.677401 + 0.735614i \(0.263106\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 28.0526 1.76017
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 19.8564 1.23861 0.619304 0.785151i \(-0.287415\pi\)
0.619304 + 0.785151i \(0.287415\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.26795 0.0781851 0.0390925 0.999236i \(-0.487553\pi\)
0.0390925 + 0.999236i \(0.487553\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −21.4641 −1.31605
\(267\) 0 0
\(268\) −6.39230 −0.390472
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) 30.9808 1.88195 0.940974 0.338480i \(-0.109913\pi\)
0.940974 + 0.338480i \(0.109913\pi\)
\(272\) 17.3205 1.05021
\(273\) 0 0
\(274\) −1.60770 −0.0971244
\(275\) 0 0
\(276\) 0 0
\(277\) 26.3923 1.58576 0.792880 0.609378i \(-0.208581\pi\)
0.792880 + 0.609378i \(0.208581\pi\)
\(278\) −21.4641 −1.28733
\(279\) 0 0
\(280\) 0 0
\(281\) −22.3923 −1.33581 −0.667906 0.744245i \(-0.732809\pi\)
−0.667906 + 0.744245i \(0.732809\pi\)
\(282\) 0 0
\(283\) −32.5885 −1.93718 −0.968591 0.248658i \(-0.920010\pi\)
−0.968591 + 0.248658i \(0.920010\pi\)
\(284\) −4.73205 −0.280796
\(285\) 0 0
\(286\) 8.19615 0.484649
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −5.07180 −0.296298 −0.148149 0.988965i \(-0.547331\pi\)
−0.148149 + 0.988965i \(0.547331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.92820 0.402694
\(297\) 0 0
\(298\) −13.6077 −0.788273
\(299\) −1.26795 −0.0733274
\(300\) 0 0
\(301\) −0.392305 −0.0226121
\(302\) 3.12436 0.179786
\(303\) 0 0
\(304\) 30.9808 1.77687
\(305\) 0 0
\(306\) 0 0
\(307\) 18.7846 1.07209 0.536047 0.844188i \(-0.319917\pi\)
0.536047 + 0.844188i \(0.319917\pi\)
\(308\) −9.46410 −0.539267
\(309\) 0 0
\(310\) 0 0
\(311\) 16.3923 0.929522 0.464761 0.885436i \(-0.346140\pi\)
0.464761 + 0.885436i \(0.346140\pi\)
\(312\) 0 0
\(313\) 14.3923 0.813501 0.406751 0.913539i \(-0.366662\pi\)
0.406751 + 0.913539i \(0.366662\pi\)
\(314\) −17.3205 −0.977453
\(315\) 0 0
\(316\) −8.39230 −0.472104
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 4.39230 0.244774
\(323\) 21.4641 1.19429
\(324\) 0 0
\(325\) 0 0
\(326\) −24.9282 −1.38065
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 2.58846 0.142274 0.0711372 0.997467i \(-0.477337\pi\)
0.0711372 + 0.997467i \(0.477337\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) 1.60770 0.0879692
\(335\) 0 0
\(336\) 0 0
\(337\) 26.3923 1.43768 0.718840 0.695175i \(-0.244673\pi\)
0.718840 + 0.695175i \(0.244673\pi\)
\(338\) −1.73205 −0.0942111
\(339\) 0 0
\(340\) 0 0
\(341\) 48.2487 2.61281
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0.339746 0.0183179
\(345\) 0 0
\(346\) −14.7846 −0.794826
\(347\) −5.66025 −0.303858 −0.151929 0.988391i \(-0.548549\pi\)
−0.151929 + 0.988391i \(0.548549\pi\)
\(348\) 0 0
\(349\) −14.3923 −0.770402 −0.385201 0.922833i \(-0.625868\pi\)
−0.385201 + 0.922833i \(0.625868\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 24.5885 1.31057
