Properties

Label 3584.2.b.i.1793.8
Level $3584$
Weight $2$
Character 3584.1793
Analytic conductor $28.618$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(1793,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.1793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.b (of order \(2\), degree \(1\), 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: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + 108 x^{3} + 68 x^{2} + 32 x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1793.8
Root \(2.93456 + 1.21553i\) of defining polynomial
Character \(\chi\) \(=\) 3584.1793
Dual form 3584.2.b.i.1793.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.01685i q^{3} +1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} +O(q^{10})\) \(q+1.01685i q^{3} +1.29145i q^{5} -1.00000 q^{7} +1.96601 q^{9} -3.99188i q^{11} -0.534938i q^{13} -1.31322 q^{15} -6.47379 q^{17} -3.19046i q^{19} -1.01685i q^{21} -2.28447 q^{23} +3.33215 q^{25} +5.04971i q^{27} +6.09287i q^{29} +5.70453 q^{31} +4.05916 q^{33} -1.29145i q^{35} +6.44008i q^{37} +0.543954 q^{39} +4.27923 q^{41} +5.11578i q^{43} +2.53901i q^{45} +7.83993 q^{47} +1.00000 q^{49} -6.58291i q^{51} +13.5814i q^{53} +5.15533 q^{55} +3.24424 q^{57} +9.71935i q^{59} +2.06814i q^{61} -1.96601 q^{63} +0.690848 q^{65} -9.06175i q^{67} -2.32298i q^{69} +16.4535 q^{71} -8.18408 q^{73} +3.38831i q^{75} +3.99188i q^{77} +0.960761 q^{79} +0.763200 q^{81} +5.40599i q^{83} -8.36061i q^{85} -6.19556 q^{87} -4.97125 q^{89} +0.534938i q^{91} +5.80068i q^{93} +4.12034 q^{95} +6.97532 q^{97} -7.84806i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.01685i 0.587081i 0.955947 + 0.293541i \(0.0948336\pi\)
−0.955947 + 0.293541i \(0.905166\pi\)
\(4\) 0 0
\(5\) 1.29145i 0.577556i 0.957396 + 0.288778i \(0.0932489\pi\)
−0.957396 + 0.288778i \(0.906751\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.96601 0.655335
\(10\) 0 0
\(11\) − 3.99188i − 1.20360i −0.798648 0.601798i \(-0.794451\pi\)
0.798648 0.601798i \(-0.205549\pi\)
\(12\) 0 0
\(13\) − 0.534938i − 0.148365i −0.997245 0.0741825i \(-0.976365\pi\)
0.997245 0.0741825i \(-0.0236347\pi\)
\(14\) 0 0
\(15\) −1.31322 −0.339072
\(16\) 0 0
\(17\) −6.47379 −1.57013 −0.785063 0.619416i \(-0.787369\pi\)
−0.785063 + 0.619416i \(0.787369\pi\)
\(18\) 0 0
\(19\) − 3.19046i − 0.731942i −0.930626 0.365971i \(-0.880737\pi\)
0.930626 0.365971i \(-0.119263\pi\)
\(20\) 0 0
\(21\) − 1.01685i − 0.221896i
\(22\) 0 0
\(23\) −2.28447 −0.476346 −0.238173 0.971223i \(-0.576548\pi\)
−0.238173 + 0.971223i \(0.576548\pi\)
\(24\) 0 0
\(25\) 3.33215 0.666429
\(26\) 0 0
\(27\) 5.04971i 0.971817i
\(28\) 0 0
\(29\) 6.09287i 1.13142i 0.824605 + 0.565709i \(0.191397\pi\)
−0.824605 + 0.565709i \(0.808603\pi\)
\(30\) 0 0
\(31\) 5.70453 1.02456 0.512282 0.858818i \(-0.328800\pi\)
0.512282 + 0.858818i \(0.328800\pi\)
\(32\) 0 0
\(33\) 4.05916 0.706609
\(34\) 0 0
\(35\) − 1.29145i − 0.218296i
\(36\) 0 0
\(37\) 6.44008i 1.05874i 0.848390 + 0.529372i \(0.177572\pi\)
−0.848390 + 0.529372i \(0.822428\pi\)
\(38\) 0 0
\(39\) 0.543954 0.0871024
\(40\) 0 0
\(41\) 4.27923 0.668303 0.334152 0.942519i \(-0.391550\pi\)
0.334152 + 0.942519i \(0.391550\pi\)
\(42\) 0 0
\(43\) 5.11578i 0.780149i 0.920783 + 0.390074i \(0.127551\pi\)
−0.920783 + 0.390074i \(0.872449\pi\)
\(44\) 0 0
\(45\) 2.53901i 0.378493i
\(46\) 0 0
\(47\) 7.83993 1.14357 0.571786 0.820403i \(-0.306251\pi\)
0.571786 + 0.820403i \(0.306251\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 6.58291i − 0.921792i
\(52\) 0 0
\(53\) 13.5814i 1.86556i 0.360452 + 0.932778i \(0.382622\pi\)
−0.360452 + 0.932778i \(0.617378\pi\)
\(54\) 0 0
\(55\) 5.15533 0.695144
\(56\) 0 0
\(57\) 3.24424 0.429710
\(58\) 0 0
\(59\) 9.71935i 1.26535i 0.774417 + 0.632676i \(0.218043\pi\)
−0.774417 + 0.632676i \(0.781957\pi\)
\(60\) 0 0
\(61\) 2.06814i 0.264798i 0.991196 + 0.132399i \(0.0422680\pi\)
−0.991196 + 0.132399i \(0.957732\pi\)
\(62\) 0 0
\(63\) −1.96601 −0.247694
\(64\) 0 0
\(65\) 0.690848 0.0856891
\(66\) 0 0
\(67\) − 9.06175i − 1.10707i −0.832826 0.553535i \(-0.813279\pi\)
0.832826 0.553535i \(-0.186721\pi\)
\(68\) 0 0
\(69\) − 2.32298i − 0.279654i
\(70\) 0 0
\(71\) 16.4535 1.95267 0.976334 0.216266i \(-0.0693880\pi\)
0.976334 + 0.216266i \(0.0693880\pi\)
\(72\) 0 0
\(73\) −8.18408 −0.957874 −0.478937 0.877849i \(-0.658978\pi\)
