Properties

Label 1152.3.j.b.161.11
Level $1152$
Weight $3$
Character 1152.161
Analytic conductor $31.390$
Analytic rank $0$
Dimension $32$
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] = [1152,3,Mod(161,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.j (of order \(4\), degree \(2\), not 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 161.11
Character \(\chi\) \(=\) 1152.161
Dual form 1152.3.j.b.737.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.84570 + 1.84570i) q^{5} +0.226665i q^{7} +O(q^{10})\) \(q+(1.84570 + 1.84570i) q^{5} +0.226665i q^{7} +(-11.5821 - 11.5821i) q^{11} +(8.11369 + 8.11369i) q^{13} +6.20082i q^{17} +(-4.43355 - 4.43355i) q^{19} +28.6638 q^{23} -18.1868i q^{25} +(15.7561 - 15.7561i) q^{29} +33.5751 q^{31} +(-0.418355 + 0.418355i) q^{35} +(-43.8384 + 43.8384i) q^{37} +62.2969 q^{41} +(18.3048 - 18.3048i) q^{43} -13.8029i q^{47} +48.9486 q^{49} +(-37.9167 - 37.9167i) q^{53} -42.7539i q^{55} +(-70.4759 - 70.4759i) q^{59} +(60.3861 + 60.3861i) q^{61} +29.9508i q^{65} +(61.9736 + 61.9736i) q^{67} -32.5259 q^{71} +130.954i q^{73} +(2.62525 - 2.62525i) q^{77} +132.387 q^{79} +(19.2783 - 19.2783i) q^{83} +(-11.4448 + 11.4448i) q^{85} +91.2401 q^{89} +(-1.83909 + 1.83909i) q^{91} -16.3660i q^{95} -20.0808 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 64 q^{19} + 128 q^{43} - 224 q^{49} - 64 q^{61} - 64 q^{67} + 512 q^{79} - 320 q^{85} - 192 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.84570 + 1.84570i 0.369140 + 0.369140i 0.867163 0.498024i \(-0.165941\pi\)
−0.498024 + 0.867163i \(0.665941\pi\)
\(6\) 0 0
\(7\) 0.226665i 0.0323807i 0.999869 + 0.0161904i \(0.00515378\pi\)
−0.999869 + 0.0161904i \(0.994846\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −11.5821 11.5821i −1.05291 1.05291i −0.998520 0.0543946i \(-0.982677\pi\)
−0.0543946 0.998520i \(-0.517323\pi\)
\(12\) 0 0
\(13\) 8.11369 + 8.11369i 0.624130 + 0.624130i 0.946585 0.322455i \(-0.104508\pi\)
−0.322455 + 0.946585i \(0.604508\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.20082i 0.364754i 0.983229 + 0.182377i \(0.0583791\pi\)
−0.983229 + 0.182377i \(0.941621\pi\)
\(18\) 0 0
\(19\) −4.43355 4.43355i −0.233345 0.233345i 0.580743 0.814087i \(-0.302762\pi\)
−0.814087 + 0.580743i \(0.802762\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 28.6638 1.24625 0.623127 0.782121i \(-0.285862\pi\)
0.623127 + 0.782121i \(0.285862\pi\)
\(24\) 0 0
\(25\) 18.1868i 0.727472i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 15.7561 15.7561i 0.543312 0.543312i −0.381186 0.924498i \(-0.624484\pi\)
0.924498 + 0.381186i \(0.124484\pi\)
\(30\) 0 0
\(31\) 33.5751 1.08307 0.541534 0.840679i \(-0.317844\pi\)
0.541534 + 0.840679i \(0.317844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.418355 + 0.418355i −0.0119530 + 0.0119530i
\(36\) 0 0
\(37\) −43.8384 + 43.8384i −1.18482 + 1.18482i −0.206342 + 0.978480i \(0.566156\pi\)
−0.978480 + 0.206342i \(0.933844\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 62.2969 1.51944 0.759719 0.650252i \(-0.225337\pi\)
0.759719 + 0.650252i \(0.225337\pi\)
\(42\) 0 0
\(43\) 18.3048 18.3048i 0.425692 0.425692i −0.461466 0.887158i \(-0.652676\pi\)
0.887158 + 0.461466i \(0.152676\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.8029i 0.293678i −0.989160 0.146839i \(-0.953090\pi\)
0.989160 0.146839i \(-0.0469099\pi\)
\(48\) 0 0
\(49\) 48.9486 0.998951
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −37.9167 37.9167i −0.715409 0.715409i 0.252252 0.967661i \(-0.418829\pi\)
−0.967661 + 0.252252i \(0.918829\pi\)
\(54\) 0 0
\(55\) 42.7539i 0.777344i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −70.4759 70.4759i −1.19451 1.19451i −0.975788 0.218719i \(-0.929812\pi\)
−0.218719 0.975788i \(-0.570188\pi\)
\(60\) 0 0
\(61\) 60.3861 + 60.3861i 0.989936 + 0.989936i 0.999950 0.0100137i \(-0.00318750\pi\)
−0.0100137 + 0.999950i \(0.503187\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 29.9508i 0.460782i
\(66\) 0 0
\(67\) 61.9736 + 61.9736i 0.924979 + 0.924979i 0.997376 0.0723967i \(-0.0230648\pi\)
−0.0723967 + 0.997376i \(0.523065\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −32.5259 −0.458111 −0.229055 0.973413i \(-0.573564\pi\)
−0.229055 + 0.973413i \(0.573564\pi\)
\(72\) 0 0
\(73\) 130.954i 1.79389i 0.442145 + 0.896944i \(0.354218\pi\)
−0.442145 + 0.896944i \(0.645782\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.62525 2.62525i 0.0340941 0.0340941i
\(78\) 0 0
\(79\) 132.387 1.67579 0.837894 0.545833i \(-0.183787\pi\)