\(353\) −27.7128 −1.47500 −0.737502 0.675345i \(-0.763995\pi\)
−0.737502 + 0.675345i \(0.763995\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.9282 0.685193
\(357\) 0 0
\(358\) −32.7846 −1.73272
\(359\) 2.19615 0.115908 0.0579542 0.998319i \(-0.481542\pi\)
0.0579542 + 0.998319i \(0.481542\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) −0.679492 −0.0357133
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −11.8038 −0.616156 −0.308078 0.951361i \(-0.599686\pi\)
−0.308078 + 0.951361i \(0.599686\pi\)
\(368\) −6.33975 −0.330482
\(369\) 0 0
\(370\) 0 0
\(371\) −20.7846 −1.07908
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 28.3923 1.46813
\(375\) 0 0
\(376\) 10.3923 0.535942
\(377\) −2.53590 −0.130605
\(378\) 0 0
\(379\) 18.9808 0.974976 0.487488 0.873130i \(-0.337913\pi\)
0.487488 + 0.873130i \(0.337913\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.78461 −0.449460
\(383\) 12.9282 0.660600 0.330300 0.943876i \(-0.392850\pi\)
0.330300 + 0.943876i \(0.392850\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −17.3205 −0.881591
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −4.39230 −0.222128
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) 22.3923 1.12811
\(395\) 0 0
\(396\) 0 0
\(397\) −28.7846 −1.44466 −0.722329 0.691549i \(-0.756928\pi\)
−0.722329 + 0.691549i \(0.756928\pi\)
\(398\) −34.6410 −1.73640
\(399\) 0 0
\(400\) 0 0
\(401\) 36.9282 1.84411 0.922053 0.387063i \(-0.126510\pi\)
0.922053 + 0.387063i \(0.126510\pi\)
\(402\) 0 0
\(403\) −10.1962 −0.507907
\(404\) 0.928203 0.0461798
\(405\) 0 0
\(406\) 8.78461 0.435973
\(407\) 18.9282 0.938236
\(408\) 0 0
\(409\) −17.6077 −0.870644 −0.435322 0.900275i \(-0.643366\pi\)
−0.435322 + 0.900275i \(0.643366\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.196152 0.00966374
\(413\) 18.2487 0.897960
\(414\) 0 0
\(415\) 0 0
\(416\) −5.19615 −0.254762
\(417\) 0 0
\(418\) 50.7846 2.48396
\(419\) 2.53590 0.123887 0.0619434 0.998080i \(-0.480270\pi\)
0.0619434 + 0.998080i \(0.480270\pi\)
\(420\) 0 0
\(421\) −30.7846 −1.50035 −0.750175 0.661239i \(-0.770031\pi\)
−0.750175 + 0.661239i \(0.770031\pi\)
\(422\) −13.8564 −0.674519
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 16.7846 0.812264
\(428\) 17.6603 0.853641
\(429\) 0 0
\(430\) 0 0
\(431\) 25.5167 1.22909 0.614547 0.788880i \(-0.289339\pi\)
0.614547 + 0.788880i \(0.289339\pi\)
\(432\) 0 0
\(433\) −34.7846 −1.67164 −0.835821 0.549002i \(-0.815008\pi\)
−0.835821 + 0.549002i \(0.815008\pi\)
\(434\) 35.3205 1.69544
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −7.85641 −0.375823
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −16.9808 −0.806780 −0.403390 0.915028i \(-0.632168\pi\)
−0.403390 + 0.915028i \(0.632168\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.46410 0.164030
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −20.5359 −0.969149 −0.484574 0.874750i \(-0.661026\pi\)
−0.484574 + 0.874750i \(0.661026\pi\)