−0.478937 + 0.877849i \(0.658978\pi\)
\(74\) 0 0
\(75\) 3.38831i 0.391248i
\(76\) 0 0
\(77\) 3.99188i 0.454917i
\(78\) 0 0
\(79\) 0.960761 0.108094 0.0540470 0.998538i \(-0.482788\pi\)
0.0540470 + 0.998538i \(0.482788\pi\)
\(80\) 0 0
\(81\) 0.763200 0.0848000
\(82\) 0 0
\(83\) 5.40599i 0.593384i 0.954973 + 0.296692i \(0.0958835\pi\)
−0.954973 + 0.296692i \(0.904116\pi\)
\(84\) 0 0
\(85\) − 8.36061i − 0.906835i
\(86\) 0 0
\(87\) −6.19556 −0.664234
\(88\) 0 0
\(89\) −4.97125 −0.526952 −0.263476 0.964666i \(-0.584869\pi\)
−0.263476 + 0.964666i \(0.584869\pi\)
\(90\) 0 0
\(91\) 0.534938i 0.0560767i
\(92\) 0 0
\(93\) 5.80068i 0.601502i
\(94\) 0 0
\(95\) 4.12034 0.422738
\(96\) 0 0
\(97\) 6.97532 0.708237 0.354118 0.935201i \(-0.384781\pi\)
0.354118 + 0.935201i \(0.384781\pi\)
\(98\) 0 0
\(99\) − 7.84806i − 0.788759i
\(100\) 0 0
\(101\) 6.19179i 0.616106i 0.951369 + 0.308053i \(0.0996775\pi\)
−0.951369 + 0.308053i \(0.900323\pi\)
\(102\) 0 0
\(103\) −9.38281 −0.924516 −0.462258 0.886745i \(-0.652961\pi\)
−0.462258 + 0.886745i \(0.652961\pi\)
\(104\) 0 0
\(105\) 1.31322 0.128157
\(106\) 0 0
\(107\) − 2.14532i − 0.207396i −0.994609 0.103698i \(-0.966933\pi\)
0.994609 0.103698i \(-0.0330675\pi\)
\(108\) 0 0
\(109\) − 8.50343i − 0.814481i −0.913321 0.407241i \(-0.866491\pi\)
0.913321 0.407241i \(-0.133509\pi\)
\(110\) 0 0
\(111\) −6.54863 −0.621569
\(112\) 0 0
\(113\) −13.0194 −1.22476 −0.612382 0.790562i \(-0.709788\pi\)
−0.612382 + 0.790562i \(0.709788\pi\)
\(114\) 0 0
\(115\) − 2.95029i − 0.275116i
\(116\) 0 0
\(117\) − 1.05169i − 0.0972289i
\(118\) 0 0
\(119\) 6.47379 0.593452
\(120\) 0 0
\(121\) −4.93509 −0.448644
\(122\) 0 0
\(123\) 4.35135i 0.392348i
\(124\) 0 0
\(125\) 10.7606i 0.962456i
\(126\) 0 0
\(127\) −2.47162 −0.219321 −0.109660 0.993969i \(-0.534976\pi\)
−0.109660 + 0.993969i \(0.534976\pi\)
\(128\) 0 0
\(129\) −5.20200 −0.458011
\(130\) 0 0
\(131\) − 3.65319i − 0.319181i −0.987183 0.159590i \(-0.948983\pi\)
0.987183 0.159590i \(-0.0510173\pi\)
\(132\) 0 0
\(133\) 3.19046i 0.276648i
\(134\) 0 0
\(135\) −6.52147 −0.561279
\(136\) 0 0
\(137\) 13.4016 1.14498 0.572490 0.819912i \(-0.305978\pi\)
0.572490 + 0.819912i \(0.305978\pi\)
\(138\) 0 0
\(139\) 16.3563i 1.38732i 0.720302 + 0.693661i \(0.244003\pi\)
−0.720302 + 0.693661i \(0.755997\pi\)
\(140\) 0 0
\(141\) 7.97207i 0.671370i
\(142\) 0 0
\(143\) −2.13541 −0.178572
\(144\) 0 0
\(145\) −7.86866 −0.653457
\(146\) 0 0
\(147\) 1.01685i 0.0838688i
\(148\) 0 0
\(149\) − 1.92053i − 0.157336i −0.996901 0.0786678i \(-0.974933\pi\)
0.996901 0.0786678i \(-0.0250666\pi\)
\(150\) 0 0
\(151\) 4.47078 0.363827 0.181913 0.983315i \(-0.441771\pi\)
0.181913 + 0.983315i \(0.441771\pi\)
\(152\) 0 0
\(153\) −12.7275 −1.02896
\(154\) 0 0
\(155\) 7.36714i 0.591743i
\(156\) 0 0
\(157\) 4.92407i 0.392984i 0.980505 + 0.196492i \(0.0629549\pi\)
−0.980505 + 0.196492i \(0.937045\pi\)
\(158\) 0 0
\(159\) −13.8104 −1.09523
\(160\) 0 0
\(161\) 2.28447 0.180042
\(162\) 0 0
\(163\) − 11.7644i − 0.921458i −0.887541 0.460729i \(-0.847588\pi\)
0.887541 0.460729i \(-0.152412\pi\)
\(164\) 0 0
\(165\) 5.24222i 0.408106i
\(166\) 0 0
\(167\) −11.5691 −0.895245 −0.447623 0.894223i \(-0.647729\pi\)
−0.447623 + 0.894223i \(0.647729\pi\)
\(168\) 0 0
\(169\) 12.7138 0.977988
\(170\) 0 0
\(171\) − 6.27247i − 0.479668i
\(172\) 0 0
\(173\) 25.6008i 1.94640i 0.229968 + 0.973198i \(0.426138\pi\)
−0.229968 + 0.973198i \(0.573862\pi\)
\(174\) 0 0
\(175\) −3.33215 −0.251887
\(176\) 0 0
\(177\) −9.88316 −0.742864
\(178\) 0 0
\(179\) − 10.2520i − 0.766267i −0.923693 0.383134i \(-0.874845\pi\)
0.923693 0.383134i \(-0.125155\pi\)
\(180\) 0 0
\(181\) 26.2524i 1.95133i 0.219270 + 0.975664i \(0.429632\pi\)
−0.219270 + 0.975664i \(0.570368\pi\)
\(182\) 0 0
\(183\) −2.10300 −0.155458
\(184\) 0 0
\(185\) −8.31708 −0.611484
\(186\) 0 0
\(187\) 25.8426i 1.88980i
\(188\) 0 0
\(189\) − 5.04971i − 0.367312i
\(190\) 0 0
\(191\) 8.18715 0.592401 0.296201 0.955126i \(-0.404280\pi\)
0.296201 + 0.955126i \(0.404280\pi\)
\(192\) 0 0
\(193\) 25.3226 1.82276 0.911382 0.411561i \(-0.135016\pi\)
0.911382 + 0.411561i \(0.135016\pi\)
\(194\) 0 0
\(195\) 0.702492i 0.0503065i
\(196\) 0 0
\(197\) 20.8467i 1.48526i 0.669699 + 0.742632i \(0.266423\pi\)
−0.669699 + 0.742632i \(0.733577\pi\)
\(198\) 0 0
