Properties

Label 378.2.d.b.377.3
Level $378$
Weight $2$
Character 378.377
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(377,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 377.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 378.377
Dual form 378.2.d.b.377.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -1.73205i q^{10} +3.00000i q^{11} +3.46410i q^{13} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -6.92820 q^{17} +1.73205i q^{19} +1.73205 q^{20} -3.00000 q^{22} +3.00000i q^{23} -2.00000 q^{25} -3.46410 q^{26} +(-2.00000 - 1.73205i) q^{28} +6.00000i q^{29} -5.19615i q^{31} +1.00000i q^{32} -6.92820i q^{34} +(-3.46410 - 3.00000i) q^{35} +7.00000 q^{37} -1.73205 q^{38} +1.73205i q^{40} +12.1244 q^{41} -2.00000 q^{43} -3.00000i q^{44} -3.00000 q^{46} +3.46410 q^{47} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} -3.46410i q^{52} -12.0000i q^{53} -5.19615i q^{55} +(1.73205 - 2.00000i) q^{56} -6.00000 q^{58} -3.46410 q^{59} +6.92820i q^{61} +5.19615 q^{62} -1.00000 q^{64} -6.00000i q^{65} +2.00000 q^{67} +6.92820 q^{68} +(3.00000 - 3.46410i) q^{70} -3.00000i q^{71} -3.46410i q^{73} +7.00000i q^{74} -1.73205i q^{76} +(-5.19615 + 6.00000i) q^{77} -10.0000 q^{79} -1.73205 q^{80} +12.1244i q^{82} +17.3205 q^{83} +12.0000 q^{85} -2.00000i q^{86} +3.00000 q^{88} +5.19615 q^{89} +(-6.00000 + 6.92820i) q^{91} -3.00000i q^{92} +3.46410i q^{94} -3.00000i q^{95} +13.8564i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} - 8 q^{28} + 28 q^{37} - 8 q^{43} - 12 q^{46} + 4 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} + 12 q^{70} - 40 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 1.73205i 0.547723i
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) −1.73205 + 2.00000i −0.462910 + 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 1.73205 0.387298
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) −2.00000 1.73205i −0.377964 0.327327i
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i −0.884454 0.466628i \(-0.845469\pi\)
0.884454 0.466628i \(-0.154531\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 6.92820i 1.18818i
\(35\) −3.46410 3.00000i −0.585540 0.507093i
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −1.73205 −0.280976
\(39\) 0 0
\(40\) 1.73205i 0.273861i
\(41\) 12.1244 1.89351 0.946753 0.321960i \(-0.104342\pi\)
0.946753 + 0.321960i \(0.104342\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 3.00000i 0.452267i
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 2.00000i 0.282843i
\(51\) 0 0
\(52\) 3.46410i 0.480384i
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 5.19615i 0.700649i
\(56\) 1.73205 2.00000i 0.231455 0.267261i
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 5.19615 0.659912
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.92820 0.840168
\(69\) 0 0
\(70\) 3.00000 3.46410i 0.358569 0.414039i
\(71\) 3.00000i 0.356034i −0.984027 0.178017i \(-0.943032\pi\)
0.984027 0.178017i \(-0.0569683\pi\)
\(72\) 0 0
\(73\) 3.46410i 0.405442i −0.979236 0.202721i \(-0.935021\pi\)
0.979236 0.202721i \(-0.0649785\pi\)
\(74\) 7.00000i 0.813733i
\(75\) 0 0
\(76\) 1.73205i 0.198680i
\(77\) −5.19615 + 6.00000i −0.592157 + 0.683763i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.73205 −0.193649
\(81\) 0 0
\(82\) 12.1244i 1.33891i
\(83\) 17.3205 1.90117 0.950586 0.310460i \(-0.100483\pi\)
0.950586 + 0.310460i \(0.100483\pi\)
\(84\) 0 0
\(85\) 12.0000 1.30158
\(86\) 2.00000i 0.215666i
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −6.00000 + 6.92820i −0.628971 + 0.726273i
\(92\) 3.00000i 0.312772i
\(93\) 0 0
\(94\) 3.46410i 0.357295i
\(95\) 3.00000i 0.307794i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −6.92820 + 1.00000i −0.699854 + 0.101015i
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 19.0526i 1.87730i −0.344865 0.938652i \(-0.612075\pi\)
0.344865 0.938652i \(-0.387925\pi\)
\(104\) 3.46410 0.339683
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 5.19615 0.495434
\(111\) 0 0