\(450\) 0 0
\(451\) −16.3923 −0.771883
\(452\) 8.53590 0.401495
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) −10.7846 −0.504483 −0.252241 0.967664i \(-0.581168\pi\)
−0.252241 + 0.967664i \(0.581168\pi\)
\(458\) −11.0718 −0.517351
\(459\) 0 0
\(460\) 0 0
\(461\) 3.46410 0.161339 0.0806696 0.996741i \(-0.474294\pi\)
0.0806696 + 0.996741i \(0.474294\pi\)
\(462\) 0 0
\(463\) 2.39230 0.111180 0.0555899 0.998454i \(-0.482296\pi\)
0.0555899 + 0.998454i \(0.482296\pi\)
\(464\) −12.6795 −0.588631
\(465\) 0 0
\(466\) 10.3923 0.481414
\(467\) 27.8038 1.28661 0.643304 0.765611i \(-0.277563\pi\)
0.643304 + 0.765611i \(0.277563\pi\)
\(468\) 0 0
\(469\) 12.7846 0.590338
\(470\) 0 0
\(471\) 0 0
\(472\) −15.8038 −0.727431
\(473\) 0.928203 0.0426788
\(474\) 0 0
\(475\) 0 0
\(476\) 6.92820 0.317554
\(477\) 0 0
\(478\) −24.5885 −1.12465
\(479\) −35.6603 −1.62936 −0.814679 0.579912i \(-0.803087\pi\)
−0.814679 + 0.579912i \(0.803087\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 4.14359 0.188736
\(483\) 0 0
\(484\) 11.3923 0.517832
\(485\) 0 0
\(486\) 0 0
\(487\) 26.3923 1.19595 0.597975 0.801515i \(-0.295972\pi\)
0.597975 + 0.801515i \(0.295972\pi\)
\(488\) −14.5359 −0.658009
\(489\) 0 0
\(490\) 0 0
\(491\) 2.53590 0.114443 0.0572217 0.998361i \(-0.481776\pi\)
0.0572217 + 0.998361i \(0.481776\pi\)
\(492\) 0 0
\(493\) −8.78461 −0.395639
\(494\) −10.7321 −0.482858
\(495\) 0 0
\(496\) −50.9808 −2.28910
\(497\) 9.46410 0.424523
\(498\) 0 0
\(499\) −38.9808 −1.74502 −0.872509 0.488598i \(-0.837509\pi\)
−0.872509 + 0.488598i \(0.837509\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −37.1769 −1.65929
\(503\) 19.5167 0.870205 0.435102 0.900381i \(-0.356712\pi\)
0.435102 + 0.900381i \(0.356712\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −10.3923 −0.461994
\(507\) 0 0
\(508\) −16.1962 −0.718588
\(509\) −39.4641 −1.74922 −0.874608 0.484831i \(-0.838881\pi\)
−0.874608 + 0.484831i \(0.838881\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) −34.3923 −1.51698
\(515\) 0 0
\(516\) 0 0
\(517\) 28.3923 1.24869
\(518\) 13.8564 0.608816
\(519\) 0 0
\(520\) 0 0
\(521\) 28.3923 1.24389 0.621945 0.783061i \(-0.286343\pi\)
0.621945 + 0.783061i \(0.286343\pi\)
\(522\) 0 0
\(523\) 24.1962 1.05802 0.529012 0.848614i \(-0.322563\pi\)
0.529012 + 0.848614i \(0.322563\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.19615 −0.0957568
\(527\) −35.3205 −1.53859
\(528\) 0 0
\(529\) −21.3923 −0.930100
\(530\) 0 0
\(531\) 0 0
\(532\) 12.3923 0.537275
\(533\) 3.46410 0.150047
\(534\) 0 0
\(535\) 0 0
\(536\) −11.0718 −0.478229
\(537\) 0 0
\(538\) −34.3923 −1.48276
\(539\) −14.1962 −0.611472
\(540\) 0 0
\(541\) −26.3923 −1.13469 −0.567347 0.823479i \(-0.692030\pi\)
−0.567347 + 0.823479i \(0.692030\pi\)
\(542\) −53.6603 −2.30491
\(543\) 0 0
\(544\) −18.0000 −0.771744
\(545\) 0 0
\(546\) 0 0
\(547\) 12.1962 0.521470 0.260735 0.965410i \(-0.416035\pi\)
0.260735 + 0.965410i \(0.416035\pi\)