\(199\) 20.2055 1.43233 0.716164 0.697932i \(-0.245896\pi\)
0.716164 + 0.697932i \(0.245896\pi\)
\(200\) 0 0
\(201\) 9.21449 0.649940
\(202\) 0 0
\(203\) − 6.09287i − 0.427636i
\(204\) 0 0
\(205\) 5.52643i 0.385982i
\(206\) 0 0
\(207\) −4.49129 −0.312166
\(208\) 0 0
\(209\) −12.7359 −0.880963
\(210\) 0 0
\(211\) 0.231640i 0.0159468i 0.999968 + 0.00797339i \(0.00253804\pi\)
−0.999968 + 0.00797339i \(0.997462\pi\)
\(212\) 0 0
\(213\) 16.7308i 1.14638i
\(214\) 0 0
\(215\) −6.60679 −0.450580
\(216\) 0 0
\(217\) −5.70453 −0.387249
\(218\) 0 0
\(219\) − 8.32202i − 0.562350i
\(220\) 0 0
\(221\) 3.46308i 0.232952i
\(222\) 0 0
\(223\) −22.3886 −1.49925 −0.749625 0.661863i \(-0.769766\pi\)
−0.749625 + 0.661863i \(0.769766\pi\)
\(224\) 0 0
\(225\) 6.55102 0.436735
\(226\) 0 0
\(227\) − 17.1167i − 1.13608i −0.823002 0.568038i \(-0.807703\pi\)
0.823002 0.568038i \(-0.192297\pi\)
\(228\) 0 0
\(229\) − 15.0638i − 0.995445i −0.867336 0.497723i \(-0.834170\pi\)
0.867336 0.497723i \(-0.165830\pi\)
\(230\) 0 0
\(231\) −4.05916 −0.267073
\(232\) 0 0
\(233\) −11.8566 −0.776752 −0.388376 0.921501i \(-0.626964\pi\)
−0.388376 + 0.921501i \(0.626964\pi\)
\(234\) 0 0
\(235\) 10.1249i 0.660477i
\(236\) 0 0
\(237\) 0.976955i 0.0634600i
\(238\) 0 0
\(239\) 8.55444 0.553341 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(240\) 0 0
\(241\) −14.3170 −0.922236 −0.461118 0.887339i \(-0.652552\pi\)
−0.461118 + 0.887339i \(0.652552\pi\)
\(242\) 0 0
\(243\) 15.9252i 1.02160i
\(244\) 0 0
\(245\) 1.29145i 0.0825080i
\(246\) 0 0
\(247\) −1.70670 −0.108595
\(248\) 0 0
\(249\) −5.49710 −0.348365
\(250\) 0 0
\(251\) − 0.346212i − 0.0218527i −0.999940 0.0109264i \(-0.996522\pi\)
0.999940 0.0109264i \(-0.00347804\pi\)
\(252\) 0 0
\(253\) 9.11934i 0.573328i
\(254\) 0 0
\(255\) 8.50153 0.532386
\(256\) 0 0
\(257\) 21.7703 1.35799 0.678996 0.734142i \(-0.262415\pi\)
0.678996 + 0.734142i \(0.262415\pi\)
\(258\) 0 0
\(259\) − 6.44008i − 0.400167i
\(260\) 0 0
\(261\) 11.9786i 0.741458i
\(262\) 0 0
\(263\) −16.0380 −0.988946 −0.494473 0.869193i \(-0.664639\pi\)
−0.494473 + 0.869193i \(0.664639\pi\)
\(264\) 0 0
\(265\) −17.5398 −1.07746
\(266\) 0 0
\(267\) − 5.05504i − 0.309363i
\(268\) 0 0
\(269\) − 6.98157i − 0.425674i −0.977088 0.212837i \(-0.931730\pi\)
0.977088 0.212837i \(-0.0682703\pi\)
\(270\) 0 0
\(271\) −3.13231 −0.190275 −0.0951373 0.995464i \(-0.530329\pi\)
−0.0951373 + 0.995464i \(0.530329\pi\)
\(272\) 0 0
\(273\) −0.543954 −0.0329216
\(274\) 0 0
\(275\) − 13.3015i − 0.802112i
\(276\) 0 0
\(277\) 2.07134i 0.124454i 0.998062 + 0.0622272i \(0.0198204\pi\)
−0.998062 + 0.0622272i \(0.980180\pi\)
\(278\) 0 0
\(279\) 11.2151 0.671433
\(280\) 0 0
\(281\) −10.1923 −0.608024 −0.304012 0.952668i \(-0.598326\pi\)
−0.304012 + 0.952668i \(0.598326\pi\)
\(282\) 0 0
\(283\) − 14.8446i − 0.882421i −0.897404 0.441210i \(-0.854549\pi\)
0.897404 0.441210i \(-0.145451\pi\)
\(284\) 0 0
\(285\) 4.18978i 0.248181i
\(286\) 0 0
\(287\) −4.27923 −0.252595
\(288\) 0 0
\(289\) 24.9100 1.46529
\(290\) 0 0
\(291\) 7.09289i 0.415792i
\(292\) 0 0
\(293\) − 10.2362i − 0.598007i −0.954252 0.299004i \(-0.903346\pi\)
0.954252 0.299004i \(-0.0966543\pi\)
\(294\) 0 0
\(295\) −12.5521 −0.730811
\(296\) 0 0
\(297\) 20.1578 1.16967
\(298\) 0 0
\(299\) 1.22205i 0.0706730i
\(300\) 0 0
\(301\) − 5.11578i − 0.294869i
\(302\) 0 0
\(303\) −6.29615 −0.361705
\(304\) 0 0
\(305\) −2.67091 −0.152936
\(306\) 0 0
\(307\) − 19.6132i − 1.11938i −0.828701 0.559691i \(-0.810920\pi\)
0.828701 0.559691i \(-0.189080\pi\)
\(308\) 0 0
\(309\) − 9.54096i − 0.542766i
\(310\) 0 0
\(311\) −9.88322 −0.560426 −0.280213 0.959938i \(-0.590405\pi\)
−0.280213 + 0.959938i \(0.590405\pi\)
\(312\) 0 0
\(313\) −0.573015 −0.0323887 −0.0161944 0.999869i \(-0.505155\pi\)
−0.0161944 + 0.999869i \(0.505155\pi\)
\(314\) 0 0
\(315\) − 2.53901i − 0.143057i
\(316\) 0 0
\(317\) − 19.7491i − 1.10922i −0.832110 0.554611i \(-0.812867\pi\)
0.832110 0.554611i \(-0.187133\pi\)
\(318\) 0 0
\(319\) 24.3220 1.36177
\(320\) 0 0
\(321\) 2.18148 0.121758
\(322\) 0 0
\(323\) 20.6544i 1.14924i
\(324\) 0 0
\(325\) − 1.78249i − 0.0988748i
\(326\) 0 0
\(327\) 8.64676 0.478167
\(328\) 0 0
\(329\) −7.83993 −0.432230
\(330\) 0 0
\(331\) 22.8436i 1.25560i 0.778376 + 0.627799i \(0.216044\pi\)
−0.778376 + 0.627799i \(0.783956\pi\)