\(112\) 2.00000 + 1.73205i 0.188982 + 0.163663i
\(113\) 12.0000i 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) 5.19615i 0.484544i
\(116\) 6.00000i 0.557086i
\(117\) 0 0
\(118\) 3.46410i 0.318896i
\(119\) −13.8564 12.0000i −1.27021 1.10004i
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) −6.92820 −0.627250
\(123\) 0 0
\(124\) 5.19615i 0.466628i
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) 10.3923 0.907980 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(132\) 0 0
\(133\) −3.00000 + 3.46410i −0.260133 + 0.300376i
\(134\) 2.00000i 0.172774i
\(135\) 0 0
\(136\) 6.92820i 0.594089i
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i −0.989150 0.146911i \(-0.953067\pi\)
0.989150 0.146911i \(-0.0469330\pi\)
\(140\) 3.46410 + 3.00000i 0.292770 + 0.253546i
\(141\) 0 0
\(142\) 3.00000 0.251754
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 3.46410 0.286691
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 1.73205 0.140488
\(153\) 0 0
\(154\) −6.00000 5.19615i −0.483494 0.418718i
\(155\) 9.00000i 0.722897i
\(156\) 0 0
\(157\) 10.3923i 0.829396i −0.909959 0.414698i \(-0.863887\pi\)
0.909959 0.414698i \(-0.136113\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 0 0
\(160\) 1.73205i 0.136931i
\(161\) −5.19615 + 6.00000i −0.409514 + 0.472866i
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −12.1244 −0.946753
\(165\) 0 0
\(166\) 17.3205i 1.34433i
\(167\) −6.92820 −0.536120 −0.268060 0.963402i \(-0.586383\pi\)
−0.268060 + 0.963402i \(0.586383\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 12.0000i 0.920358i
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −5.19615 −0.395056 −0.197528 0.980297i \(-0.563291\pi\)
−0.197528 + 0.980297i \(0.563291\pi\)
\(174\) 0 0
\(175\) −4.00000 3.46410i −0.302372 0.261861i
\(176\) 3.00000i 0.226134i
\(177\) 0 0
\(178\) 5.19615i 0.389468i
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) −6.92820 6.00000i −0.513553 0.444750i
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) −12.1244 −0.891400
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) −3.46410 −0.252646
\(189\) 0 0
\(190\) 3.00000 0.217643
\(191\) 21.0000i 1.51951i −0.650211 0.759753i \(-0.725320\pi\)
0.650211 0.759753i \(-0.274680\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −13.8564 −0.994832
\(195\) 0 0
\(196\) −1.00000 6.92820i −0.0714286 0.494872i
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) 1.73205i 0.122782i −0.998114 0.0613909i \(-0.980446\pi\)
0.998114 0.0613909i \(-0.0195536\pi\)
\(200\) 2.00000i 0.141421i
\(201\) 0 0
\(202\) 0 0
\(203\) −10.3923 + 12.0000i −0.729397 + 0.842235i
\(204\) 0 0
\(205\) −21.0000 −1.46670
\(206\) 19.0526 1.32745
\(207\) 0 0
\(208\) 3.46410i 0.240192i
\(209\) −5.19615 −0.359425
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 0 0
\(214\) 0 0
\(215\) 3.46410 0.236250
\(216\) 0 0
\(217\) 9.00000 10.3923i 0.610960 0.705476i
\(218\) 13.0000i 0.880471i
\(219\) 0 0
\(220\) 5.19615i 0.350325i
\(221\) 24.0000i 1.61441i
\(222\) 0 0
\(223\) 15.5885i 1.04388i −0.852982 0.521940i \(-0.825208\pi\)
0.852982 0.521940i \(-0.174792\pi\)
\(224\) −1.73205 + 2.00000i −0.115728 + 0.133631i
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −20.7846 −1.37952 −0.689761 0.724037i \(-0.742285\pi\)
−0.689761 + 0.724037i \(0.742285\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 5.19615 0.342624
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 3.46410 0.225494
\(237\) 0 0
\(238\) 12.0000 13.8564i 0.777844 0.898177i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 17.3205i 1.11571i 0.829938 + 0.557856i \(0.188376\pi\)
−0.829938 + 0.557856i \(0.811624\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 0 0
\(244\) 6.92820i 0.443533i
\(245\) −1.73205 12.0000i −0.110657 0.766652i
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) −5.19615 −0.329956
\(249\) 0 0
\(250\) 12.1244i 0.766812i
\(251\) 13.8564 0.874609 0.437304 0.899314i \(-0.355933\pi\)