\(548\) 0.928203 0.0396509
\(549\) 0 0
\(550\) 0 0
\(551\) −15.7128 −0.669388
\(552\) 0 0
\(553\) 16.7846 0.713754
\(554\) −45.7128 −1.94215
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) 1.85641 0.0786585 0.0393292 0.999226i \(-0.487478\pi\)
0.0393292 + 0.999226i \(0.487478\pi\)
\(558\) 0 0
\(559\) −0.196152 −0.00829636
\(560\) 0 0
\(561\) 0 0
\(562\) 38.7846 1.63603
\(563\) −22.0526 −0.929405 −0.464702 0.885467i \(-0.653839\pi\)
−0.464702 + 0.885467i \(0.653839\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 56.4449 2.37255
\(567\) 0 0
\(568\) −8.19615 −0.343903
\(569\) 2.53590 0.106310 0.0531552 0.998586i \(-0.483072\pi\)
0.0531552 + 0.998586i \(0.483072\pi\)
\(570\) 0 0
\(571\) 36.3923 1.52297 0.761485 0.648182i \(-0.224470\pi\)
0.761485 + 0.648182i \(0.224470\pi\)
\(572\) −4.73205 −0.197857
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 8.66025 0.360219
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 49.1769 2.03670
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) 8.78461 0.362889
\(587\) −8.53590 −0.352314 −0.176157 0.984362i \(-0.556367\pi\)
−0.176157 + 0.984362i \(0.556367\pi\)
\(588\) 0 0
\(589\) −63.1769 −2.60316
\(590\) 0 0
\(591\) 0 0
\(592\) −20.0000 −0.821995
\(593\) 26.7846 1.09991 0.549956 0.835194i \(-0.314644\pi\)
0.549956 + 0.835194i \(0.314644\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.85641 0.321811
\(597\) 0 0
\(598\) 2.19615 0.0898074
\(599\) 7.60770 0.310842 0.155421 0.987848i \(-0.450327\pi\)
0.155421 + 0.987848i \(0.450327\pi\)
\(600\) 0 0
\(601\) 43.5692 1.77723 0.888613 0.458658i \(-0.151670\pi\)
0.888613 + 0.458658i \(0.151670\pi\)
\(602\) 0.679492 0.0276940
\(603\) 0 0
\(604\) −1.80385 −0.0733975
\(605\) 0 0
\(606\) 0 0
\(607\) −24.9808 −1.01394 −0.506969 0.861964i \(-0.669234\pi\)
−0.506969 + 0.861964i \(0.669234\pi\)
\(608\) −32.1962 −1.30573
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −32.5359 −1.31304
\(615\) 0 0
\(616\) −16.3923 −0.660465
\(617\) 33.7128 1.35723 0.678613 0.734496i \(-0.262581\pi\)
0.678613 + 0.734496i \(0.262581\pi\)
\(618\) 0 0
\(619\) 6.98076 0.280581 0.140290 0.990110i \(-0.455196\pi\)
0.140290 + 0.990110i \(0.455196\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −28.3923 −1.13843
\(623\) −25.8564 −1.03592
\(624\) 0 0
\(625\) 0 0
\(626\) −24.9282 −0.996331
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) −13.8564 −0.552491
\(630\) 0 0
\(631\) 5.80385 0.231048 0.115524 0.993305i \(-0.463145\pi\)
0.115524 + 0.993305i \(0.463145\pi\)
\(632\) −14.5359 −0.578207
\(633\) 0 0
\(634\) −41.5692 −1.65092
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) −20.7846 −0.822871
\(639\) 0 0
\(640\) 0 0
\(641\) −12.9282 −0.510633 −0.255317 0.966857i \(-0.582180\pi\)
−0.255317 + 0.966857i \(0.582180\pi\)
\(642\) 0 0
\(643\) 6.78461 0.267559 0.133779 0.991011i \(-0.457289\pi\)
0.133779 + 0.991011i \(0.457289\pi\)
\(644\) −2.53590 −0.0999284
\(645\) 0 0
\(646\) −37.1769 −1.46271
