Properties

Label 1840.2.e.c.369.2
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(369,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1840.369");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (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: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.c.369.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.618034i q^{3} +2.23607 q^{5} -4.85410i q^{7} +2.61803 q^{9} +O(q^{10})\) \(q-0.618034i q^{3} +2.23607 q^{5} -4.85410i q^{7} +2.61803 q^{9} +3.38197 q^{11} -0.381966i q^{13} -1.38197i q^{15} -5.85410i q^{17} -6.85410 q^{19} -3.00000 q^{21} +1.00000i q^{23} +5.00000 q^{25} -3.47214i q^{27} -3.70820 q^{29} +8.85410 q^{31} -2.09017i q^{33} -10.8541i q^{35} +3.70820i q^{37} -0.236068 q^{39} -3.38197 q^{41} +6.76393i q^{43} +5.85410 q^{45} +11.7082i q^{47} -16.5623 q^{49} -3.61803 q^{51} +2.00000i q^{53} +7.56231 q^{55} +4.23607i q^{57} -6.00000 q^{59} -3.85410 q^{61} -12.7082i q^{63} -0.854102i q^{65} -0.763932i q^{67} +0.618034 q^{69} -2.61803 q^{71} +7.52786i q^{73} -3.09017i q^{75} -16.4164i q^{77} +5.70820 q^{79} +5.70820 q^{81} -5.70820i q^{83} -13.0902i q^{85} +2.29180i q^{87} +9.70820 q^{89} -1.85410 q^{91} -5.47214i q^{93} -15.3262 q^{95} -16.0344i q^{97} +8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71} - 4 q^{79} - 4 q^{81} + 12 q^{89} + 6 q^{91} - 30 q^{95} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 0.618034i − 0.356822i −0.983956 0.178411i \(-0.942904\pi\)
0.983956 0.178411i \(-0.0570957\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) − 4.85410i − 1.83468i −0.398107 0.917339i \(-0.630333\pi\)
0.398107 0.917339i \(-0.369667\pi\)
\(8\) 0 0
\(9\) 2.61803 0.872678
\(10\) 0 0
\(11\) 3.38197 1.01970 0.509851 0.860263i \(-0.329701\pi\)
0.509851 + 0.860263i \(0.329701\pi\)
\(12\) 0 0
\(13\) − 0.381966i − 0.105938i −0.998596 0.0529692i \(-0.983131\pi\)
0.998596 0.0529692i \(-0.0168685\pi\)
\(14\) 0 0
\(15\) − 1.38197i − 0.356822i
\(16\) 0 0
\(17\) − 5.85410i − 1.41983i −0.704288 0.709914i \(-0.748734\pi\)
0.704288 0.709914i \(-0.251266\pi\)
\(18\) 0 0
\(19\) −6.85410 −1.57244 −0.786219 0.617947i \(-0.787964\pi\)
−0.786219 + 0.617947i \(0.787964\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 3.47214i − 0.668213i
\(28\) 0 0
\(29\) −3.70820 −0.688596 −0.344298 0.938860i \(-0.611883\pi\)
−0.344298 + 0.938860i \(0.611883\pi\)
\(30\) 0 0
\(31\) 8.85410 1.59024 0.795122 0.606450i \(-0.207407\pi\)
0.795122 + 0.606450i \(0.207407\pi\)
\(32\) 0 0