0.437304 + 0.899314i \(0.355933\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 22.0000i 1.38040i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.19615 −0.324127 −0.162064 0.986780i \(-0.551815\pi\)
−0.162064 + 0.986780i \(0.551815\pi\)
\(258\) 0 0
\(259\) 14.0000 + 12.1244i 0.869918 + 0.753371i
\(260\) 6.00000i 0.372104i
\(261\) 0 0
\(262\) 10.3923i 0.642039i
\(263\) 27.0000i 1.66489i 0.554107 + 0.832446i \(0.313060\pi\)
−0.554107 + 0.832446i \(0.686940\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) −3.46410 3.00000i −0.212398 0.183942i
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 1.73205 0.105605 0.0528025 0.998605i \(-0.483185\pi\)
0.0528025 + 0.998605i \(0.483185\pi\)
\(270\) 0 0
\(271\) 17.3205i 1.05215i −0.850439 0.526073i \(-0.823664\pi\)
0.850439 0.526073i \(-0.176336\pi\)
\(272\) −6.92820 −0.420084
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 6.00000i 0.361814i
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 3.46410 0.207763
\(279\) 0 0
\(280\) −3.00000 + 3.46410i −0.179284 + 0.207020i
\(281\) 30.0000i 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) 31.1769i 1.85328i 0.375956 + 0.926638i \(0.377314\pi\)
−0.375956 + 0.926638i \(0.622686\pi\)
\(284\) 3.00000i 0.178017i
\(285\) 0 0
\(286\) 10.3923i 0.614510i
\(287\) 24.2487 + 21.0000i 1.43136 + 1.23959i
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 10.3923 0.610257
\(291\) 0 0
\(292\) 3.46410i 0.202721i
\(293\) 6.92820 0.404750 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 7.00000i 0.406867i
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) −10.3923 −0.601003
\(300\) 0 0
\(301\) −4.00000 3.46410i −0.230556 0.199667i
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) 1.73205i 0.0993399i
\(305\) 12.0000i 0.687118i
\(306\) 0 0
\(307\) 15.5885i 0.889680i 0.895610 + 0.444840i \(0.146740\pi\)
−0.895610 + 0.444840i \(0.853260\pi\)
\(308\) 5.19615 6.00000i 0.296078 0.341882i
\(309\) 0 0
\(310\) −9.00000 −0.511166
\(311\) 24.2487 1.37502 0.687509 0.726176i \(-0.258704\pi\)
0.687509 + 0.726176i \(0.258704\pi\)
\(312\) 0 0
\(313\) 17.3205i 0.979013i −0.872000 0.489506i \(-0.837177\pi\)
0.872000 0.489506i \(-0.162823\pi\)
\(314\) 10.3923 0.586472
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 1.73205 0.0968246
\(321\) 0 0
\(322\) −6.00000 5.19615i −0.334367 0.289570i
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 6.92820i 0.384308i
\(326\) 2.00000i 0.110770i
\(327\) 0 0
\(328\) 12.1244i 0.669456i
\(329\) 6.92820 + 6.00000i 0.381964 + 0.330791i
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −17.3205 −0.950586
\(333\) 0 0
\(334\) 6.92820i 0.379094i
\(335\) −3.46410 −0.189264
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) 15.5885 0.844162
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 2.00000i 0.107833i
\(345\) 0 0
\(346\) 5.19615i 0.279347i
\(347\) 27.0000i 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) 27.7128i 1.48343i 0.670714 + 0.741716i \(0.265988\pi\)
−0.670714 + 0.741716i \(0.734012\pi\)
\(350\) 3.46410 4.00000i 0.185164 0.213809i
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 19.0526 1.01407 0.507033 0.861927i \(-0.330742\pi\)
0.507033 + 0.861927i \(0.330742\pi\)
\(354\) 0 0
\(355\) 5.19615i 0.275783i
\(356\) −5.19615 −0.275396
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) −13.8564 −0.728277
\(363\) 0 0
\(364\) 6.00000 6.92820i 0.314485 0.363137i
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) 1.73205i 0.0904123i −0.998978 0.0452062i \(-0.985606\pi\)
0.998978 0.0452062i \(-0.0143945\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 12.1244i 0.630315i
\(371\) 20.7846 24.0000i 1.07908 1.24602i
\(372\) 0 0
\(373\) 5.00000 0.258890 0.129445 0.991587i \(-0.458680\pi\)
0.129445 + 0.991587i \(0.458680\pi\)
\(374\) 20.7846 1.07475
\(375\) 0 0
\(376\) 3.46410i 0.178647i
\(377\) −20.7846 −1.07046
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 3.00000i 0.153897i
\(381\) 0 0
\(382\) 21.0000 1.07445
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 0 0