0.837894 + 0.545833i \(0.183787\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 19.2783 19.2783i 0.232268 0.232268i −0.581371 0.813639i \(-0.697483\pi\)
0.813639 + 0.581371i \(0.197483\pi\)
\(84\) 0 0
\(85\) −11.4448 + 11.4448i −0.134645 + 0.134645i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 91.2401 1.02517 0.512585 0.858637i \(-0.328688\pi\)
0.512585 + 0.858637i \(0.328688\pi\)
\(90\) 0 0
\(91\) −1.83909 + 1.83909i −0.0202098 + 0.0202098i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.3660i 0.172274i
\(96\) 0 0
\(97\) −20.0808 −0.207019 −0.103509 0.994628i \(-0.533007\pi\)
−0.103509 + 0.994628i \(0.533007\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 29.7693 + 29.7693i 0.294745 + 0.294745i 0.838951 0.544206i \(-0.183169\pi\)
−0.544206 + 0.838951i \(0.683169\pi\)
\(102\) 0 0
\(103\) 111.006i 1.07773i −0.842392 0.538865i \(-0.818853\pi\)
0.842392 0.538865i \(-0.181147\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.1409 13.1409i −0.122812 0.122812i 0.643029 0.765842i \(-0.277677\pi\)
−0.765842 + 0.643029i \(0.777677\pi\)
\(108\) 0 0
\(109\) −3.89888 3.89888i −0.0357696 0.0357696i 0.688996 0.724765i \(-0.258052\pi\)
−0.724765 + 0.688996i \(0.758052\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 117.686i 1.04147i −0.853718 0.520735i \(-0.825658\pi\)
0.853718 0.520735i \(-0.174342\pi\)
\(114\) 0 0
\(115\) 52.9048 + 52.9048i 0.460041 + 0.460041i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.40551 −0.0118110
\(120\) 0 0
\(121\) 147.288i 1.21726i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 79.7098 79.7098i 0.637678 0.637678i
\(126\) 0 0
\(127\) 43.1157 0.339494 0.169747 0.985488i \(-0.445705\pi\)
0.169747 + 0.985488i \(0.445705\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 97.5874 97.5874i 0.744942 0.744942i −0.228583 0.973524i \(-0.573409\pi\)
0.973524 + 0.228583i \(0.0734092\pi\)
\(132\) 0 0
\(133\) 1.00493 1.00493i 0.00755588 0.00755588i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 49.2731 0.359658 0.179829 0.983698i \(-0.442446\pi\)
0.179829 + 0.983698i \(0.442446\pi\)
\(138\) 0 0
\(139\) 172.904 172.904i 1.24392 1.24392i 0.285552 0.958363i \(-0.407823\pi\)
0.958363 0.285552i \(-0.0921769\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 187.946i 1.31431i
\(144\) 0 0
\(145\) 58.1618 0.401116
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 128.537 + 128.537i 0.862663 + 0.862663i 0.991647 0.128984i \(-0.0411716\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(150\) 0 0
\(151\) 89.9523i 0.595710i 0.954611 + 0.297855i \(0.0962713\pi\)
−0.954611 + 0.297855i \(0.903729\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 61.9695 + 61.9695i 0.399803 + 0.399803i
\(156\) 0 0
\(157\) 95.2413 + 95.2413i 0.606633 + 0.606633i 0.942065 0.335432i \(-0.108882\pi\)
−0.335432 + 0.942065i \(0.608882\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.49709i 0.0403546i
\(162\) 0 0
\(163\) −84.2958 84.2958i −0.517152 0.517152i 0.399556 0.916709i \(-0.369164\pi\)
−0.916709 + 0.399556i \(0.869164\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −193.153 −1.15661 −0.578304 0.815821i \(-0.696285\pi\)
−0.578304 + 0.815821i \(0.696285\pi\)
\(168\) 0 0
\(169\) 37.3362i 0.220924i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 60.1097 60.1097i 0.347455 0.347455i −0.511706 0.859161i \(-0.670986\pi\)
0.859161 + 0.511706i \(0.170986\pi\)
\(174\) 0 0
\(175\) 4.12231 0.0235561
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −103.680 + 103.680i −0.579217 + 0.579217i −0.934687 0.355471i \(-0.884321\pi\)
0.355471 + 0.934687i \(0.384321\pi\)
\(180\) 0 0
\(181\) 3.08029 3.08029i 0.0170182 0.0170182i −0.698547 0.715565i \(-0.746169\pi\)
0.715565 + 0.698547i \(0.246169\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −161.825 −0.874729
\(186\) 0 0
\(187\) 71.8182 71.8182i 0.384054 0.384054i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 235.549i 1.23324i −0.787261 0.616620i \(-0.788501\pi\)
0.787261 0.616620i \(-0.211499\pi\)
\(192\) 0 0
\(193\) −9.38546 −0.0486293 −0.0243147 0.999704i \(-0.507740\pi\)
−0.0243147 + 0.999704i \(0.507740\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 75.8630 + 75.8630i 0.385091 + 0.385091i 0.872932 0.487841i \(-0.162215\pi\)
−0.487841 + 0.872932i \(0.662215\pi\)
\(198\) 0 0
\(199\) 315.653i 1.58620i 0.609093 + 0.793099i \(0.291534\pi\)
−0.609093 + 0.793099i \(0.708466\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.57135 + 3.57135i 0.0175928 + 0.0175928i
\(204\) 0 0
\(205\) 114.981 + 114.981i 0.560884 + 0.560884i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 102.699i 0.491384i