\(332\) 0 0
\(333\) 12.6612i 0.693832i
\(334\) 0 0
\(335\) 11.7028 0.639395
\(336\) 0 0
\(337\) −2.69392 −0.146747 −0.0733736 0.997305i \(-0.523377\pi\)
−0.0733736 + 0.997305i \(0.523377\pi\)
\(338\) 0 0
\(339\) − 13.2389i − 0.719036i
\(340\) 0 0
\(341\) − 22.7718i − 1.23316i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.00002 0.161516
\(346\) 0 0
\(347\) 6.08073i 0.326431i 0.986590 + 0.163215i \(0.0521865\pi\)
−0.986590 + 0.163215i \(0.947813\pi\)
\(348\) 0 0
\(349\) − 17.0896i − 0.914787i −0.889264 0.457394i \(-0.848783\pi\)
0.889264 0.457394i \(-0.151217\pi\)
\(350\) 0 0
\(351\) 2.70128 0.144184
\(352\) 0 0
\(353\) 28.3967 1.51140 0.755702 0.654916i \(-0.227296\pi\)
0.755702 + 0.654916i \(0.227296\pi\)
\(354\) 0 0
\(355\) 21.2489i 1.12778i
\(356\) 0 0
\(357\) 6.58291i 0.348404i
\(358\) 0 0
\(359\) −5.18843 −0.273835 −0.136917 0.990582i \(-0.543719\pi\)
−0.136917 + 0.990582i \(0.543719\pi\)
\(360\) 0 0
\(361\) 8.82095 0.464260
\(362\) 0 0
\(363\) − 5.01826i − 0.263391i
\(364\) 0 0
\(365\) − 10.5694i − 0.553226i
\(366\) 0 0
\(367\) −20.9256 −1.09231 −0.546153 0.837685i \(-0.683908\pi\)
−0.546153 + 0.837685i \(0.683908\pi\)
\(368\) 0 0
\(369\) 8.41299 0.437963
\(370\) 0 0
\(371\) − 13.5814i − 0.705114i
\(372\) 0 0
\(373\) 5.93536i 0.307321i 0.988124 + 0.153661i \(0.0491063\pi\)
−0.988124 + 0.153661i \(0.950894\pi\)
\(374\) 0 0
\(375\) −10.9420 −0.565040
\(376\) 0 0
\(377\) 3.25931 0.167863
\(378\) 0 0
\(379\) 6.04243i 0.310379i 0.987885 + 0.155189i \(0.0495988\pi\)
−0.987885 + 0.155189i \(0.950401\pi\)
\(380\) 0 0
\(381\) − 2.51328i − 0.128759i
\(382\) 0 0
\(383\) 3.53306 0.180531 0.0902653 0.995918i \(-0.471228\pi\)
0.0902653 + 0.995918i \(0.471228\pi\)
\(384\) 0 0
\(385\) −5.15533 −0.262740
\(386\) 0 0
\(387\) 10.0577i 0.511259i
\(388\) 0 0
\(389\) 2.75123i 0.139493i 0.997565 + 0.0697466i \(0.0222191\pi\)
−0.997565 + 0.0697466i \(0.977781\pi\)
\(390\) 0 0
\(391\) 14.7892 0.747922
\(392\) 0 0
\(393\) 3.71477 0.187385
\(394\) 0 0
\(395\) 1.24078i 0.0624304i
\(396\) 0 0
\(397\) 27.5791i 1.38416i 0.721822 + 0.692079i \(0.243305\pi\)
−0.721822 + 0.692079i \(0.756695\pi\)
\(398\) 0 0
\(399\) −3.24424 −0.162415
\(400\) 0 0
\(401\) −1.51409 −0.0756099 −0.0378050 0.999285i \(-0.512037\pi\)
−0.0378050 + 0.999285i \(0.512037\pi\)
\(402\) 0 0
\(403\) − 3.05157i − 0.152009i
\(404\) 0 0
\(405\) 0.985638i 0.0489767i
\(406\) 0 0
\(407\) 25.7080 1.27430
\(408\) 0 0
\(409\) 3.50489 0.173306 0.0866529 0.996239i \(-0.472383\pi\)
0.0866529 + 0.996239i \(0.472383\pi\)
\(410\) 0 0
\(411\) 13.6275i 0.672196i
\(412\) 0 0
\(413\) − 9.71935i − 0.478258i
\(414\) 0 0
\(415\) −6.98159 −0.342713
\(416\) 0 0
\(417\) −16.6320 −0.814471
\(418\) 0 0
\(419\) − 0.0423241i − 0.00206767i −0.999999 0.00103383i \(-0.999671\pi\)
0.999999 0.00103383i \(-0.000329080\pi\)
\(420\) 0 0
\(421\) − 25.8137i − 1.25808i −0.777372 0.629041i \(-0.783448\pi\)
0.777372 0.629041i \(-0.216552\pi\)
\(422\) 0 0
\(423\) 15.4134 0.749423
\(424\) 0 0
\(425\) −21.5716 −1.04638
\(426\) 0 0
\(427\) − 2.06814i − 0.100084i
\(428\) 0 0
\(429\) − 2.17140i − 0.104836i
\(430\) 0 0
\(431\) 33.0398 1.59147 0.795735 0.605645i \(-0.207085\pi\)
0.795735 + 0.605645i \(0.207085\pi\)
\(432\) 0 0
\(433\) −10.7807 −0.518086 −0.259043 0.965866i \(-0.583407\pi\)
−0.259043 + 0.965866i \(0.583407\pi\)
\(434\) 0 0
\(435\) − 8.00129i − 0.383632i
\(436\) 0 0
\(437\) 7.28853i 0.348657i
\(438\) 0 0
\(439\) −23.5740 −1.12513 −0.562564 0.826754i \(-0.690185\pi\)
−0.562564 + 0.826754i \(0.690185\pi\)
\(440\) 0 0
\(441\) 1.96601 0.0936193
\(442\) 0 0
\(443\) − 35.7764i − 1.69979i −0.526955 0.849893i \(-0.676666\pi\)
0.526955 0.849893i \(-0.323334\pi\)
\(444\) 0 0
\(445\) − 6.42014i − 0.304344i
\(446\) 0 0
\(447\) 1.95290 0.0923688
\(448\) 0 0
\(449\) −12.9320 −0.610300 −0.305150 0.952304i \(-0.598707\pi\)
−0.305150 + 0.952304i \(0.598707\pi\)
\(450\) 0 0
\(451\) − 17.0822i − 0.804367i
\(452\) 0 0
\(453\) 4.54613i 0.213596i
\(454\) 0 0
\(455\) −0.690848 −0.0323875
\(456\) 0 0
\(457\) 30.5150 1.42743 0.713715 0.700436i \(-0.247011\pi\)
0.713715 + 0.700436i \(0.247011\pi\)
\(458\) 0 0
\(459\) − 32.6908i − 1.52587i
\(460\) 0 0
\(461\) 19.6155i 0.913585i 0.889573 + 0.456793i \(0.151002\pi\)
−0.889573 + 0.456793i \(0.848998\pi\)
\(462\) 0 0
\(463\) 15.3254 0.712231 0.356116 0.934442i \(-0.384101\pi\)