\(33\) − 2.09017i − 0.363852i
\(34\) 0 0
\(35\) − 10.8541i − 1.83468i
\(36\) 0 0
\(37\) 3.70820i 0.609625i 0.952412 + 0.304812i \(0.0985938\pi\)
−0.952412 + 0.304812i \(0.901406\pi\)
\(38\) 0 0
\(39\) −0.236068 −0.0378011
\(40\) 0 0
\(41\) −3.38197 −0.528174 −0.264087 0.964499i \(-0.585071\pi\)
−0.264087 + 0.964499i \(0.585071\pi\)
\(42\) 0 0
\(43\) 6.76393i 1.03149i 0.856742 + 0.515745i \(0.172485\pi\)
−0.856742 + 0.515745i \(0.827515\pi\)
\(44\) 0 0
\(45\) 5.85410 0.872678
\(46\) 0 0
\(47\) 11.7082i 1.70782i 0.520423 + 0.853909i \(0.325774\pi\)
−0.520423 + 0.853909i \(0.674226\pi\)
\(48\) 0 0
\(49\) −16.5623 −2.36604
\(50\) 0 0
\(51\) −3.61803 −0.506626
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 7.56231 1.01970
\(56\) 0 0
\(57\) 4.23607i 0.561081i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −3.85410 −0.493467 −0.246734 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(62\) 0 0
\(63\) − 12.7082i − 1.60108i
\(64\) 0 0
\(65\) − 0.854102i − 0.105938i
\(66\) 0 0
\(67\) − 0.763932i − 0.0933292i −0.998911 0.0466646i \(-0.985141\pi\)
0.998911 0.0466646i \(-0.0148592\pi\)
\(68\) 0 0
\(69\) 0.618034 0.0744025
\(70\) 0 0
\(71\) −2.61803 −0.310703 −0.155352 0.987859i \(-0.549651\pi\)
−0.155352 + 0.987859i \(0.549651\pi\)
\(72\) 0 0
\(73\) 7.52786i 0.881070i 0.897735 + 0.440535i \(0.145211\pi\)
−0.897735 + 0.440535i \(0.854789\pi\)
\(74\) 0 0
\(75\) − 3.09017i − 0.356822i
\(76\) 0 0
\(77\) − 16.4164i − 1.87082i
\(78\) 0 0
\(79\) 5.70820 0.642223 0.321112 0.947041i \(-0.395944\pi\)
0.321112 + 0.947041i \(0.395944\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) − 5.70820i − 0.626557i −0.949661 0.313278i \(-0.898573\pi\)
0.949661 0.313278i \(-0.101427\pi\)
\(84\) 0 0
\(85\) − 13.0902i − 1.41983i
\(86\) 0 0
\(87\) 2.29180i 0.245706i
\(88\) 0 0
\(89\) 9.70820 1.02907 0.514534 0.857470i \(-0.327965\pi\)
0.514534 + 0.857470i \(0.327965\pi\)
\(90\) 0 0
\(91\) −1.85410 −0.194363
\(92\) 0 0
\(93\) − 5.47214i − 0.567434i
\(94\) 0 0
\(95\) −15.3262 −1.57244
\(96\) 0 0
\(97\) − 16.0344i − 1.62805i −0.580829 0.814025i \(-0.697272\pi\)
0.580829 0.814025i \(-0.302728\pi\)
\(98\) 0 0
\(99\) 8.85410 0.889871
\(100\) 0 0
\(101\) −10.4721 −1.04202 −0.521008 0.853552i \(-0.674444\pi\)
−0.521008 + 0.853552i \(0.674444\pi\)
\(102\) 0 0
\(103\) 4.14590i 0.408507i 0.978918 + 0.204254i \(0.0654768\pi\)
−0.978918 + 0.204254i \(0.934523\pi\)
\(104\) 0 0
\(105\) −6.70820 −0.654654
\(106\) 0 0
\(107\) 1.70820i 0.165138i 0.996585 + 0.0825692i \(0.0263125\pi\)