\(385\) 9.00000 10.3923i 0.458682 0.529641i
\(386\) 14.0000i 0.712581i
\(387\) 0 0
\(388\) 13.8564i 0.703452i
\(389\) 24.0000i 1.21685i 0.793612 + 0.608424i \(0.208198\pi\)
−0.793612 + 0.608424i \(0.791802\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 6.92820 1.00000i 0.349927 0.0505076i
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 17.3205 0.871489
\(396\) 0 0
\(397\) 20.7846i 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 1.73205 0.0868199
\(399\) 0 0
\(400\) −2.00000 −0.100000
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 0 0
\(406\) −12.0000 10.3923i −0.595550 0.515761i
\(407\) 21.0000i 1.04093i
\(408\) 0 0
\(409\) 3.46410i 0.171289i −0.996326 0.0856444i \(-0.972705\pi\)
0.996326 0.0856444i \(-0.0272949\pi\)
\(410\) 21.0000i 1.03712i
\(411\) 0 0
\(412\) 19.0526i 0.938652i
\(413\) −6.92820 6.00000i −0.340915 0.295241i
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) −3.46410 −0.169842
\(417\) 0 0
\(418\) 5.19615i 0.254152i
\(419\) −6.92820 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 16.0000i 0.778868i
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 13.8564 0.672134
\(426\) 0 0
\(427\) −12.0000 + 13.8564i −0.580721 + 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 3.46410i 0.167054i
\(431\) 21.0000i 1.01153i −0.862670 0.505767i \(-0.831209\pi\)
0.862670 0.505767i \(-0.168791\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 10.3923 + 9.00000i 0.498847 + 0.432014i
\(435\) 0 0
\(436\) −13.0000 −0.622587
\(437\) −5.19615 −0.248566
\(438\) 0 0
\(439\) 10.3923i 0.495998i −0.968760 0.247999i \(-0.920227\pi\)
0.968760 0.247999i \(-0.0797729\pi\)
\(440\) −5.19615 −0.247717
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 9.00000i 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 15.5885 0.738135
\(447\) 0 0
\(448\) −2.00000 1.73205i −0.0944911 0.0818317i
\(449\) 12.0000i 0.566315i −0.959073 0.283158i \(-0.908618\pi\)
0.959073 0.283158i \(-0.0913819\pi\)
\(450\) 0 0
\(451\) 36.3731i 1.71274i
\(452\) 12.0000i 0.564433i
\(453\) 0 0
\(454\) 20.7846i 0.975470i
\(455\) 10.3923 12.0000i 0.487199 0.562569i
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 5.19615i 0.242272i
\(461\) −19.0526 −0.887366 −0.443683 0.896184i \(-0.646328\pi\)
−0.443683 + 0.896184i \(0.646328\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 6.00000i 0.278543i
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −31.1769 −1.44270 −0.721348 0.692573i \(-0.756477\pi\)
−0.721348 + 0.692573i \(0.756477\pi\)
\(468\) 0 0
\(469\) 4.00000 + 3.46410i 0.184703 + 0.159957i
\(470\) 6.00000i 0.276759i
\(471\) 0 0
\(472\) 3.46410i 0.159448i
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 3.46410i 0.158944i
\(476\) 13.8564 + 12.0000i 0.635107 + 0.550019i
\(477\) 0 0
\(478\) 0 0
\(479\) 27.7128 1.26623 0.633115 0.774057i \(-0.281776\pi\)
0.633115 + 0.774057i \(0.281776\pi\)
\(480\) 0 0
\(481\) 24.2487i 1.10565i
\(482\) −17.3205 −0.788928
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 24.0000i 1.08978i
\(486\) 0 0
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 6.92820 0.313625
\(489\) 0 0
\(490\) 12.0000 1.73205i 0.542105 0.0782461i
\(491\) 9.00000i 0.406164i 0.979162 + 0.203082i \(0.0650959\pi\)
−0.979162 + 0.203082i \(0.934904\pi\)
\(492\) 0 0
\(493\) 41.5692i 1.87218i
\(494\) 6.00000i 0.269953i
\(495\) 0 0
\(496\) 5.19615i 0.233314i
\(497\) 5.19615 6.00000i 0.233079 0.269137i
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) −12.1244 −0.542218
\(501\) 0 0
\(502\) 13.8564i 0.618442i
\(503\) −24.2487 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.00000i 0.400099i
\(507\) 0 0
\(508\) 22.0000 0.976092
\(509\) −13.8564 −0.614174 −0.307087 0.951681i \(-0.599354\pi\)
−0.307087 + 0.951681i \(0.599354\pi\)
\(510\) 0 0
\(511\) 6.00000 6.92820i 0.265424 0.306486i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 5.19615i 0.229192i
\(515\) 33.0000i 1.45415i
\(516\) 0 0
\(517\) 10.3923i 0.457053i
\(518\) −12.1244 + 14.0000i −0.532714 + 0.615125i