\(210\) 0 0
\(211\) −20.9669 20.9669i −0.0993692 0.0993692i 0.655675 0.755044i \(-0.272384\pi\)
−0.755044 + 0.655675i \(0.772384\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 67.5701 0.314280
\(216\) 0 0
\(217\) 7.61030i 0.0350705i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −50.3115 + 50.3115i −0.227654 + 0.227654i
\(222\) 0 0
\(223\) −317.091 −1.42193 −0.710966 0.703226i \(-0.751742\pi\)
−0.710966 + 0.703226i \(0.751742\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 101.358 101.358i 0.446513 0.446513i −0.447681 0.894193i \(-0.647750\pi\)
0.894193 + 0.447681i \(0.147750\pi\)
\(228\) 0 0
\(229\) 83.7775 83.7775i 0.365841 0.365841i −0.500117 0.865958i \(-0.666710\pi\)
0.865958 + 0.500117i \(0.166710\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −289.948 −1.24441 −0.622207 0.782853i \(-0.713764\pi\)
−0.622207 + 0.782853i \(0.713764\pi\)
\(234\) 0 0
\(235\) 25.4759 25.4759i 0.108408 0.108408i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 441.865i 1.84881i 0.381413 + 0.924405i \(0.375438\pi\)
−0.381413 + 0.924405i \(0.624562\pi\)
\(240\) 0 0
\(241\) −336.119 −1.39469 −0.697343 0.716738i \(-0.745634\pi\)
−0.697343 + 0.716738i \(0.745634\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 90.3444 + 90.3444i 0.368752 + 0.368752i
\(246\) 0 0
\(247\) 71.9449i 0.291275i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 73.9741 + 73.9741i 0.294717 + 0.294717i 0.838941 0.544223i \(-0.183176\pi\)
−0.544223 + 0.838941i \(0.683176\pi\)
\(252\) 0 0
\(253\) −331.986 331.986i −1.31220 1.31220i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 441.535i 1.71803i −0.511947 0.859017i \(-0.671076\pi\)
0.511947 0.859017i \(-0.328924\pi\)
\(258\) 0 0
\(259\) −9.93664 9.93664i −0.0383654 0.0383654i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 116.115 0.441503 0.220752 0.975330i \(-0.429149\pi\)
0.220752 + 0.975330i \(0.429149\pi\)
\(264\) 0 0
\(265\) 139.965i 0.528171i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 46.5610 46.5610i 0.173089 0.173089i −0.615246 0.788335i \(-0.710943\pi\)
0.788335 + 0.615246i \(0.210943\pi\)
\(270\) 0 0
\(271\) 343.577 1.26781 0.633907 0.773410i \(-0.281450\pi\)
0.633907 + 0.773410i \(0.281450\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −210.641 + 210.641i −0.765966 + 0.765966i
\(276\) 0 0
\(277\) −10.4668 + 10.4668i −0.0377863 + 0.0377863i −0.725747 0.687961i \(-0.758506\pi\)
0.687961 + 0.725747i \(0.258506\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 56.0486 0.199461 0.0997306 0.995014i \(-0.468202\pi\)
0.0997306 + 0.995014i \(0.468202\pi\)
\(282\) 0 0
\(283\) −248.129 + 248.129i −0.876782 + 0.876782i −0.993200 0.116418i \(-0.962859\pi\)
0.116418 + 0.993200i \(0.462859\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.1205i 0.0492005i
\(288\) 0 0
\(289\) 250.550 0.866955
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −243.305 243.305i −0.830392 0.830392i 0.157178 0.987570i \(-0.449760\pi\)
−0.987570 + 0.157178i \(0.949760\pi\)
\(294\) 0 0
\(295\) 260.154i 0.881880i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 232.569 + 232.569i 0.777824 + 0.777824i
\(300\) 0 0
\(301\) 4.14905 + 4.14905i 0.0137842 + 0.0137842i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 222.909i 0.730849i
\(306\) 0 0
\(307\) −29.1965 29.1965i −0.0951028 0.0951028i 0.657955 0.753057i \(-0.271422\pi\)
−0.753057 + 0.657955i \(0.771422\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.31281 −0.0170830 −0.00854150 0.999964i \(-0.502719\pi\)
−0.00854150 + 0.999964i \(0.502719\pi\)
\(312\) 0 0
\(313\) 71.0570i 0.227019i −0.993537 0.113510i \(-0.963791\pi\)
0.993537 0.113510i \(-0.0362093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −340.594 + 340.594i −1.07443 + 1.07443i −0.0774326 + 0.996998i \(0.524672\pi\)
−0.996998 + 0.0774326i \(0.975328\pi\)
\(318\) 0 0
\(319\) −364.975 −1.14412
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.4916 27.4916i 0.0851134 0.0851134i
\(324\) 0 0
\(325\) 147.562 147.562i 0.454037 0.454037i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.12863 0.00950951
\(330\) 0 0
\(331\) −283.647 + 283.647i −0.856939 + 0.856939i −0.990976 0.134038i \(-0.957206\pi\)
0.134038 + 0.990976i \(0.457206\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 228.769i 0.682893i
\(336\) 0 0
\(337\) −44.9447 −0.133367 −0.0666835 0.997774i \(-0.521242\pi\)
−0.0666835 + 0.997774i \(0.521242\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −388.869 388.869i −1.14038 1.14038i
\(342\) 0 0