0.356116 + 0.934442i \(0.384101\pi\)
\(464\) 0 0
\(465\) −7.49131 −0.347401
\(466\) 0 0
\(467\) 30.2233i 1.39857i 0.714845 + 0.699283i \(0.246497\pi\)
−0.714845 + 0.699283i \(0.753503\pi\)
\(468\) 0 0
\(469\) 9.06175i 0.418433i
\(470\) 0 0
\(471\) −5.00706 −0.230713
\(472\) 0 0
\(473\) 20.4216 0.938984
\(474\) 0 0
\(475\) − 10.6311i − 0.487788i
\(476\) 0 0
\(477\) 26.7012i 1.22256i
\(478\) 0 0
\(479\) −18.2121 −0.832131 −0.416066 0.909335i \(-0.636591\pi\)
−0.416066 + 0.909335i \(0.636591\pi\)
\(480\) 0 0
\(481\) 3.44505 0.157081
\(482\) 0 0
\(483\) 2.32298i 0.105699i
\(484\) 0 0
\(485\) 9.00831i 0.409046i
\(486\) 0 0
\(487\) −33.1641 −1.50281 −0.751404 0.659843i \(-0.770623\pi\)
−0.751404 + 0.659843i \(0.770623\pi\)
\(488\) 0 0
\(489\) 11.9627 0.540971
\(490\) 0 0
\(491\) 38.1547i 1.72190i 0.508690 + 0.860950i \(0.330130\pi\)
−0.508690 + 0.860950i \(0.669870\pi\)
\(492\) 0 0
\(493\) − 39.4440i − 1.77647i
\(494\) 0 0
\(495\) 10.1354 0.455553
\(496\) 0 0
\(497\) −16.4535 −0.738039
\(498\) 0 0
\(499\) − 1.44819i − 0.0648298i −0.999474 0.0324149i \(-0.989680\pi\)
0.999474 0.0324149i \(-0.0103198\pi\)
\(500\) 0 0
\(501\) − 11.7641i − 0.525582i
\(502\) 0 0
\(503\) −14.4905 −0.646098 −0.323049 0.946382i \(-0.604708\pi\)
−0.323049 + 0.946382i \(0.604708\pi\)
\(504\) 0 0
\(505\) −7.99642 −0.355836
\(506\) 0 0
\(507\) 12.9281i 0.574158i
\(508\) 0 0
\(509\) 29.1421i 1.29170i 0.763464 + 0.645850i \(0.223497\pi\)
−0.763464 + 0.645850i \(0.776503\pi\)
\(510\) 0 0
\(511\) 8.18408 0.362042
\(512\) 0 0
\(513\) 16.1109 0.711314
\(514\) 0 0
\(515\) − 12.1175i − 0.533960i
\(516\) 0 0
\(517\) − 31.2961i − 1.37640i
\(518\) 0 0
\(519\) −26.0323 −1.14269
\(520\) 0 0
\(521\) −1.25214 −0.0548573 −0.0274287 0.999624i \(-0.508732\pi\)
−0.0274287 + 0.999624i \(0.508732\pi\)
\(522\) 0 0
\(523\) 34.3401i 1.50159i 0.660536 + 0.750795i \(0.270329\pi\)
−0.660536 + 0.750795i \(0.729671\pi\)
\(524\) 0 0
\(525\) − 3.38831i − 0.147878i
\(526\) 0 0
\(527\) −36.9299 −1.60869
\(528\) 0 0
\(529\) −17.7812 −0.773095
\(530\) 0 0
\(531\) 19.1083i 0.829229i
\(532\) 0 0
\(533\) − 2.28912i − 0.0991529i
\(534\) 0 0
\(535\) 2.77058 0.119783
\(536\) 0 0
\(537\) 10.4247 0.449861
\(538\) 0 0
\(539\) − 3.99188i − 0.171942i
\(540\) 0 0
\(541\) − 37.9611i − 1.63207i −0.578000 0.816037i \(-0.696167\pi\)
0.578000 0.816037i \(-0.303833\pi\)
\(542\) 0 0
\(543\) −26.6949 −1.14559
\(544\) 0 0
\(545\) 10.9818 0.470409
\(546\) 0 0
\(547\) 9.82521i 0.420096i 0.977691 + 0.210048i \(0.0673620\pi\)
−0.977691 + 0.210048i \(0.932638\pi\)
\(548\) 0 0
\(549\) 4.06598i 0.173532i
\(550\) 0 0
\(551\) 19.4391 0.828132
\(552\) 0 0
\(553\) −0.960761 −0.0408557
\(554\) 0 0
\(555\) − 8.45726i − 0.358991i
\(556\) 0 0
\(557\) − 32.5759i − 1.38028i −0.723674 0.690142i \(-0.757548\pi\)
0.723674 0.690142i \(-0.242452\pi\)
\(558\) 0 0
\(559\) 2.73662 0.115747
\(560\) 0 0
\(561\) −26.2782 −1.10946
\(562\) 0 0
\(563\) 40.1157i 1.69067i 0.534234 + 0.845337i \(0.320600\pi\)
−0.534234 + 0.845337i \(0.679400\pi\)
\(564\) 0 0
\(565\) − 16.8140i − 0.707369i
\(566\) 0 0
\(567\) −0.763200 −0.0320514
\(568\) 0 0
\(569\) −0.461975 −0.0193670 −0.00968351 0.999953i \(-0.503082\pi\)
−0.00968351 + 0.999953i \(0.503082\pi\)
\(570\) 0 0
\(571\) − 41.6492i − 1.74296i −0.490428 0.871482i \(-0.663159\pi\)
0.490428 0.871482i \(-0.336841\pi\)
\(572\) 0 0
\(573\) 8.32514i 0.347788i
\(574\) 0 0
\(575\) −7.61220 −0.317451
\(576\) 0 0
\(577\) −12.1029 −0.503851 −0.251925 0.967747i \(-0.581064\pi\)
−0.251925 + 0.967747i \(0.581064\pi\)
\(578\) 0 0
\(579\) 25.7495i 1.07011i
\(580\) 0 0
\(581\) − 5.40599i − 0.224278i
\(582\) 0 0
\(583\) 54.2155 2.24538
\(584\) 0 0
\(585\) 1.35821 0.0561551
\(586\) 0 0
\(587\) − 14.2165i − 0.586779i −0.955993 0.293389i \(-0.905217\pi\)
0.955993 0.293389i \(-0.0947833\pi\)
\(588\) 0 0
\(589\) − 18.2001i − 0.749921i
\(590\) 0 0
\(591\) −21.1980 −0.871971
\(592\) 0 0
\(593\) −12.7044 −0.521705 −0.260853 0.965379i \(-0.584004\pi\)
−0.260853 + 0.965379i \(0.584004\pi\)
\(594\) 0 0
\(595\) 8.36061i 0.342752i
\(596\) 0 0
\(597\) 20.5460i 0.840894i
\(598\) 0 0
\(599\) 14.0608 0.574510 0.287255 0.957854i \(-0.407257\pi\)
0.287255 + 0.957854i \(0.407257\pi\)
\(600\) 0 0
\(601\) −26.0142 −1.06114 −0.530570 0.847641i \(-0.678022\pi\)
−0.530570 + 0.847641i \(0.678022\pi\)