−0.996585 + 0.0825692i \(0.973687\pi\)
\(108\) 0 0
\(109\) 12.5623 1.20325 0.601625 0.798778i \(-0.294520\pi\)
0.601625 + 0.798778i \(0.294520\pi\)
\(110\) 0 0
\(111\) 2.29180 0.217528
\(112\) 0 0
\(113\) − 13.4164i − 1.26211i −0.775738 0.631055i \(-0.782622\pi\)
0.775738 0.631055i \(-0.217378\pi\)
\(114\) 0 0
\(115\) 2.23607i 0.208514i
\(116\) 0 0
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) −28.4164 −2.60493
\(120\) 0 0
\(121\) 0.437694 0.0397904
\(122\) 0 0
\(123\) 2.09017i 0.188464i
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) − 3.70820i − 0.329050i −0.986373 0.164525i \(-0.947391\pi\)
0.986373 0.164525i \(-0.0526091\pi\)
\(128\) 0 0
\(129\) 4.18034 0.368058
\(130\) 0 0
\(131\) −8.18034 −0.714720 −0.357360 0.933967i \(-0.616323\pi\)
−0.357360 + 0.933967i \(0.616323\pi\)
\(132\) 0 0
\(133\) 33.2705i 2.88492i
\(134\) 0 0
\(135\) − 7.76393i − 0.668213i
\(136\) 0 0
\(137\) − 7.14590i − 0.610515i −0.952270 0.305258i \(-0.901257\pi\)
0.952270 0.305258i \(-0.0987426\pi\)
\(138\) 0 0
\(139\) 17.7082 1.50199 0.750995 0.660308i \(-0.229574\pi\)
0.750995 + 0.660308i \(0.229574\pi\)
\(140\) 0 0
\(141\) 7.23607 0.609387
\(142\) 0 0
\(143\) − 1.29180i − 0.108025i
\(144\) 0 0
\(145\) −8.29180 −0.688596
\(146\) 0 0
\(147\) 10.2361i 0.844257i
\(148\) 0 0
\(149\) 0.381966 0.0312919 0.0156459 0.999878i \(-0.495020\pi\)
0.0156459 + 0.999878i \(0.495020\pi\)
\(150\) 0 0
\(151\) 19.2705 1.56821 0.784106 0.620627i \(-0.213122\pi\)
0.784106 + 0.620627i \(0.213122\pi\)
\(152\) 0 0
\(153\) − 15.3262i − 1.23905i
\(154\) 0 0
\(155\) 19.7984 1.59024
\(156\) 0 0
\(157\) 15.7082i 1.25365i 0.779160 + 0.626826i \(0.215646\pi\)
−0.779160 + 0.626826i \(0.784354\pi\)
\(158\) 0 0
\(159\) 1.23607 0.0980266
\(160\) 0 0
\(161\) 4.85410 0.382557
\(162\) 0 0
\(163\) 7.03444i 0.550980i 0.961304 + 0.275490i \(0.0888401\pi\)
−0.961304 + 0.275490i \(0.911160\pi\)
\(164\) 0 0
\(165\) − 4.67376i − 0.363852i
\(166\) 0 0
\(167\) − 3.70820i − 0.286949i −0.989654 0.143475i \(-0.954172\pi\)
0.989654 0.143475i \(-0.0458276\pi\)
\(168\) 0 0
\(169\) 12.8541 0.988777
\(170\) 0 0
\(171\) −17.9443 −1.37223
\(172\) 0 0
\(173\) 3.56231i 0.270837i 0.990788 + 0.135419i \(0.0432379\pi\)
−0.990788 + 0.135419i \(0.956762\pi\)
\(174\) 0 0
\(175\) − 24.2705i − 1.83468i
\(176\) 0 0
\(177\) 3.70820i 0.278726i
\(178\) 0 0
\(179\) 16.4721 1.23119 0.615593 0.788065i \(-0.288917\pi\)
0.615593 + 0.788065i \(0.288917\pi\)
\(180\) 0 0
\(181\) −4.56231 −0.339114 −0.169557 0.985520i \(-0.554234\pi\)