\(343\) 22.2015i 0.0647275i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −272.333 272.333i −0.784821 0.784821i 0.195819 0.980640i \(-0.437263\pi\)
−0.980640 + 0.195819i \(0.937263\pi\)
\(348\) 0 0
\(349\) −178.956 178.956i −0.512768 0.512768i 0.402605 0.915374i \(-0.368105\pi\)
−0.915374 + 0.402605i \(0.868105\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 698.328i 1.97827i 0.147025 + 0.989133i \(0.453030\pi\)
−0.147025 + 0.989133i \(0.546970\pi\)
\(354\) 0 0
\(355\) −60.0329 60.0329i −0.169107 0.169107i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −364.766 −1.01606 −0.508030 0.861339i \(-0.669626\pi\)
−0.508030 + 0.861339i \(0.669626\pi\)
\(360\) 0 0
\(361\) 321.687i 0.891100i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −241.701 + 241.701i −0.662195 + 0.662195i
\(366\) 0 0
\(367\) −632.894 −1.72451 −0.862254 0.506476i \(-0.830948\pi\)
−0.862254 + 0.506476i \(0.830948\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.59439 8.59439i 0.0231655 0.0231655i
\(372\) 0 0
\(373\) −33.3679 + 33.3679i −0.0894582 + 0.0894582i −0.750420 0.660962i \(-0.770149\pi\)
0.660962 + 0.750420i \(0.270149\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 255.679 0.678194
\(378\) 0 0
\(379\) −439.149 + 439.149i −1.15870 + 1.15870i −0.173950 + 0.984755i \(0.555653\pi\)
−0.984755 + 0.173950i \(0.944347\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 168.628i 0.440283i 0.975468 + 0.220142i \(0.0706519\pi\)
−0.975468 + 0.220142i \(0.929348\pi\)
\(384\) 0 0
\(385\) 9.69083 0.0251710
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −54.4610 54.4610i −0.140003 0.140003i 0.633632 0.773635i \(-0.281563\pi\)
−0.773635 + 0.633632i \(0.781563\pi\)
\(390\) 0 0
\(391\) 177.739i 0.454576i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 244.347 + 244.347i 0.618600 + 0.618600i
\(396\) 0 0
\(397\) −446.859 446.859i −1.12559 1.12559i −0.990886 0.134704i \(-0.956992\pi\)
−0.134704 0.990886i \(-0.543008\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.95350i 0.0198342i −0.999951 0.00991709i \(-0.996843\pi\)
0.999951 0.00991709i \(-0.00315676\pi\)
\(402\) 0 0
\(403\) 272.418 + 272.418i 0.675975 + 0.675975i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1015.48 2.49503
\(408\) 0 0
\(409\) 624.006i 1.52569i −0.646583 0.762843i \(-0.723803\pi\)
0.646583 0.762843i \(-0.276197\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 15.9744 15.9744i 0.0386790 0.0386790i
\(414\) 0 0
\(415\) 71.1637 0.171479
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 57.4026 57.4026i 0.136999 0.136999i −0.635282 0.772281i \(-0.719116\pi\)
0.772281 + 0.635282i \(0.219116\pi\)
\(420\) 0 0
\(421\) 4.45855 4.45855i 0.0105904 0.0105904i −0.701792 0.712382i \(-0.747616\pi\)
0.712382 + 0.701792i \(0.247616\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 112.773 0.265348
\(426\) 0 0
\(427\) −13.6874 + 13.6874i −0.0320549 + 0.0320549i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 515.590i 1.19626i 0.801397 + 0.598132i \(0.204090\pi\)
−0.801397 + 0.598132i \(0.795910\pi\)
\(432\) 0 0
\(433\) 301.416 0.696111 0.348056 0.937474i \(-0.386842\pi\)
0.348056 + 0.937474i \(0.386842\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −127.083 127.083i −0.290807 0.290807i
\(438\) 0 0
\(439\) 656.561i 1.49558i −0.663933 0.747792i \(-0.731114\pi\)
0.663933 0.747792i \(-0.268886\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −229.799 229.799i −0.518733 0.518733i 0.398455 0.917188i \(-0.369546\pi\)
−0.917188 + 0.398455i \(0.869546\pi\)
\(444\) 0 0
\(445\) 168.402 + 168.402i 0.378431 + 0.378431i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 184.489i 0.410888i −0.978669 0.205444i \(-0.934136\pi\)
0.978669 0.205444i \(-0.0658639\pi\)
\(450\) 0 0
\(451\) −721.526 721.526i −1.59984 1.59984i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.78881 −0.0149205
\(456\) 0 0
\(457\) 376.876i 0.824673i −0.911032 0.412336i \(-0.864713\pi\)
0.911032 0.412336i \(-0.135287\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.5551 + 28.5551i −0.0619418 + 0.0619418i −0.737399 0.675457i \(-0.763946\pi\)
0.675457 + 0.737399i \(0.263946\pi\)
\(462\) 0 0
\(463\) 144.828 0.312803 0.156401 0.987694i \(-0.450011\pi\)
0.156401 + 0.987694i \(0.450011\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −366.519 + 366.519i −0.784837 + 0.784837i −0.980643 0.195806i \(-0.937268\pi\)
0.195806 + 0.980643i \(0.437268\pi\)
\(468\) 0 0
\(469\) −14.0473 + 14.0473i −0.0299515 + 0.0299515i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −424.014 −0.896435
\(474\) 0 0