\(647\) 22.0526 0.866976 0.433488 0.901159i \(-0.357283\pi\)
0.433488 + 0.901159i \(0.357283\pi\)
\(648\) 0 0
\(649\) −43.1769 −1.69484
\(650\) 0 0
\(651\) 0 0
\(652\) 14.3923 0.563646
\(653\) 7.85641 0.307445 0.153722 0.988114i \(-0.450874\pi\)
0.153722 + 0.988114i \(0.450874\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) 0 0
\(658\) 20.7846 0.810268
\(659\) 21.4641 0.836123 0.418061 0.908419i \(-0.362710\pi\)
0.418061 + 0.908419i \(0.362710\pi\)
\(660\) 0 0
\(661\) 10.7846 0.419473 0.209736 0.977758i \(-0.432739\pi\)
0.209736 + 0.977758i \(0.432739\pi\)
\(662\) −4.48334 −0.174250
\(663\) 0 0
\(664\) −10.3923 −0.403300
\(665\) 0 0
\(666\) 0 0
\(667\) 3.21539 0.124500
\(668\) −0.928203 −0.0359133
\(669\) 0 0
\(670\) 0 0
\(671\) −39.7128 −1.53310
\(672\) 0 0
\(673\) 14.3923 0.554783 0.277391 0.960757i \(-0.410530\pi\)
0.277391 + 0.960757i \(0.410530\pi\)
\(674\) −45.7128 −1.76079
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) −83.5692 −3.20003
\(683\) 32.5359 1.24495 0.622476 0.782639i \(-0.286127\pi\)
0.622476 + 0.782639i \(0.286127\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) −0.980762 −0.0373912
\(689\) −10.3923 −0.395915
\(690\) 0 0
\(691\) −47.7654 −1.81708 −0.908540 0.417797i \(-0.862802\pi\)
−0.908540 + 0.417797i \(0.862802\pi\)
\(692\) 8.53590 0.324486
\(693\) 0 0
\(694\) 9.80385 0.372149
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 24.9282 0.943546
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −24.7846 −0.934769
\(704\) 4.73205 0.178346
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) −1.85641 −0.0698174
\(708\) 0 0
\(709\) 30.3923 1.14141 0.570703 0.821156i \(-0.306671\pi\)
0.570703 + 0.821156i \(0.306671\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 22.3923 0.839187
\(713\) 12.9282 0.484165
\(714\) 0 0
\(715\) 0 0
\(716\) 18.9282 0.707380
\(717\) 0 0
\(718\) −3.80385 −0.141958
\(719\) 25.8564 0.964281 0.482141 0.876094i \(-0.339859\pi\)
0.482141 + 0.876094i \(0.339859\pi\)
\(720\) 0 0
\(721\) −0.392305 −0.0146102
\(722\) −33.5885 −1.25003
\(723\) 0 0
\(724\) 0.392305 0.0145799
\(725\) 0 0
\(726\) 0 0
\(727\) −44.5885 −1.65369 −0.826847 0.562427i \(-0.809868\pi\)
−0.826847 + 0.562427i \(0.809868\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) 0 0
\(731\) −0.679492 −0.0251319
\(732\) 0 0
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 20.4449 0.754634
\(735\) 0 0
\(736\) 6.58846 0.242854
\(737\) −30.2487 −1.11423
\(738\) 0 0
\(739\) −18.1962 −0.669356 −0.334678 0.942332i \(-0.608628\pi\)
−0.334678 + 0.942332i \(0.608628\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) −16.1436 −0.592251 −0.296126 0.955149i \(-0.595695\pi\)
−0.296126 + 0.955149i \(0.595695\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −17.3205 −0.634149
\(747\) 0 0
\(748\) −16.3923 −0.599362
\(749\) −35.3205 −1.29058
\(750\) 0 0
\(751\) 36.3923 1.32797 0.663987 0.747744i \(-0.268863\pi\)
0.663987 + 0.747744i \(0.268863\pi\)