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 22.5167 0.986473 0.493236 0.869895i \(-0.335814\pi\)
0.493236 + 0.869895i \(0.335814\pi\)
\(522\) 0 0
\(523\) 39.8372i 1.74196i 0.491320 + 0.870979i \(0.336514\pi\)
−0.491320 + 0.870979i \(0.663486\pi\)
\(524\) −10.3923 −0.453990
\(525\) 0 0
\(526\) −27.0000 −1.17726
\(527\) 36.0000i 1.56818i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) −20.7846 −0.902826
\(531\) 0 0
\(532\) 3.00000 3.46410i 0.130066 0.150188i
\(533\) 42.0000i 1.81922i
\(534\) 0 0
\(535\) 0 0
\(536\) 2.00000i 0.0863868i
\(537\) 0 0
\(538\) 1.73205i 0.0746740i
\(539\) −20.7846 + 3.00000i −0.895257 + 0.129219i
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 17.3205 0.743980
\(543\) 0 0
\(544\) 6.92820i 0.297044i
\(545\) −22.5167 −0.964508
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 0 0
\(550\) 6.00000 0.255841
\(551\) −10.3923 −0.442727
\(552\) 0 0
\(553\) −20.0000 17.3205i −0.850487 0.736543i
\(554\) 11.0000i 0.467345i
\(555\) 0 0
\(556\) 3.46410i 0.146911i
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) −3.46410 3.00000i −0.146385 0.126773i
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −10.3923 −0.437983 −0.218992 0.975727i \(-0.570277\pi\)
−0.218992 + 0.975727i \(0.570277\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) −31.1769 −1.31046
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 10.3923 0.434524
\(573\) 0 0
\(574\) −21.0000 + 24.2487i −0.876523 + 1.01212i
\(575\) 6.00000i 0.250217i
\(576\) 0 0
\(577\) 34.6410i 1.44212i −0.692870 0.721062i \(-0.743654\pi\)
0.692870 0.721062i \(-0.256346\pi\)
\(578\) 31.0000i 1.28943i
\(579\) 0 0
\(580\) 10.3923i 0.431517i
\(581\) 34.6410 + 30.0000i 1.43715 + 1.24461i
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) −3.46410 −0.143346
\(585\) 0 0
\(586\) 6.92820i 0.286201i
\(587\) −41.5692 −1.71575 −0.857873 0.513862i \(-0.828214\pi\)
−0.857873 + 0.513862i \(0.828214\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 6.00000i 0.247016i
\(591\) 0 0
\(592\) 7.00000 0.287698
\(593\) −19.0526 −0.782395 −0.391197 0.920307i \(-0.627939\pi\)
−0.391197 + 0.920307i \(0.627939\pi\)
\(594\) 0 0
\(595\) 24.0000 + 20.7846i 0.983904 + 0.852086i
\(596\) 12.0000i 0.491539i
\(597\) 0 0
\(598\) 10.3923i 0.424973i
\(599\) 9.00000i 0.367730i −0.982952 0.183865i \(-0.941139\pi\)
0.982952 0.183865i \(-0.0588609\pi\)
\(600\) 0 0
\(601\) 10.3923i 0.423911i −0.977279 0.211955i \(-0.932017\pi\)
0.977279 0.211955i \(-0.0679832\pi\)
\(602\) 3.46410 4.00000i 0.141186 0.163028i
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −3.46410 −0.140836
\(606\) 0 0
\(607\) 17.3205i 0.703018i −0.936185 0.351509i \(-0.885669\pi\)
0.936185 0.351509i \(-0.114331\pi\)
\(608\) −1.73205 −0.0702439
\(609\) 0 0
\(610\) 12.0000 0.485866
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) −19.0000 −0.767403 −0.383701 0.923457i \(-0.625351\pi\)
−0.383701 + 0.923457i \(0.625351\pi\)
\(614\) −15.5885 −0.629099
\(615\) 0 0
\(616\) 6.00000 + 5.19615i 0.241747 + 0.209359i
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 19.0526i 0.765787i 0.923792 + 0.382893i \(0.125072\pi\)
−0.923792 + 0.382893i \(0.874928\pi\)
\(620\) 9.00000i 0.361449i
\(621\) 0 0
\(622\) 24.2487i 0.972285i
\(623\) 10.3923 + 9.00000i 0.416359 + 0.360577i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 17.3205 0.692267
\(627\) 0 0
\(628\) 10.3923i 0.414698i
\(629\) −48.4974 −1.93372
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 38.1051 1.51216
\(636\) 0 0
\(637\) −24.0000 + 3.46410i −0.950915 + 0.137253i
\(638\) 18.0000i 0.712627i
\(639\) 0 0
\(640\) 1.73205i 0.0684653i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 5.19615i 0.204916i −0.994737 0.102458i \(-0.967329\pi\)
0.994737 0.102458i \(-0.0326708\pi\)
\(644\) 5.19615 6.00000i 0.204757 0.236433i
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −10.3923 −0.408564 −0.204282 0.978912i \(-0.565486\pi\)
−0.204282 + 0.978912i \(0.565486\pi\)
\(648\) 0 0