\(475\) −80.6321 + 80.6321i −0.169752 + 0.169752i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 119.680i 0.249853i −0.992166 0.124927i \(-0.960130\pi\)
0.992166 0.124927i \(-0.0398696\pi\)
\(480\) 0 0
\(481\) −711.382 −1.47896
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −37.0631 37.0631i −0.0764188 0.0764188i
\(486\) 0 0
\(487\) 637.829i 1.30971i 0.755754 + 0.654855i \(0.227270\pi\)
−0.755754 + 0.654855i \(0.772730\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 130.249 + 130.249i 0.265273 + 0.265273i 0.827192 0.561919i \(-0.189937\pi\)
−0.561919 + 0.827192i \(0.689937\pi\)
\(492\) 0 0
\(493\) 97.7004 + 97.7004i 0.198175 + 0.198175i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.37248i 0.0148340i
\(498\) 0 0
\(499\) 7.28612 + 7.28612i 0.0146014 + 0.0146014i 0.714370 0.699768i \(-0.246713\pi\)
−0.699768 + 0.714370i \(0.746713\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 311.009 0.618307 0.309154 0.951012i \(-0.399954\pi\)
0.309154 + 0.951012i \(0.399954\pi\)
\(504\) 0 0
\(505\) 109.890i 0.217604i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 431.114 431.114i 0.846983 0.846983i −0.142773 0.989756i \(-0.545602\pi\)
0.989756 + 0.142773i \(0.0456017\pi\)
\(510\) 0 0
\(511\) −29.6827 −0.0580874
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 204.884 204.884i 0.397832 0.397832i
\(516\) 0 0
\(517\) −159.866 + 159.866i −0.309218 + 0.309218i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −241.849 −0.464201 −0.232101 0.972692i \(-0.574560\pi\)
−0.232101 + 0.972692i \(0.574560\pi\)
\(522\) 0 0
\(523\) 183.752 183.752i 0.351342 0.351342i −0.509267 0.860609i \(-0.670083\pi\)
0.860609 + 0.509267i \(0.170083\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 208.193i 0.395053i
\(528\) 0 0
\(529\) 292.615 0.553147
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 505.458 + 505.458i 0.948326 + 0.948326i
\(534\) 0 0
\(535\) 48.5083i 0.0906698i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −566.926 566.926i −1.05181 1.05181i
\(540\) 0 0
\(541\) 186.010 + 186.010i 0.343826 + 0.343826i 0.857803 0.513978i \(-0.171829\pi\)
−0.513978 + 0.857803i \(0.671829\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.3923i 0.0264079i
\(546\) 0 0
\(547\) 244.192 + 244.192i 0.446421 + 0.446421i 0.894163 0.447742i \(-0.147772\pi\)
−0.447742 + 0.894163i \(0.647772\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −139.711 −0.253558
\(552\) 0 0
\(553\) 30.0076i 0.0542632i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −367.416 + 367.416i −0.659634 + 0.659634i −0.955293 0.295659i \(-0.904461\pi\)
0.295659 + 0.955293i \(0.404461\pi\)
\(558\) 0 0
\(559\) 297.038 0.531374
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −418.290 + 418.290i −0.742967 + 0.742967i −0.973148 0.230181i \(-0.926068\pi\)
0.230181 + 0.973148i \(0.426068\pi\)
\(564\) 0 0
\(565\) 217.213 217.213i 0.384448 0.384448i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −439.363 −0.772166 −0.386083 0.922464i \(-0.626172\pi\)
−0.386083 + 0.922464i \(0.626172\pi\)
\(570\) 0 0
\(571\) −81.8476 + 81.8476i −0.143341 + 0.143341i −0.775136 0.631795i \(-0.782318\pi\)
0.631795 + 0.775136i \(0.282318\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 521.303i 0.906614i
\(576\) 0 0
\(577\) 303.497 0.525992 0.262996 0.964797i \(-0.415289\pi\)
0.262996 + 0.964797i \(0.415289\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.36971 + 4.36971i 0.00752101 + 0.00752101i
\(582\) 0 0
\(583\) 878.306i 1.50653i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 670.665 + 670.665i 1.14253 + 1.14253i 0.987986 + 0.154543i \(0.0493905\pi\)
0.154543 + 0.987986i \(0.450609\pi\)
\(588\) 0 0
\(589\) −148.857 148.857i −0.252728 0.252728i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 390.656i 0.658779i −0.944194 0.329390i \(-0.893157\pi\)
0.944194 0.329390i \(-0.106843\pi\)
\(594\) 0 0
\(595\) −2.59414 2.59414i −0.00435991 0.00435991i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −788.713 −1.31672 −0.658358 0.752705i \(-0.728749\pi\)
−0.658358 + 0.752705i \(0.728749\pi\)
\(600\) 0 0
\(601\) 30.8170i 0.0512762i −0.999671 0.0256381i \(-0.991838\pi\)
0.999671 0.0256381i \(-0.00816176\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −271.849 + 271.849i −0.449337 + 0.449337i
\(606\) 0 0
\(607\) −607.774 −1.00128 −0.500638 0.865657i \(-0.666901\pi\)
−0.500638 + 0.865657i \(0.666901\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 111.992 111.992i 0.183293 0.183293i
\(612\) 0 0
\(613\) −507.089 + 507.089i −0.827225 + 0.827225i −0.987132 0.159907i \(-0.948881\pi\)