\(752\) −30.0000 −1.09399
\(753\) 0 0
\(754\) 4.39230 0.159958
\(755\) 0 0
\(756\) 0 0
\(757\) 2.39230 0.0869498 0.0434749 0.999055i \(-0.486157\pi\)
0.0434749 + 0.999055i \(0.486157\pi\)
\(758\) −32.8756 −1.19410
\(759\) 0 0
\(760\) 0 0
\(761\) 19.8564 0.719794 0.359897 0.932992i \(-0.382812\pi\)
0.359897 + 0.932992i \(0.382812\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 5.07180 0.183491
\(765\) 0 0
\(766\) −22.3923 −0.809067
\(767\) 9.12436 0.329461
\(768\) 0 0
\(769\) 34.7846 1.25437 0.627183 0.778872i \(-0.284208\pi\)
0.627183 + 0.778872i \(0.284208\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) 6.92820 0.249190 0.124595 0.992208i \(-0.460237\pi\)
0.124595 + 0.992208i \(0.460237\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.46410 −0.124354
\(777\) 0 0
\(778\) 10.3923 0.372582
\(779\) 21.4641 0.769031
\(780\) 0 0
\(781\) −22.3923 −0.801260
\(782\) 7.60770 0.272051
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) −31.5692 −1.12532 −0.562661 0.826688i \(-0.690222\pi\)
−0.562661 + 0.826688i \(0.690222\pi\)
\(788\) −12.9282 −0.460548
\(789\) 0 0
\(790\) 0 0
\(791\) −17.0718 −0.607003
\(792\) 0 0
\(793\) 8.39230 0.298019
\(794\) 49.8564 1.76934
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −40.6410 −1.43958 −0.719789 0.694193i \(-0.755762\pi\)
−0.719789 + 0.694193i \(0.755762\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) 0 0
\(802\) −63.9615 −2.25856
\(803\) 18.9282 0.667962
\(804\) 0 0
\(805\) 0 0
\(806\) 17.6603 0.622056
\(807\) 0 0
\(808\) 1.60770 0.0565585
\(809\) 2.53590 0.0891574 0.0445787 0.999006i \(-0.485805\pi\)
0.0445787 + 0.999006i \(0.485805\pi\)
\(810\) 0 0
\(811\) 17.8038 0.625178 0.312589 0.949889i \(-0.398804\pi\)
0.312589 + 0.949889i \(0.398804\pi\)
\(812\) −5.07180 −0.177985
\(813\) 0 0
\(814\) −32.7846 −1.14910
\(815\) 0 0
\(816\) 0 0
\(817\) −1.21539 −0.0425211
\(818\) 30.4974 1.06632
\(819\) 0 0
\(820\) 0 0
\(821\) −28.6410 −0.999578 −0.499789 0.866147i \(-0.666589\pi\)
−0.499789 + 0.866147i \(0.666589\pi\)
\(822\) 0 0
\(823\) 15.4115 0.537213 0.268606 0.963250i \(-0.413437\pi\)
0.268606 + 0.963250i \(0.413437\pi\)
\(824\) 0.339746 0.0118356
\(825\) 0 0
\(826\) −31.6077 −1.09977
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 0 0
\(829\) 0.392305 0.0136253 0.00681266 0.999977i \(-0.497831\pi\)
0.00681266 + 0.999977i \(0.497831\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 10.3923 0.360072
\(834\) 0 0
\(835\) 0 0
\(836\) −29.3205 −1.01407
\(837\) 0 0
\(838\) −4.39230 −0.151730
\(839\) −0.339746 −0.0117293 −0.00586467 0.999983i \(-0.501867\pi\)
−0.00586467 + 0.999983i \(0.501867\pi\)
\(840\) 0 0
\(841\) −22.5692 −0.778249
\(842\) 53.3205 1.83755
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) −22.7846 −0.782888
\(848\) −51.9615 −1.78437
\(849\) 0 0
\(850\) 0 0
\(851\) 5.07180 0.173859
\(852\) 0 0
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) −29.0718 −0.994816
\(855\) 0 0