\(602\) 0 0
\(603\) − 17.8155i − 0.725502i
\(604\) 0 0
\(605\) − 6.37344i − 0.259117i
\(606\) 0 0
\(607\) 28.0201 1.13730 0.568650 0.822579i \(-0.307466\pi\)
0.568650 + 0.822579i \(0.307466\pi\)
\(608\) 0 0
\(609\) 6.19556 0.251057
\(610\) 0 0
\(611\) − 4.19388i − 0.169666i
\(612\) 0 0
\(613\) 18.1315i 0.732323i 0.930551 + 0.366161i \(0.119328\pi\)
−0.930551 + 0.366161i \(0.880672\pi\)
\(614\) 0 0
\(615\) −5.61957 −0.226603
\(616\) 0 0
\(617\) −24.2692 −0.977042 −0.488521 0.872552i \(-0.662463\pi\)
−0.488521 + 0.872552i \(0.662463\pi\)
\(618\) 0 0
\(619\) − 34.4384i − 1.38420i −0.721803 0.692099i \(-0.756686\pi\)
0.721803 0.692099i \(-0.243314\pi\)
\(620\) 0 0
\(621\) − 11.5359i − 0.462921i
\(622\) 0 0
\(623\) 4.97125 0.199169
\(624\) 0 0
\(625\) 2.76392 0.110557
\(626\) 0 0
\(627\) − 12.9506i − 0.517197i
\(628\) 0 0
\(629\) − 41.6918i − 1.66236i
\(630\) 0 0
\(631\) 45.3042 1.80353 0.901766 0.432224i \(-0.142271\pi\)
0.901766 + 0.432224i \(0.142271\pi\)
\(632\) 0 0
\(633\) −0.235545 −0.00936206
\(634\) 0 0
\(635\) − 3.19199i − 0.126670i
\(636\) 0 0
\(637\) − 0.534938i − 0.0211950i
\(638\) 0 0
\(639\) 32.3476 1.27965
\(640\) 0 0
\(641\) 46.2702 1.82756 0.913782 0.406204i \(-0.133148\pi\)
0.913782 + 0.406204i \(0.133148\pi\)
\(642\) 0 0
\(643\) 20.9674i 0.826872i 0.910533 + 0.413436i \(0.135671\pi\)
−0.910533 + 0.413436i \(0.864329\pi\)
\(644\) 0 0
\(645\) − 6.71815i − 0.264527i
\(646\) 0 0
\(647\) 44.7211 1.75817 0.879083 0.476668i \(-0.158156\pi\)
0.879083 + 0.476668i \(0.158156\pi\)
\(648\) 0 0
\(649\) 38.7984 1.52297
\(650\) 0 0
\(651\) − 5.80068i − 0.227346i
\(652\) 0 0
\(653\) − 2.23312i − 0.0873886i −0.999045 0.0436943i \(-0.986087\pi\)
0.999045 0.0436943i \(-0.0139128\pi\)
\(654\) 0 0
\(655\) 4.71793 0.184345
\(656\) 0 0
\(657\) −16.0899 −0.627729
\(658\) 0 0
\(659\) 45.3644i 1.76715i 0.468294 + 0.883573i \(0.344869\pi\)
−0.468294 + 0.883573i \(0.655131\pi\)
\(660\) 0 0
\(661\) − 36.0951i − 1.40394i −0.712208 0.701969i \(-0.752305\pi\)
0.712208 0.701969i \(-0.247695\pi\)
\(662\) 0 0
\(663\) −3.52145 −0.136762
\(664\) 0 0
\(665\) −4.12034 −0.159780
\(666\) 0 0
\(667\) − 13.9190i − 0.538946i
\(668\) 0 0
\(669\) − 22.7659i − 0.880181i
\(670\) 0 0
\(671\) 8.25576 0.318710
\(672\) 0 0
\(673\) 9.23324 0.355915 0.177958 0.984038i \(-0.443051\pi\)
0.177958 + 0.984038i \(0.443051\pi\)
\(674\) 0 0
\(675\) 16.8264i 0.647647i
\(676\) 0 0
\(677\) − 42.4578i − 1.63179i −0.578202 0.815893i \(-0.696246\pi\)
0.578202 0.815893i \(-0.303754\pi\)
\(678\) 0 0
\(679\) −6.97532 −0.267688
\(680\) 0 0
\(681\) 17.4052 0.666969
\(682\) 0 0
\(683\) − 29.5156i − 1.12938i −0.825302 0.564692i \(-0.808995\pi\)
0.825302 0.564692i \(-0.191005\pi\)
\(684\) 0 0
\(685\) 17.3076i 0.661289i
\(686\) 0 0
\(687\) 15.3177 0.584407
\(688\) 0 0
\(689\) 7.26523 0.276783
\(690\) 0 0
\(691\) 21.1911i 0.806147i 0.915168 + 0.403073i \(0.132058\pi\)
−0.915168 + 0.403073i \(0.867942\pi\)
\(692\) 0 0
\(693\) 7.84806i 0.298123i
\(694\) 0 0
\(695\) −21.1234 −0.801256
\(696\) 0 0
\(697\) −27.7028 −1.04932
\(698\) 0 0
\(699\) − 12.0564i − 0.456017i
\(700\) 0 0
\(701\) 10.3625i 0.391386i 0.980665 + 0.195693i \(0.0626956\pi\)
−0.980665 + 0.195693i \(0.937304\pi\)
\(702\) 0 0
\(703\) 20.5468 0.774939
\(704\) 0 0
\(705\) −10.2956 −0.387754
\(706\) 0 0
\(707\) − 6.19179i − 0.232866i
\(708\) 0 0
\(709\) − 14.9196i − 0.560318i −0.959954 0.280159i \(-0.909613\pi\)
0.959954 0.280159i \(-0.0903871\pi\)
\(710\) 0 0
\(711\) 1.88886 0.0708379
\(712\) 0 0
\(713\) −13.0318 −0.488046
\(714\) 0 0
\(715\) − 2.75778i − 0.103135i
\(716\) 0 0
\(717\) 8.69863i 0.324856i
\(718\) 0 0
\(719\) −19.3570 −0.721896 −0.360948 0.932586i \(-0.617547\pi\)
−0.360948 + 0.932586i \(0.617547\pi\)
\(720\) 0 0
\(721\) 9.38281 0.349434
\(722\) 0 0
\(723\) − 14.5583i − 0.541428i
\(724\) 0 0
\(725\) 20.3023i 0.754010i
\(726\) 0 0
\(727\) 11.1685 0.414216 0.207108 0.978318i \(-0.433595\pi\)
0.207108 + 0.978318i \(0.433595\pi\)
\(728\) 0 0
\(729\) −13.9040 −0.514963
\(730\) 0 0
\(731\) − 33.1185i − 1.22493i
\(732\) 0 0
\(733\) − 27.4759i − 1.01484i −0.861698 0.507422i \(-0.830599\pi\)
0.861698 0.507422i \(-0.169401\pi\)
\(734\) 0 0
\(735\) −1.31322 −0.0484389
\(736\) 0 0
\(737\) −36.1734 −1.33246
\(738\) 0 0
\(739\) − 47.1619i − 1.73488i −0.497542 0.867440i \(-0.665764\pi\)