0.159907 + 0.987132i \(0.448881\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 202.840 0.328752 0.164376 0.986398i \(-0.447439\pi\)
0.164376 + 0.986398i \(0.447439\pi\)
\(618\) 0 0
\(619\) 699.181 699.181i 1.12953 1.12953i 0.139280 0.990253i \(-0.455521\pi\)
0.990253 0.139280i \(-0.0444787\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 20.6809i 0.0331957i
\(624\) 0 0
\(625\) −160.430 −0.256687
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −271.834 271.834i −0.432168 0.432168i
\(630\) 0 0
\(631\) 269.488i 0.427081i −0.976934 0.213540i \(-0.931501\pi\)
0.976934 0.213540i \(-0.0684995\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 79.5786 + 79.5786i 0.125321 + 0.125321i
\(636\) 0 0
\(637\) 397.154 + 397.154i 0.623475 + 0.623475i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 638.869i 0.996676i 0.866983 + 0.498338i \(0.166056\pi\)
−0.866983 + 0.498338i \(0.833944\pi\)
\(642\) 0 0
\(643\) 161.282 + 161.282i 0.250827 + 0.250827i 0.821310 0.570483i \(-0.193244\pi\)
−0.570483 + 0.821310i \(0.693244\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −921.712 −1.42459 −0.712297 0.701878i \(-0.752345\pi\)
−0.712297 + 0.701878i \(0.752345\pi\)
\(648\) 0 0
\(649\) 1632.51i 2.51543i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 661.262 661.262i 1.01265 1.01265i 0.0127331 0.999919i \(-0.495947\pi\)
0.999919 0.0127331i \(-0.00405319\pi\)
\(654\) 0 0
\(655\) 360.234 0.549975
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 702.485 702.485i 1.06599 1.06599i 0.0683229 0.997663i \(-0.478235\pi\)
0.997663 0.0683229i \(-0.0217648\pi\)
\(660\) 0 0
\(661\) 451.292 451.292i 0.682741 0.682741i −0.277876 0.960617i \(-0.589630\pi\)
0.960617 + 0.277876i \(0.0896304\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.70960 0.00557835
\(666\) 0 0
\(667\) 451.629 451.629i 0.677104 0.677104i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1398.79i 2.08464i
\(672\) 0 0
\(673\) −626.462 −0.930849 −0.465425 0.885088i \(-0.654098\pi\)
−0.465425 + 0.885088i \(0.654098\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 652.167 + 652.167i 0.963318 + 0.963318i 0.999351 0.0360323i \(-0.0114719\pi\)
−0.0360323 + 0.999351i \(0.511472\pi\)
\(678\) 0 0
\(679\) 4.55162i 0.00670342i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 331.471 + 331.471i 0.485316 + 0.485316i 0.906824 0.421509i \(-0.138499\pi\)
−0.421509 + 0.906824i \(0.638499\pi\)
\(684\) 0 0
\(685\) 90.9433 + 90.9433i 0.132764 + 0.132764i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 615.288i 0.893016i
\(690\) 0 0
\(691\) 648.386 + 648.386i 0.938331 + 0.938331i 0.998206 0.0598752i \(-0.0190703\pi\)
−0.0598752 + 0.998206i \(0.519070\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 638.258 0.918357
\(696\) 0 0
\(697\) 386.292i 0.554221i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 557.872 557.872i 0.795823 0.795823i −0.186611 0.982434i \(-0.559750\pi\)
0.982434 + 0.186611i \(0.0597503\pi\)
\(702\) 0 0
\(703\) 388.720 0.552944
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.74765 + 6.74765i −0.00954407 + 0.00954407i
\(708\) 0 0
\(709\) 409.754 409.754i 0.577932 0.577932i −0.356401 0.934333i \(-0.615996\pi\)
0.934333 + 0.356401i \(0.115996\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 962.391 1.34978
\(714\) 0 0
\(715\) 346.892 346.892i 0.485164 0.485164i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 700.847i 0.974752i 0.873192 + 0.487376i \(0.162046\pi\)
−0.873192 + 0.487376i \(0.837954\pi\)
\(720\) 0 0
\(721\) 25.1612 0.0348977
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −286.552 286.552i −0.395244 0.395244i
\(726\) 0 0
\(727\) 1106.88i 1.52253i −0.648439 0.761267i \(-0.724578\pi\)
0.648439 0.761267i \(-0.275422\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 113.504 + 113.504i 0.155273 + 0.155273i
\(732\) 0 0
\(733\) −185.291 185.291i −0.252784 0.252784i 0.569327 0.822111i \(-0.307204\pi\)
−0.822111 + 0.569327i \(0.807204\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1435.56i 1.94785i
\(738\) 0 0
\(739\) −764.710 764.710i −1.03479 1.03479i −0.999373 0.0354181i \(-0.988724\pi\)
−0.0354181 0.999373i \(-0.511276\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −860.679 −1.15838 −0.579192 0.815191i \(-0.696632\pi\)
−0.579192 + 0.815191i \(0.696632\pi\)
\(744\) 0 0
\(745\) 474.480i 0.636886i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.97859 2.97859i 0.00397675 0.00397675i
\(750\) 0 0
\(751\) 1060.33 1.41189 0.705943 0.708268i \(-0.250523\pi\)
0.705943 + 0.708268i \(0.250523\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −166.025 + 166.025i −0.219900 + 0.219900i