\(649\) 10.3923i 0.407934i
\(650\) 6.92820 0.271746
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 18.0000i 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 12.1244 0.473377
\(657\) 0 0
\(658\) −6.00000 + 6.92820i −0.233904 + 0.270089i
\(659\) 21.0000i 0.818044i −0.912525 0.409022i \(-0.865870\pi\)
0.912525 0.409022i \(-0.134130\pi\)
\(660\) 0 0
\(661\) 38.1051i 1.48212i −0.671440 0.741059i \(-0.734324\pi\)
0.671440 0.741059i \(-0.265676\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 0 0
\(664\) 17.3205i 0.672166i
\(665\) 5.19615 6.00000i 0.201498 0.232670i
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 6.92820 0.268060
\(669\) 0 0
\(670\) 3.46410i 0.133830i
\(671\) −20.7846 −0.802381
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 5.00000i 0.192593i
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 25.9808 0.998522 0.499261 0.866452i \(-0.333605\pi\)
0.499261 + 0.866452i \(0.333605\pi\)
\(678\) 0 0
\(679\) −24.0000 + 27.7128i −0.921035 + 1.06352i
\(680\) 12.0000i 0.460179i
\(681\) 0 0
\(682\) 15.5885i 0.596913i
\(683\) 33.0000i 1.26271i −0.775494 0.631355i \(-0.782499\pi\)
0.775494 0.631355i \(-0.217501\pi\)
\(684\) 0 0
\(685\) 31.1769i 1.19121i
\(686\) −15.5885 10.0000i −0.595170 0.381802i
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) 41.5692 1.58366
\(690\) 0 0
\(691\) 10.3923i 0.395342i 0.980268 + 0.197671i \(0.0633378\pi\)
−0.980268 + 0.197671i \(0.936662\pi\)
\(692\) 5.19615 0.197528
\(693\) 0 0
\(694\) 27.0000 1.02491
\(695\) 6.00000i 0.227593i
\(696\) 0 0
\(697\) −84.0000 −3.18173
\(698\) −27.7128 −1.04895
\(699\) 0 0
\(700\) 4.00000 + 3.46410i 0.151186 + 0.130931i
\(701\) 6.00000i 0.226617i −0.993560 0.113308i \(-0.963855\pi\)
0.993560 0.113308i \(-0.0361448\pi\)
\(702\) 0 0
\(703\) 12.1244i 0.457279i
\(704\) 3.00000i 0.113067i
\(705\) 0 0
\(706\) 19.0526i 0.717053i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) −5.19615 −0.195008
\(711\) 0 0
\(712\) 5.19615i 0.194734i
\(713\) 15.5885 0.583792
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 12.0000i 0.448461i
\(717\) 0 0
\(718\) 0 0
\(719\) −3.46410 −0.129189 −0.0645946 0.997912i \(-0.520575\pi\)
−0.0645946 + 0.997912i \(0.520575\pi\)
\(720\) 0 0
\(721\) 33.0000 38.1051i 1.22898 1.41911i
\(722\) 16.0000i 0.595458i
\(723\) 0 0
\(724\) 13.8564i 0.514969i
\(725\) 12.0000i 0.445669i
\(726\) 0 0
\(727\) 51.9615i 1.92715i −0.267445 0.963573i \(-0.586179\pi\)
0.267445 0.963573i \(-0.413821\pi\)
\(728\) 6.92820 + 6.00000i 0.256776 + 0.222375i
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) 13.8564 0.512498
\(732\) 0 0
\(733\) 10.3923i 0.383849i −0.981410 0.191924i \(-0.938527\pi\)
0.981410 0.191924i \(-0.0614728\pi\)
\(734\) 1.73205 0.0639312
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 12.1244 0.445700
\(741\) 0 0
\(742\) 24.0000 + 20.7846i 0.881068 + 0.763027i
\(743\) 27.0000i 0.990534i −0.868741 0.495267i \(-0.835070\pi\)
0.868741 0.495267i \(-0.164930\pi\)
\(744\) 0 0
\(745\) 20.7846i 0.761489i
\(746\) 5.00000i 0.183063i
\(747\) 0 0
\(748\) 20.7846i 0.759961i
\(749\) 0 0
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 3.46410 0.126323
\(753\) 0 0
\(754\) 20.7846i 0.756931i
\(755\) 13.8564 0.504286
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 2.00000i 0.0726433i
\(759\) 0 0
\(760\) −3.00000 −0.108821
\(761\) −13.8564 −0.502294 −0.251147 0.967949i \(-0.580808\pi\)
−0.251147 + 0.967949i \(0.580808\pi\)
\(762\) 0 0
\(763\) 26.0000 + 22.5167i 0.941263 + 0.815158i
\(764\) 21.0000i 0.759753i
\(765\) 0 0
\(766\) 13.8564i 0.500652i
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) 38.1051i 1.37411i 0.726607 + 0.687053i \(0.241096\pi\)
−0.726607 + 0.687053i \(0.758904\pi\)
\(770\) 10.3923 + 9.00000i 0.374513 + 0.324337i
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 19.0526 0.685273 0.342636 0.939468i \(-0.388680\pi\)
0.342636 + 0.939468i \(0.388680\pi\)
\(774\) 0 0
\(775\) 10.3923i 0.373303i
\(776\) 13.8564 0.497416