\(856\) 30.5885 1.04549
\(857\) −35.5692 −1.21502 −0.607511 0.794311i \(-0.707832\pi\)
−0.607511 + 0.794311i \(0.707832\pi\)
\(858\) 0 0
\(859\) −17.1769 −0.586069 −0.293034 0.956102i \(-0.594665\pi\)
−0.293034 + 0.956102i \(0.594665\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −44.1962 −1.50533
\(863\) −38.7846 −1.32024 −0.660122 0.751159i \(-0.729495\pi\)
−0.660122 + 0.751159i \(0.729495\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 60.2487 2.04733
\(867\) 0 0
\(868\) −20.3923 −0.692160
\(869\) −39.7128 −1.34716
\(870\) 0 0
\(871\) 6.39230 0.216595
\(872\) 3.46410 0.117309
\(873\) 0 0
\(874\) 13.6077 0.460287
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −55.4256 −1.87052
\(879\) 0 0
\(880\) 0 0
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 0 0
\(883\) −23.8038 −0.801063 −0.400532 0.916283i \(-0.631175\pi\)
−0.400532 + 0.916283i \(0.631175\pi\)
\(884\) 3.46410 0.116510
\(885\) 0 0
\(886\) 29.4115 0.988100
\(887\) −47.9090 −1.60863 −0.804313 0.594206i \(-0.797466\pi\)
−0.804313 + 0.594206i \(0.797466\pi\)
\(888\) 0 0
\(889\) 32.3923 1.08640
\(890\) 0 0
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) −37.1769 −1.24408
\(894\) 0 0
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) 35.5692 1.18696
\(899\) 25.8564 0.862359
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 28.3923 0.945360
\(903\) 0 0
\(904\) 14.7846 0.491729
\(905\) 0 0
\(906\) 0 0
\(907\) 53.7654 1.78525 0.892625 0.450800i \(-0.148861\pi\)
0.892625 + 0.450800i \(0.148861\pi\)
\(908\) −3.46410 −0.114960
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −28.3923 −0.939648
\(914\) 18.6795 0.617863
\(915\) 0 0
\(916\) 6.39230 0.211208
\(917\) 0 0
\(918\) 0 0
\(919\) 9.17691 0.302718 0.151359 0.988479i \(-0.451635\pi\)
0.151359 + 0.988479i \(0.451635\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 4.73205 0.155757
\(924\) 0 0
\(925\) 0 0
\(926\) −4.14359 −0.136167
\(927\) 0 0
\(928\) 13.1769 0.432553
\(929\) −44.5359 −1.46118 −0.730588 0.682819i \(-0.760754\pi\)
−0.730588 + 0.682819i \(0.760754\pi\)
\(930\) 0 0
\(931\) 18.5885 0.609212
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −48.1577 −1.57577
\(935\) 0 0
\(936\) 0 0
\(937\) −34.7846 −1.13636 −0.568182 0.822903i \(-0.692353\pi\)
−0.568182 + 0.822903i \(0.692353\pi\)
\(938\) −22.1436 −0.723014
\(939\) 0 0
\(940\) 0 0
\(941\) −31.1769 −1.01634 −0.508169 0.861257i \(-0.669678\pi\)
−0.508169 + 0.861257i \(0.669678\pi\)
\(942\) 0 0
\(943\) −4.39230 −0.143033
\(944\) 45.6218 1.48486
\(945\) 0 0
\(946\) −1.60770 −0.0522707
\(947\) 40.6410 1.32066 0.660328 0.750978i \(-0.270417\pi\)
0.660328 + 0.750978i \(0.270417\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 0.928203 0.0300675 0.0150337 0.999887i \(-0.495214\pi\)
0.0150337 + 0.999887i \(0.495214\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.1962 0.459136
\(957\) 0 0
\(958\) 61.7654 1.99555
\(959\) −1.85641 −0.0599465
\(960\) 0 0
\(961\) 72.9615 2.35360