−0.169557 + 0.985520i \(0.554234\pi\)
\(182\) 0 0
\(183\) 2.38197i 0.176080i
\(184\) 0 0
\(185\) 8.29180i 0.609625i
\(186\) 0 0
\(187\) − 19.7984i − 1.44780i
\(188\) 0 0
\(189\) −16.8541 −1.22596
\(190\) 0 0
\(191\) 1.41641 0.102488 0.0512438 0.998686i \(-0.483681\pi\)
0.0512438 + 0.998686i \(0.483681\pi\)
\(192\) 0 0
\(193\) 2.29180i 0.164967i 0.996592 + 0.0824835i \(0.0262852\pi\)
−0.996592 + 0.0824835i \(0.973715\pi\)
\(194\) 0 0
\(195\) −0.527864 −0.0378011
\(196\) 0 0
\(197\) − 0.437694i − 0.0311844i −0.999878 0.0155922i \(-0.995037\pi\)
0.999878 0.0155922i \(-0.00496335\pi\)
\(198\) 0 0
\(199\) −15.4164 −1.09284 −0.546420 0.837511i \(-0.684010\pi\)
−0.546420 + 0.837511i \(0.684010\pi\)
\(200\) 0 0
\(201\) −0.472136 −0.0333019
\(202\) 0 0
\(203\) 18.0000i 1.26335i
\(204\) 0 0
\(205\) −7.56231 −0.528174
\(206\) 0 0
\(207\) 2.61803i 0.181966i
\(208\) 0 0
\(209\) −23.1803 −1.60342
\(210\) 0 0
\(211\) −5.70820 −0.392969 −0.196484 0.980507i \(-0.562953\pi\)
−0.196484 + 0.980507i \(0.562953\pi\)
\(212\) 0 0
\(213\) 1.61803i 0.110866i
\(214\) 0 0
\(215\) 15.1246i 1.03149i
\(216\) 0 0
\(217\) − 42.9787i − 2.91759i
\(218\) 0 0
\(219\) 4.65248 0.314385
\(220\) 0 0
\(221\) −2.23607 −0.150414
\(222\) 0 0
\(223\) − 22.3607i − 1.49738i −0.662919 0.748691i \(-0.730683\pi\)
0.662919 0.748691i \(-0.269317\pi\)
\(224\) 0 0
\(225\) 13.0902 0.872678
\(226\) 0 0
\(227\) − 23.7082i − 1.57357i −0.617228 0.786784i \(-0.711744\pi\)
0.617228 0.786784i \(-0.288256\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −10.1459 −0.667551
\(232\) 0 0
\(233\) 19.1246i 1.25289i 0.779464 + 0.626447i \(0.215492\pi\)
−0.779464 + 0.626447i \(0.784508\pi\)
\(234\) 0 0
\(235\) 26.1803i 1.70782i
\(236\) 0 0
\(237\) − 3.52786i − 0.229159i
\(238\) 0 0
\(239\) −1.52786 −0.0988293 −0.0494147 0.998778i \(-0.515736\pi\)
−0.0494147 + 0.998778i \(0.515736\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) − 13.9443i − 0.894525i
\(244\) 0 0
\(245\) −37.0344 −2.36604
\(246\) 0 0
\(247\) 2.61803i 0.166582i
\(248\) 0 0
\(249\) −3.52786 −0.223569
\(250\) 0 0
\(251\) −7.79837 −0.492229 −0.246114 0.969241i \(-0.579154\pi\)
−0.246114 + 0.969241i \(0.579154\pi\)
\(252\) 0 0
\(253\) 3.38197i 0.212622i
\(254\) 0 0
\(255\) −8.09017 −0.506626
\(256\) 0 0
\(257\) 15.4164i 0.961649i 0.876817 + 0.480825i \(0.159663\pi\)
−0.876817 + 0.480825i \(0.840337\pi\)
\(258\) 0 0
\(259\) 18.0000 1.11847
\(260\) 0 0
\(261\) −9.70820 −0.600923
\(262\) 0 0
\(263\) 15.2705i 0.941620i 0.882235 + 0.470810i \(0.156038\pi\)