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 21.0000i 0.752403i
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 20.7846 0.743256
\(783\) 0 0
\(784\) 1.00000 + 6.92820i 0.0357143 + 0.247436i
\(785\) 18.0000i 0.642448i
\(786\) 0 0
\(787\) 17.3205i 0.617409i −0.951158 0.308705i \(-0.900105\pi\)
0.951158 0.308705i \(-0.0998955\pi\)
\(788\) 12.0000i 0.427482i
\(789\) 0 0
\(790\) 17.3205i 0.616236i
\(791\) 20.7846 24.0000i 0.739016 0.853342i
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 20.7846 0.737618
\(795\) 0 0
\(796\) 1.73205i 0.0613909i
\(797\) −15.5885 −0.552171 −0.276086 0.961133i \(-0.589037\pi\)
−0.276086 + 0.961133i \(0.589037\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 2.00000i 0.0707107i
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 10.3923 0.366736
\(804\) 0 0
\(805\) 9.00000 10.3923i 0.317208 0.366281i
\(806\) 18.0000i 0.634023i
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000i 0.210949i −0.994422 0.105474i \(-0.966364\pi\)
0.994422 0.105474i \(-0.0336361\pi\)
\(810\) 0 0
\(811\) 36.3731i 1.27723i −0.769526 0.638616i \(-0.779507\pi\)
0.769526 0.638616i \(-0.220493\pi\)
\(812\) 10.3923 12.0000i 0.364698 0.421117i
\(813\) 0 0
\(814\) −21.0000 −0.736050
\(815\) −3.46410 −0.121342
\(816\) 0 0
\(817\) 3.46410i 0.121194i
\(818\) 3.46410 0.121119
\(819\) 0 0
\(820\) 21.0000 0.733352
\(821\) 30.0000i 1.04701i −0.852023 0.523504i \(-0.824625\pi\)
0.852023 0.523504i \(-0.175375\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −19.0526 −0.663727
\(825\) 0 0
\(826\) 6.00000 6.92820i 0.208767 0.241063i
\(827\) 3.00000i 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) 0 0
\(829\) 38.1051i 1.32345i −0.749749 0.661723i \(-0.769826\pi\)
0.749749 0.661723i \(-0.230174\pi\)
\(830\) 30.0000i 1.04132i
\(831\) 0 0
\(832\) 3.46410i 0.120096i
\(833\) −6.92820 48.0000i −0.240048 1.66310i
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 5.19615 0.179713
\(837\) 0 0
\(838\) 6.92820i 0.239331i
\(839\) 24.2487 0.837158 0.418579 0.908180i \(-0.362528\pi\)
0.418579 + 0.908180i \(0.362528\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 13.0000i 0.448010i
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) −1.73205 −0.0595844
\(846\) 0 0
\(847\) 4.00000 + 3.46410i 0.137442 + 0.119028i
\(848\) 12.0000i 0.412082i
\(849\) 0 0
\(850\) 13.8564i 0.475271i
\(851\) 21.0000i 0.719871i
\(852\) 0 0
\(853\) 13.8564i 0.474434i 0.971457 + 0.237217i \(0.0762353\pi\)
−0.971457 + 0.237217i \(0.923765\pi\)
\(854\) −13.8564 12.0000i −0.474156 0.410632i
\(855\) 0 0
\(856\) 0 0
\(857\) −43.3013 −1.47914 −0.739572 0.673078i \(-0.764972\pi\)
−0.739572 + 0.673078i \(0.764972\pi\)
\(858\) 0 0
\(859\) 29.4449i 1.00465i 0.864680 + 0.502323i \(0.167521\pi\)
−0.864680 + 0.502323i \(0.832479\pi\)
\(860\) −3.46410 −0.118125
\(861\) 0 0
\(862\) 21.0000 0.715263
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) −10.3923 −0.353145
\(867\) 0 0
\(868\) −9.00000 + 10.3923i −0.305480 + 0.352738i
\(869\) 30.0000i 1.01768i
\(870\) 0 0
\(871\) 6.92820i 0.234753i
\(872\) 13.0000i 0.440236i
\(873\) 0 0
\(874\) 5.19615i 0.175762i
\(875\) 24.2487 + 21.0000i 0.819756 + 0.709930i
\(876\) 0 0
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 10.3923 0.350723
\(879\) 0 0
\(880\) 5.19615i 0.175162i
\(881\) 25.9808 0.875314 0.437657 0.899142i \(-0.355808\pi\)
0.437657 + 0.899142i \(0.355808\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 24.0000i 0.807207i
\(885\) 0 0
\(886\) 9.00000 0.302361
\(887\) 24.2487 0.814192 0.407096 0.913385i \(-0.366541\pi\)
0.407096 + 0.913385i \(0.366541\pi\)
\(888\) 0 0
\(889\) −44.0000 38.1051i −1.47571 1.27800i
\(890\) 9.00000i 0.301681i
\(891\) 0 0
\(892\) 15.5885i 0.521940i
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 20.7846i 0.694753i
\(896\) 1.73205 2.00000i 0.0578638 0.0668153i
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 31.1769 1.03981
\(900\) 0 0
\(901\) 83.1384i 2.76974i
\(902\) −36.3731 −1.21109
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 24.0000i 0.797787i