\(756\) 0 0
\(757\) −85.8730 + 85.8730i −0.113439 + 0.113439i −0.761548 0.648109i \(-0.775560\pi\)
0.648109 + 0.761548i \(0.275560\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 587.026 0.771388 0.385694 0.922627i \(-0.373962\pi\)
0.385694 + 0.922627i \(0.373962\pi\)
\(762\) 0 0
\(763\) 0.883741 0.883741i 0.00115825 0.00115825i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1143.64i 1.49105i
\(768\) 0 0
\(769\) 1048.36 1.36328 0.681642 0.731686i \(-0.261266\pi\)
0.681642 + 0.731686i \(0.261266\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1027.31 + 1027.31i 1.32899 + 1.32899i 0.906249 + 0.422744i \(0.138933\pi\)
0.422744 + 0.906249i \(0.361067\pi\)
\(774\) 0 0
\(775\) 610.623i 0.787901i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −276.197 276.197i −0.354553 0.354553i
\(780\) 0 0
\(781\) 376.716 + 376.716i 0.482351 + 0.482351i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 351.573i 0.447864i
\(786\) 0 0
\(787\) 998.571 + 998.571i 1.26883 + 1.26883i 0.946692 + 0.322141i \(0.104402\pi\)
0.322141 + 0.946692i \(0.395598\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 26.6754 0.0337236
\(792\) 0 0
\(793\) 979.908i 1.23570i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −261.834 + 261.834i −0.328524 + 0.328524i −0.852025 0.523501i \(-0.824626\pi\)
0.523501 + 0.852025i \(0.324626\pi\)
\(798\) 0 0
\(799\) 85.5890 0.107120
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1516.71 1516.71i 1.88881 1.88881i
\(804\) 0 0
\(805\) −11.9917 + 11.9917i −0.0148965 + 0.0148965i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −652.165 −0.806137 −0.403069 0.915170i \(-0.632056\pi\)
−0.403069 + 0.915170i \(0.632056\pi\)
\(810\) 0 0
\(811\) −150.173 + 150.173i −0.185171 + 0.185171i −0.793605 0.608434i \(-0.791798\pi\)
0.608434 + 0.793605i \(0.291798\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 311.169i 0.381803i
\(816\) 0 0
\(817\) −162.310 −0.198666
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 979.062 + 979.062i 1.19252 + 1.19252i 0.976356 + 0.216167i \(0.0693557\pi\)
0.216167 + 0.976356i \(0.430644\pi\)
\(822\) 0 0
\(823\) 453.787i 0.551381i 0.961246 + 0.275691i \(0.0889065\pi\)
−0.961246 + 0.275691i \(0.911093\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1162.48 1162.48i −1.40566 1.40566i −0.780488 0.625171i \(-0.785029\pi\)
−0.625171 0.780488i \(-0.714971\pi\)
\(828\) 0 0
\(829\) −126.113 126.113i −0.152127 0.152127i 0.626940 0.779067i \(-0.284307\pi\)
−0.779067 + 0.626940i \(0.784307\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 303.521i 0.364371i
\(834\) 0 0
\(835\) −356.503 356.503i −0.426950 0.426950i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −82.9161 −0.0988273 −0.0494137 0.998778i \(-0.515735\pi\)
−0.0494137 + 0.998778i \(0.515735\pi\)
\(840\) 0 0
\(841\) 344.494i 0.409624i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 68.9113 68.9113i 0.0815519 0.0815519i
\(846\) 0 0
\(847\) −33.3851 −0.0394157
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1256.58 + 1256.58i −1.47659 + 1.47659i
\(852\) 0 0
\(853\) −558.406 + 558.406i −0.654638 + 0.654638i −0.954106 0.299468i \(-0.903191\pi\)
0.299468 + 0.954106i \(0.403191\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 429.366 0.501011 0.250505 0.968115i \(-0.419403\pi\)
0.250505 + 0.968115i \(0.419403\pi\)
\(858\) 0 0
\(859\) −63.4781 + 63.4781i −0.0738977 + 0.0738977i −0.743090 0.669192i \(-0.766640\pi\)
0.669192 + 0.743090i \(0.266640\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 176.985i 0.205081i −0.994729 0.102540i \(-0.967303\pi\)
0.994729 0.102540i \(-0.0326971\pi\)
\(864\) 0 0
\(865\) 221.889 0.256519
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1533.32 1533.32i −1.76446 1.76446i
\(870\) 0 0
\(871\) 1005.67i 1.15461i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18.0674 + 18.0674i 0.0206485 + 0.0206485i
\(876\) 0 0
\(877\) 852.573 + 852.573i 0.972147 + 0.972147i 0.999622 0.0274758i \(-0.00874693\pi\)
−0.0274758 + 0.999622i \(0.508747\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 756.244i 0.858393i 0.903211 + 0.429196i \(0.141203\pi\)
−0.903211 + 0.429196i \(0.858797\pi\)
\(882\) 0 0
\(883\) 617.889 + 617.889i 0.699761 + 0.699761i 0.964359 0.264598i \(-0.0852393\pi\)
−0.264598 + 0.964359i \(0.585239\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 311.194 0.350838 0.175419 0.984494i \(-0.443872\pi\)
0.175419 + 0.984494i \(0.443872\pi\)
\(888\) 0 0