−0.882235 + 0.470810i \(0.843962\pi\)
\(264\) 0 0
\(265\) 4.47214i 0.274721i
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) 10.4721 0.638497 0.319249 0.947671i \(-0.396569\pi\)
0.319249 + 0.947671i \(0.396569\pi\)
\(270\) 0 0
\(271\) −19.1459 −1.16303 −0.581515 0.813536i \(-0.697540\pi\)
−0.581515 + 0.813536i \(0.697540\pi\)
\(272\) 0 0
\(273\) 1.14590i 0.0693529i
\(274\) 0 0
\(275\) 16.9098 1.01970
\(276\) 0 0
\(277\) 16.4721i 0.989715i 0.868974 + 0.494857i \(0.164780\pi\)
−0.868974 + 0.494857i \(0.835220\pi\)
\(278\) 0 0
\(279\) 23.1803 1.38777
\(280\) 0 0
\(281\) −0.652476 −0.0389234 −0.0194617 0.999811i \(-0.506195\pi\)
−0.0194617 + 0.999811i \(0.506195\pi\)
\(282\) 0 0
\(283\) 3.05573i 0.181644i 0.995867 + 0.0908221i \(0.0289495\pi\)
−0.995867 + 0.0908221i \(0.971051\pi\)
\(284\) 0 0
\(285\) 9.47214i 0.561081i
\(286\) 0 0
\(287\) 16.4164i 0.969030i
\(288\) 0 0
\(289\) −17.2705 −1.01591
\(290\) 0 0
\(291\) −9.90983 −0.580925
\(292\) 0 0
\(293\) 27.7082i 1.61873i 0.587306 + 0.809365i \(0.300189\pi\)
−0.587306 + 0.809365i \(0.699811\pi\)
\(294\) 0 0
\(295\) −13.4164 −0.781133
\(296\) 0 0
\(297\) − 11.7426i − 0.681377i
\(298\) 0 0
\(299\) 0.381966 0.0220897
\(300\) 0 0
\(301\) 32.8328 1.89245
\(302\) 0 0
\(303\) 6.47214i 0.371814i
\(304\) 0 0
\(305\) −8.61803 −0.493467
\(306\) 0 0
\(307\) − 10.1459i − 0.579057i −0.957169 0.289528i \(-0.906502\pi\)
0.957169 0.289528i \(-0.0934985\pi\)
\(308\) 0 0
\(309\) 2.56231 0.145764
\(310\) 0 0
\(311\) 13.5279 0.767095 0.383547 0.923521i \(-0.374702\pi\)
0.383547 + 0.923521i \(0.374702\pi\)
\(312\) 0 0
\(313\) − 19.1459i − 1.08219i −0.840961 0.541095i \(-0.818010\pi\)
0.840961 0.541095i \(-0.181990\pi\)
\(314\) 0 0
\(315\) − 28.4164i − 1.60108i
\(316\) 0 0
\(317\) 26.2705i 1.47550i 0.675075 + 0.737749i \(0.264111\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(318\) 0 0
\(319\) −12.5410 −0.702162
\(320\) 0 0
\(321\) 1.05573 0.0589250
\(322\) 0 0
\(323\) 40.1246i 2.23259i
\(324\) 0 0
\(325\) − 1.90983i − 0.105938i
\(326\) 0 0
\(327\) − 7.76393i − 0.429346i
\(328\) 0 0
\(329\) 56.8328 3.13329
\(330\) 0 0
\(331\) −15.1246 −0.831324 −0.415662 0.909519i \(-0.636450\pi\)
−0.415662 + 0.909519i \(0.636450\pi\)
\(332\) 0 0
\(333\) 9.70820i 0.532006i
\(334\) 0 0
\(335\) − 1.70820i − 0.0933292i
\(336\) 0 0
\(337\) − 5.61803i − 0.306034i −0.988224 0.153017i \(-0.951101\pi\)
0.988224 0.153017i \(-0.0488989\pi\)
\(338\) 0 0
\(339\) −8.29180 −0.450349
\(340\) 0 0
\(341\) 29.9443 1.62157