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 20.7846 0.689761
\(909\) 0 0
\(910\) 12.0000 + 10.3923i 0.397796 + 0.344502i
\(911\) 12.0000i 0.397578i −0.980042 0.198789i \(-0.936299\pi\)
0.980042 0.198789i \(-0.0637008\pi\)
\(912\) 0 0
\(913\) 51.9615i 1.71968i
\(914\) 17.0000i 0.562310i
\(915\) 0 0
\(916\) 0 0
\(917\) 20.7846 + 18.0000i 0.686368 + 0.594412i
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) −5.19615 −0.171312
\(921\) 0 0
\(922\) 19.0526i 0.627463i
\(923\) 10.3923 0.342067
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) 32.0000i 1.05159i
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −27.7128 −0.909228 −0.454614 0.890689i \(-0.650223\pi\)
−0.454614 + 0.890689i \(0.650223\pi\)
\(930\) 0 0
\(931\) −12.0000 + 1.73205i −0.393284 + 0.0567657i
\(932\) 24.0000i 0.786146i
\(933\) 0 0
\(934\) 31.1769i 1.02014i
\(935\) 36.0000i 1.17733i
\(936\) 0 0
\(937\) 6.92820i 0.226335i 0.993576 + 0.113167i \(0.0360996\pi\)
−0.993576 + 0.113167i \(0.963900\pi\)
\(938\) −3.46410 + 4.00000i −0.113107 + 0.130605i
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 12.1244 0.395243 0.197621 0.980278i \(-0.436678\pi\)
0.197621 + 0.980278i \(0.436678\pi\)
\(942\) 0 0
\(943\) 36.3731i 1.18447i
\(944\) −3.46410 −0.112747
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) 15.0000i 0.487435i −0.969846 0.243717i \(-0.921633\pi\)
0.969846 0.243717i \(-0.0783669\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 3.46410 0.112390
\(951\) 0 0
\(952\) −12.0000 + 13.8564i −0.388922 + 0.449089i
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 36.3731i 1.17700i
\(956\) 0 0
\(957\) 0 0
\(958\) 27.7128i 0.895360i
\(959\) −31.1769 + 36.0000i −1.00676 + 1.16250i
\(960\) 0 0
\(961\) 4.00000 0.129032
\(962\) −24.2487 −0.781810
\(963\) 0 0
\(964\) 17.3205i 0.557856i
\(965\) 24.2487 0.780594
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −3.46410 −0.111168 −0.0555842 0.998454i \(-0.517702\pi\)
−0.0555842 + 0.998454i \(0.517702\pi\)
\(972\) 0 0
\(973\) 6.00000 6.92820i 0.192351 0.222108i
\(974\) 10.0000i 0.320421i
\(975\) 0 0
\(976\) 6.92820i 0.221766i
\(977\) 48.0000i 1.53566i 0.640656 + 0.767828i \(0.278662\pi\)
−0.640656 + 0.767828i \(0.721338\pi\)
\(978\) 0 0
\(979\) 15.5885i 0.498209i
\(980\) 1.73205 + 12.0000i 0.0553283 + 0.383326i
\(981\) 0 0
\(982\) −9.00000 −0.287202
\(983\) −3.46410 −0.110488 −0.0552438 0.998473i \(-0.517594\pi\)
−0.0552438 + 0.998473i \(0.517594\pi\)
\(984\) 0 0
\(985\) 20.7846i 0.662253i
\(986\) 41.5692 1.32383
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 6.00000i 0.190789i
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 5.19615 0.164978
\(993\) 0 0
\(994\) 6.00000 + 5.19615i 0.190308 + 0.164812i
\(995\) 3.00000i 0.0951064i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 10.0000i 0.316544i
\(999\) 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 378.2.d.b.377.3 yes 4
3.2 odd 2 inner 378.2.d.b.377.2 yes 4
4.3 odd 2 3024.2.k.e.1889.1 4
7.6 odd 2 inner 378.2.d.b.377.4 yes 4
9.2 odd 6 1134.2.m.c.377.1 4
9.4 even 3 1134.2.m.b.755.1 4
9.5 odd 6 1134.2.m.b.755.2 4
9.7 even 3 1134.2.m.c.377.2 4
12.11 even 2 3024.2.k.e.1889.3 4
21.20 even 2 inner 378.2.d.b.377.1 4
28.27 even 2 3024.2.k.e.1889.4 4
63.13 odd 6 1134.2.m.c.755.1 4
63.20 even 6 1134.2.m.b.377.1 4
63.34 odd 6 1134.2.m.b.377.2 4
63.41 even 6 1134.2.m.c.755.2 4
84.83 odd 2 3024.2.k.e.1889.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.d.b.377.1 4 21.20 even 2 inner
378.2.d.b.377.2 yes 4 3.2 odd 2 inner
378.2.d.b.377.3 yes 4 1.1 even 1 trivial
378.2.d.b.377.4 yes 4 7.6 odd 2 inner
1134.2.m.b.377.1 4 63.20 even 6
1134.2.m.b.377.2 4 63.34 odd 6
1134.2.m.b.755.1 4 9.4 even 3
1134.2.m.b.755.2 4 9.5 odd 6
1134.2.m.c.377.1 4 9.2 odd 6
1134.2.m.c.377.2 4 9.7 even 3
1134.2.m.c.755.1 4 63.13 odd 6
1134.2.m.c.755.2 4 63.41 even 6
3024.2.k.e.1889.1 4 4.3 odd 2
3024.2.k.e.1889.2 4 84.83 odd 2
3024.2.k.e.1889.3 4 12.11 even 2
3024.2.k.e.1889.4 4 28.27 even 2