0.497542 0.867440i \(-0.334236\pi\)
\(740\) 0 0
\(741\) − 1.73547i − 0.0637539i
\(742\) 0 0
\(743\) −47.0796 −1.72718 −0.863591 0.504193i \(-0.831790\pi\)
−0.863591 + 0.504193i \(0.831790\pi\)
\(744\) 0 0
\(745\) 2.48027 0.0908701
\(746\) 0 0
\(747\) 10.6282i 0.388866i
\(748\) 0 0
\(749\) 2.14532i 0.0783882i
\(750\) 0 0
\(751\) 44.5767 1.62663 0.813313 0.581827i \(-0.197662\pi\)
0.813313 + 0.581827i \(0.197662\pi\)
\(752\) 0 0
\(753\) 0.352047 0.0128293
\(754\) 0 0
\(755\) 5.77381i 0.210130i
\(756\) 0 0
\(757\) − 28.9656i − 1.05277i −0.850246 0.526386i \(-0.823547\pi\)
0.850246 0.526386i \(-0.176453\pi\)
\(758\) 0 0
\(759\) −9.27304 −0.336590
\(760\) 0 0
\(761\) 9.89164 0.358572 0.179286 0.983797i \(-0.442621\pi\)
0.179286 + 0.983797i \(0.442621\pi\)
\(762\) 0 0
\(763\) 8.50343i 0.307845i
\(764\) 0 0
\(765\) − 16.4370i − 0.594281i
\(766\) 0 0
\(767\) 5.19925 0.187734
\(768\) 0 0
\(769\) −22.1642 −0.799260 −0.399630 0.916677i \(-0.630861\pi\)
−0.399630 + 0.916677i \(0.630861\pi\)
\(770\) 0 0
\(771\) 22.1372i 0.797252i
\(772\) 0 0
\(773\) 40.6860i 1.46337i 0.681641 + 0.731686i \(0.261266\pi\)
−0.681641 + 0.731686i \(0.738734\pi\)
\(774\) 0 0
\(775\) 19.0083 0.682799
\(776\) 0 0
\(777\) 6.54863 0.234931
\(778\) 0 0
\(779\) − 13.6527i − 0.489159i
\(780\) 0 0
\(781\) − 65.6803i − 2.35022i
\(782\) 0 0
\(783\) −30.7672 −1.09953
\(784\) 0 0
\(785\) −6.35921 −0.226970
\(786\) 0 0
\(787\) − 50.6622i − 1.80591i −0.429734 0.902955i \(-0.641393\pi\)
0.429734 0.902955i \(-0.358607\pi\)
\(788\) 0 0
\(789\) − 16.3083i − 0.580592i
\(790\) 0 0
\(791\) 13.0194 0.462917
\(792\) 0 0
\(793\) 1.10633 0.0392868
\(794\) 0 0
\(795\) − 17.8355i − 0.632558i
\(796\) 0 0
\(797\) 4.92249i 0.174364i 0.996192 + 0.0871818i \(0.0277861\pi\)
−0.996192 + 0.0871818i \(0.972214\pi\)
\(798\) 0 0
\(799\) −50.7541 −1.79555
\(800\) 0 0
\(801\) −9.77351 −0.345330
\(802\) 0 0
\(803\) 32.6698i 1.15289i
\(804\) 0 0
\(805\) 2.95029i 0.103984i
\(806\) 0 0
\(807\) 7.09924 0.249905
\(808\) 0 0
\(809\) −22.1155 −0.777538 −0.388769 0.921335i \(-0.627100\pi\)
−0.388769 + 0.921335i \(0.627100\pi\)
\(810\) 0 0
\(811\) 46.3318i 1.62693i 0.581615 + 0.813464i \(0.302421\pi\)
−0.581615 + 0.813464i \(0.697579\pi\)
\(812\) 0 0
\(813\) − 3.18511i − 0.111707i
\(814\) 0 0
\(815\) 15.1932 0.532194
\(816\) 0 0
\(817\) 16.3217 0.571024
\(818\) 0 0
\(819\) 1.05169i 0.0367491i
\(820\) 0 0
\(821\) 41.9810i 1.46515i 0.680689 + 0.732573i \(0.261681\pi\)
−0.680689 + 0.732573i \(0.738319\pi\)
\(822\) 0 0
\(823\) 33.7030 1.17481 0.587406 0.809292i \(-0.300149\pi\)
0.587406 + 0.809292i \(0.300149\pi\)
\(824\) 0 0
\(825\) 13.5257 0.470905
\(826\) 0 0
\(827\) 7.51764i 0.261414i 0.991421 + 0.130707i \(0.0417247\pi\)
−0.991421 + 0.130707i \(0.958275\pi\)
\(828\) 0 0
\(829\) 12.0080i 0.417055i 0.978016 + 0.208528i \(0.0668671\pi\)
−0.978016 + 0.208528i \(0.933133\pi\)
\(830\) 0 0
\(831\) −2.10625 −0.0730649
\(832\) 0 0
\(833\) −6.47379 −0.224304
\(834\) 0 0
\(835\) − 14.9410i − 0.517054i
\(836\) 0 0
\(837\) 28.8062i 0.995688i
\(838\) 0 0
\(839\) −0.308970 −0.0106668 −0.00533341 0.999986i \(-0.501698\pi\)
−0.00533341 + 0.999986i \(0.501698\pi\)
\(840\) 0 0
\(841\) −8.12306 −0.280105
\(842\) 0 0
\(843\) − 10.3641i − 0.356959i
\(844\) 0 0
\(845\) 16.4193i 0.564843i
\(846\) 0 0
\(847\) 4.93509 0.169572
\(848\) 0 0
\(849\) 15.0948 0.518053
\(850\) 0 0
\(851\) − 14.7122i − 0.504328i
\(852\) 0 0
\(853\) − 48.4197i − 1.65786i −0.559354 0.828929i \(-0.688951\pi\)
0.559354 0.828929i \(-0.311049\pi\)
\(854\) 0 0
\(855\) 8.10061 0.277035
\(856\) 0 0
\(857\) −8.09941 −0.276671 −0.138335 0.990385i \(-0.544175\pi\)
−0.138335 + 0.990385i \(0.544175\pi\)
\(858\) 0 0
\(859\) 40.2757i 1.37419i 0.726568 + 0.687094i \(0.241114\pi\)
−0.726568 + 0.687094i \(0.758886\pi\)
\(860\) 0 0
\(861\) − 4.35135i − 0.148294i
\(862\) 0 0
\(863\) −2.93426 −0.0998835 −0.0499417 0.998752i \(-0.515904\pi\)
−0.0499417 + 0.998752i \(0.515904\pi\)
\(864\) 0 0
\(865\) −33.0623 −1.12415
\(866\) 0 0
\(867\) 25.3299i 0.860247i
\(868\) 0 0
\(869\) − 3.83524i − 0.130102i
\(870\) 0 0
\(871\) −4.84748 −0.164250
\(872\) 0 0
\(873\) 13.7135 0.464132
\(874\) 0 0
\(875\) − 10.7606i − 0.363774i
\(876\) 0 0
\(877\) − 54.3937i − 1.83674i −0.395719 0.918372i \(-0.629505\pi\)
0.395719 0.918372i \(-0.370495\pi\)
\(878\) 0 0
\(879\) 10.4088 0.351079