\(889\) 9.77283i 0.0109931i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −61.1957 + 61.1957i −0.0685282 + 0.0685282i
\(894\) 0 0
\(895\) −382.723 −0.427624
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 529.011 529.011i 0.588444 0.588444i
\(900\) 0 0
\(901\) 235.114 235.114i 0.260948 0.260948i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.3706 0.0125641
\(906\) 0 0
\(907\) 593.896 593.896i 0.654792 0.654792i −0.299351 0.954143i \(-0.596770\pi\)
0.954143 + 0.299351i \(0.0967702\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 499.219i 0.547991i 0.961731 + 0.273995i \(0.0883453\pi\)
−0.961731 + 0.273995i \(0.911655\pi\)
\(912\) 0 0
\(913\) −446.564 −0.489117
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 22.1197 + 22.1197i 0.0241218 + 0.0241218i
\(918\) 0 0
\(919\) 405.667i 0.441422i −0.975339 0.220711i \(-0.929162\pi\)
0.975339 0.220711i \(-0.0708378\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −263.905 263.905i −0.285921 0.285921i
\(924\) 0 0
\(925\) 797.280 + 797.280i 0.861925 + 0.861925i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 640.107i 0.689028i 0.938781 + 0.344514i \(0.111956\pi\)
−0.938781 + 0.344514i \(0.888044\pi\)
\(930\) 0 0
\(931\) −217.016 217.016i −0.233100 0.233100i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 265.109 0.283539
\(936\) 0 0
\(937\) 1648.87i 1.75973i −0.475224 0.879865i \(-0.657633\pi\)
0.475224 0.879865i \(-0.342367\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 783.146 783.146i 0.832248 0.832248i −0.155576 0.987824i \(-0.549723\pi\)
0.987824 + 0.155576i \(0.0497233\pi\)
\(942\) 0 0
\(943\) 1785.67 1.89360
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −510.366 + 510.366i −0.538929 + 0.538929i −0.923214 0.384286i \(-0.874448\pi\)
0.384286 + 0.923214i \(0.374448\pi\)
\(948\) 0 0
\(949\) −1062.52 + 1062.52i −1.11962 + 1.11962i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1381.65 1.44979 0.724897 0.688857i \(-0.241887\pi\)
0.724897 + 0.688857i \(0.241887\pi\)
\(954\) 0 0
\(955\) 434.752 434.752i 0.455238 0.455238i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.1685i 0.0116460i
\(960\) 0 0
\(961\) 166.287 0.173035
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −17.3227 17.3227i −0.0179510 0.0179510i
\(966\) 0 0
\(967\) 783.542i 0.810281i −0.914254 0.405141i \(-0.867223\pi\)
0.914254 0.405141i \(-0.132777\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 73.9020 + 73.9020i 0.0761091 + 0.0761091i 0.744137 0.668027i \(-0.232861\pi\)
−0.668027 + 0.744137i \(0.732861\pi\)
\(972\) 0 0
\(973\) 39.1914 + 39.1914i 0.0402789 + 0.0402789i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 584.085i 0.597835i −0.954279 0.298918i \(-0.903374\pi\)
0.954279 0.298918i \(-0.0966256\pi\)
\(978\) 0 0
\(979\) −1056.75 1056.75i −1.07942 1.07942i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1608.50 1.63632 0.818158 0.574993i \(-0.194995\pi\)
0.818158 + 0.574993i \(0.194995\pi\)
\(984\) 0 0
\(985\) 280.040i 0.284305i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 524.685 524.685i 0.530520 0.530520i
\(990\) 0 0
\(991\) 477.041 0.481374 0.240687 0.970603i \(-0.422627\pi\)
0.240687 + 0.970603i \(0.422627\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −582.600 + 582.600i −0.585528 + 0.585528i
\(996\) 0 0
\(997\) −783.852 + 783.852i −0.786210 + 0.786210i −0.980871 0.194660i \(-0.937640\pi\)
0.194660 + 0.980871i \(0.437640\pi\)
\(998\) 0 0
\(999\) 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 1152.3.j.b.161.11 32
3.2 odd 2 inner 1152.3.j.b.161.6 32
4.3 odd 2 1152.3.j.a.161.11 32
8.3 odd 2 144.3.j.a.125.4 yes 32
8.5 even 2 576.3.j.a.17.6 32
12.11 even 2 1152.3.j.a.161.6 32
16.3 odd 4 144.3.j.a.53.13 yes 32
16.5 even 4 inner 1152.3.j.b.737.6 32
16.11 odd 4 1152.3.j.a.737.6 32
16.13 even 4 576.3.j.a.305.11 32
24.5 odd 2 576.3.j.a.17.11 32
24.11 even 2 144.3.j.a.125.13 yes 32
48.5 odd 4 inner 1152.3.j.b.737.11 32
48.11 even 4 1152.3.j.a.737.11 32
48.29 odd 4 576.3.j.a.305.6 32
48.35 even 4 144.3.j.a.53.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.j.a.53.4 32 48.35 even 4
144.3.j.a.53.13 yes 32 16.3 odd 4
144.3.j.a.125.4 yes 32 8.3 odd 2
144.3.j.a.125.13 yes 32 24.11 even 2
576.3.j.a.17.6 32 8.5 even 2
576.3.j.a.17.11 32 24.5 odd 2
576.3.j.a.305.6 32 48.29 odd 4
576.3.j.a.305.11 32 16.13 even 4
1152.3.j.a.161.6 32 12.11 even 2
1152.3.j.a.161.11 32 4.3 odd 2
1152.3.j.a.737.6 32 16.11 odd 4
1152.3.j.a.737.11 32 48.11 even 4
1152.3.j.b.161.6 32 3.2 odd 2 inner
1152.3.j.b.161.11 32 1.1 even 1 trivial
1152.3.j.b.737.6 32 16.5 even 4 inner
1152.3.j.b.737.11 32 48.5 odd 4 inner