\(962\) 6.92820 0.223374
\(963\) 0 0
\(964\) −2.39230 −0.0770510
\(965\) 0 0
\(966\) 0 0
\(967\) 50.3923 1.62051 0.810254 0.586079i \(-0.199329\pi\)
0.810254 + 0.586079i \(0.199329\pi\)
\(968\) 19.7321 0.634212
\(969\) 0 0
\(970\) 0 0
\(971\) −18.9282 −0.607435 −0.303717 0.952762i \(-0.598228\pi\)
−0.303717 + 0.952762i \(0.598228\pi\)
\(972\) 0 0
\(973\) −24.7846 −0.794558
\(974\) −45.7128 −1.46473
\(975\) 0 0
\(976\) 41.9615 1.34316
\(977\) −15.7128 −0.502697 −0.251349 0.967897i \(-0.580874\pi\)
−0.251349 + 0.967897i \(0.580874\pi\)
\(978\) 0 0
\(979\) 61.1769 1.95522
\(980\) 0 0
\(981\) 0 0
\(982\) −4.39230 −0.140164
\(983\) −34.3923 −1.09694 −0.548472 0.836169i \(-0.684790\pi\)
−0.548472 + 0.836169i \(0.684790\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 15.2154 0.484557
\(987\) 0 0
\(988\) 6.19615 0.197126
\(989\) 0.248711 0.00790856
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 52.9808 1.68214
\(993\) 0 0
\(994\) −16.3923 −0.519932
\(995\) 0 0
\(996\) 0 0
\(997\) −33.6077 −1.06437 −0.532183 0.846629i \(-0.678628\pi\)
−0.532183 + 0.846629i \(0.678628\pi\)
\(998\) 67.5167 2.13720
\(999\) 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 2925.2.a.z.1.1 2
3.2 odd 2 325.2.a.g.1.2 2
5.2 odd 4 2925.2.c.v.2224.1 4
5.3 odd 4 2925.2.c.v.2224.4 4
5.4 even 2 585.2.a.k.1.2 2
12.11 even 2 5200.2.a.ca.1.2 2
15.2 even 4 325.2.b.e.274.4 4
15.8 even 4 325.2.b.e.274.1 4
15.14 odd 2 65.2.a.c.1.1 2
20.19 odd 2 9360.2.a.cm.1.1 2
39.38 odd 2 4225.2.a.w.1.1 2
60.59 even 2 1040.2.a.h.1.1 2
65.64 even 2 7605.2.a.be.1.1 2
105.104 even 2 3185.2.a.k.1.1 2
120.29 odd 2 4160.2.a.y.1.1 2
120.59 even 2 4160.2.a.bj.1.2 2
165.164 even 2 7865.2.a.h.1.2 2
195.29 odd 6 845.2.e.e.191.2 4
195.44 even 4 845.2.c.e.506.4 4
195.59 even 12 845.2.m.a.361.1 4
195.74 odd 6 845.2.e.e.146.2 4
195.89 even 12 845.2.m.c.316.1 4
195.119 even 12 845.2.m.a.316.1 4
195.134 odd 6 845.2.e.f.146.1 4
195.149 even 12 845.2.m.c.361.1 4
195.164 even 4 845.2.c.e.506.2 4
195.179 odd 6 845.2.e.f.191.1 4
195.194 odd 2 845.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.1 2 15.14 odd 2
325.2.a.g.1.2 2 3.2 odd 2
325.2.b.e.274.1 4 15.8 even 4
325.2.b.e.274.4 4 15.2 even 4
585.2.a.k.1.2 2 5.4 even 2
845.2.a.d.1.2 2 195.194 odd 2
845.2.c.e.506.2 4 195.164 even 4
845.2.c.e.506.4 4 195.44 even 4
845.2.e.e.146.2 4 195.74 odd 6
845.2.e.e.191.2 4 195.29 odd 6
845.2.e.f.146.1 4 195.134 odd 6
845.2.e.f.191.1 4 195.179 odd 6
845.2.m.a.316.1 4 195.119 even 12
845.2.m.a.361.1 4 195.59 even 12
845.2.m.c.316.1 4 195.89 even 12
845.2.m.c.361.1 4 195.149 even 12
1040.2.a.h.1.1 2 60.59 even 2
2925.2.a.z.1.1 2 1.1 even 1 trivial
2925.2.c.v.2224.1 4 5.2 odd 4
2925.2.c.v.2224.4 4 5.3 odd 4
3185.2.a.k.1.1 2 105.104 even 2
4160.2.a.y.1.1 2 120.29 odd 2
4160.2.a.bj.1.2 2 120.59 even 2
4225.2.a.w.1.1 2 39.38 odd 2
5200.2.a.ca.1.2 2 12.11 even 2
7605.2.a.be.1.1 2 65.64 even 2
7865.2.a.h.1.2 2 165.164 even 2
9360.2.a.cm.1.1 2 20.19 odd 2