\(880\) 0 0
\(881\) 27.0359 0.910862 0.455431 0.890271i \(-0.349485\pi\)
0.455431 + 0.890271i \(0.349485\pi\)
\(882\) 0 0
\(883\) 13.3882i 0.450549i 0.974295 + 0.225274i \(0.0723279\pi\)
−0.974295 + 0.225274i \(0.927672\pi\)
\(884\) 0 0
\(885\) − 12.7637i − 0.429046i
\(886\) 0 0
\(887\) −12.6712 −0.425457 −0.212729 0.977111i \(-0.568235\pi\)
−0.212729 + 0.977111i \(0.568235\pi\)
\(888\) 0 0
\(889\) 2.47162 0.0828955
\(890\) 0 0
\(891\) − 3.04660i − 0.102065i
\(892\) 0 0
\(893\) − 25.0130i − 0.837029i
\(894\) 0 0
\(895\) 13.2399 0.442562
\(896\) 0 0
\(897\) −1.24265 −0.0414908
\(898\) 0 0
\(899\) 34.7569i 1.15921i
\(900\) 0 0
\(901\) − 87.9235i − 2.92916i
\(902\) 0 0
\(903\) 5.20200 0.173112
\(904\) 0 0
\(905\) −33.9038 −1.12700
\(906\) 0 0
\(907\) − 22.0547i − 0.732313i −0.930553 0.366157i \(-0.880673\pi\)
0.930553 0.366157i \(-0.119327\pi\)
\(908\) 0 0
\(909\) 12.1731i 0.403756i
\(910\) 0 0
\(911\) −43.3934 −1.43769 −0.718844 0.695171i \(-0.755328\pi\)
−0.718844 + 0.695171i \(0.755328\pi\)
\(912\) 0 0
\(913\) 21.5800 0.714195
\(914\) 0 0
\(915\) − 2.71593i − 0.0897857i
\(916\) 0 0
\(917\) 3.65319i 0.120639i
\(918\) 0 0
\(919\) −52.4028 −1.72861 −0.864304 0.502970i \(-0.832241\pi\)
−0.864304 + 0.502970i \(0.832241\pi\)
\(920\) 0 0
\(921\) 19.9437 0.657168
\(922\) 0 0
\(923\) − 8.80159i − 0.289708i
\(924\) 0 0
\(925\) 21.4593i 0.705577i
\(926\) 0 0
\(927\) −18.4467 −0.605868
\(928\) 0 0
\(929\) −8.43753 −0.276826 −0.138413 0.990375i \(-0.544200\pi\)
−0.138413 + 0.990375i \(0.544200\pi\)
\(930\) 0 0
\(931\) − 3.19046i − 0.104563i
\(932\) 0 0
\(933\) − 10.0498i − 0.329016i
\(934\) 0 0
\(935\) −33.3745 −1.09146
\(936\) 0 0
\(937\) 35.8332 1.17062 0.585310 0.810810i \(-0.300973\pi\)
0.585310 + 0.810810i \(0.300973\pi\)
\(938\) 0 0
\(939\) − 0.582673i − 0.0190148i
\(940\) 0 0
\(941\) − 21.7906i − 0.710352i −0.934800 0.355176i \(-0.884421\pi\)
0.934800 0.355176i \(-0.115579\pi\)
\(942\) 0 0
\(943\) −9.77578 −0.318343
\(944\) 0 0
\(945\) 6.52147 0.212143
\(946\) 0 0
\(947\) 11.1570i 0.362555i 0.983432 + 0.181277i \(0.0580232\pi\)
−0.983432 + 0.181277i \(0.941977\pi\)
\(948\) 0 0
\(949\) 4.37797i 0.142115i
\(950\) 0 0
\(951\) 20.0820 0.651203
\(952\) 0 0
\(953\) 49.6642 1.60878 0.804391 0.594101i \(-0.202492\pi\)
0.804391 + 0.594101i \(0.202492\pi\)
\(954\) 0 0
\(955\) 10.5733i 0.342145i
\(956\) 0 0
\(957\) 24.7319i 0.799470i
\(958\) 0 0
\(959\) −13.4016 −0.432761
\(960\) 0 0
\(961\) 1.54163 0.0497299
\(962\) 0 0
\(963\) − 4.21771i − 0.135914i
\(964\) 0 0
\(965\) 32.7030i 1.05275i
\(966\) 0 0
\(967\) −15.8683 −0.510291 −0.255145 0.966903i \(-0.582123\pi\)
−0.255145 + 0.966903i \(0.582123\pi\)
\(968\) 0 0
\(969\) −21.0025 −0.674698
\(970\) 0 0
\(971\) − 1.01819i − 0.0326752i −0.999867 0.0163376i \(-0.994799\pi\)
0.999867 0.0163376i \(-0.00520065\pi\)
\(972\) 0 0
\(973\) − 16.3563i − 0.524358i
\(974\) 0 0
\(975\) 1.81253 0.0580476
\(976\) 0 0
\(977\) 34.8975 1.11647 0.558235 0.829683i \(-0.311479\pi\)
0.558235 + 0.829683i \(0.311479\pi\)
\(978\) 0 0
\(979\) 19.8446i 0.634237i
\(980\) 0 0
\(981\) − 16.7178i − 0.533758i
\(982\) 0 0
\(983\) −56.4029 −1.79897 −0.899487 0.436947i \(-0.856060\pi\)
−0.899487 + 0.436947i \(0.856060\pi\)
\(984\) 0 0
\(985\) −26.9225 −0.857823
\(986\) 0 0
\(987\) − 7.97207i − 0.253754i
\(988\) 0 0
\(989\) − 11.6869i − 0.371620i
\(990\) 0 0
\(991\) −24.2515 −0.770374 −0.385187 0.922839i \(-0.625863\pi\)
−0.385187 + 0.922839i \(0.625863\pi\)
\(992\) 0 0
\(993\) −23.2286 −0.737138
\(994\) 0 0
\(995\) 26.0945i 0.827250i
\(996\) 0 0
\(997\) − 34.7021i − 1.09902i −0.835486 0.549512i \(-0.814813\pi\)
0.835486 0.549512i \(-0.185187\pi\)
\(998\) 0 0
\(999\) −32.5205 −1.02890
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 3584.2.b.i.1793.8 12
4.3 odd 2 3584.2.b.k.1793.5 12
8.3 odd 2 3584.2.b.k.1793.8 12
8.5 even 2 inner 3584.2.b.i.1793.5 12
16.3 odd 4 3584.2.a.e.1.5 6
16.5 even 4 3584.2.a.f.1.5 yes 6
16.11 odd 4 3584.2.a.k.1.2 yes 6
16.13 even 4 3584.2.a.l.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3584.2.a.e.1.5 6 16.3 odd 4
3584.2.a.f.1.5 yes 6 16.5 even 4
3584.2.a.k.1.2 yes 6 16.11 odd 4
3584.2.a.l.1.2 yes 6 16.13 even 4
3584.2.b.i.1793.5 12 8.5 even 2 inner
3584.2.b.i.1793.8 12 1.1 even 1 trivial
3584.2.b.k.1793.5 12 4.3 odd 2
3584.2.b.k.1793.8 12 8.3 odd 2