\(342\) 0 0
\(343\) 46.4164i 2.50625i
\(344\) 0 0
\(345\) 1.38197 0.0744025
\(346\) 0 0
\(347\) 5.56231i 0.298600i 0.988792 + 0.149300i \(0.0477021\pi\)
−0.988792 + 0.149300i \(0.952298\pi\)
\(348\) 0 0
\(349\) −14.2918 −0.765022 −0.382511 0.923951i \(-0.624941\pi\)
−0.382511 + 0.923951i \(0.624941\pi\)
\(350\) 0 0
\(351\) −1.32624 −0.0707893
\(352\) 0 0
\(353\) − 8.00000i − 0.425797i −0.977074 0.212899i \(-0.931710\pi\)
0.977074 0.212899i \(-0.0682904\pi\)
\(354\) 0 0
\(355\) −5.85410 −0.310703
\(356\) 0 0
\(357\) 17.5623i 0.929496i
\(358\) 0 0
\(359\) 13.4164 0.708091 0.354045 0.935228i \(-0.384806\pi\)
0.354045 + 0.935228i \(0.384806\pi\)
\(360\) 0 0
\(361\) 27.9787 1.47256
\(362\) 0 0
\(363\) − 0.270510i − 0.0141981i
\(364\) 0 0
\(365\) 16.8328i 0.881070i
\(366\) 0 0
\(367\) − 12.0000i − 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) −8.85410 −0.460926
\(370\) 0 0
\(371\) 9.70820 0.504025
\(372\) 0 0
\(373\) 3.05573i 0.158220i 0.996866 + 0.0791098i \(0.0252078\pi\)
−0.996866 + 0.0791098i \(0.974792\pi\)
\(374\) 0 0
\(375\) − 6.90983i − 0.356822i
\(376\) 0 0
\(377\) 1.41641i 0.0729487i
\(378\) 0 0
\(379\) 9.27051 0.476194 0.238097 0.971241i \(-0.423476\pi\)
0.238097 + 0.971241i \(0.423476\pi\)
\(380\) 0 0
\(381\) −2.29180 −0.117412
\(382\) 0 0
\(383\) 27.4164i 1.40091i 0.713695 + 0.700456i \(0.247020\pi\)
−0.713695 + 0.700456i \(0.752980\pi\)
\(384\) 0 0
\(385\) − 36.7082i − 1.87082i
\(386\) 0 0
\(387\) 17.7082i 0.900159i
\(388\) 0 0
\(389\) 5.67376 0.287671 0.143836 0.989602i \(-0.454056\pi\)
0.143836 + 0.989602i \(0.454056\pi\)
\(390\) 0 0
\(391\) 5.85410 0.296055
\(392\) 0 0
\(393\) 5.05573i 0.255028i
\(394\) 0 0
\(395\) 12.7639 0.642223
\(396\) 0 0
\(397\) 35.5623i 1.78482i 0.451224 + 0.892410i \(0.350987\pi\)
−0.451224 + 0.892410i \(0.649013\pi\)
\(398\) 0 0
\(399\) 20.5623 1.02940
\(400\) 0 0
\(401\) −34.4721 −1.72146 −0.860728 0.509065i \(-0.829991\pi\)
−0.860728 + 0.509065i \(0.829991\pi\)
\(402\) 0 0
\(403\) − 3.38197i − 0.168468i
\(404\) 0 0
\(405\) 12.7639 0.634245
\(406\) 0 0
\(407\) 12.5410i 0.621635i
\(408\) 0 0
\(409\) 6.56231 0.324485 0.162243 0.986751i \(-0.448127\pi\)
0.162243 + 0.986751i \(0.448127\pi\)
\(410\) 0 0
\(411\) −4.41641 −0.217845
\(412\) 0 0
\(413\) 29.1246i 1.43313i
\(414\) 0 0
\(415\) − 12.7639i − 0.626557i
\(416\) 0 0
\(417\) − 10.9443i − 0.535943i
\(418\) 0 0
\(419\) 5.88854 0.287674 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(420\) 0 0
\(421\) 6.85410 0.334048 0.167024 0.985953i \(-0.446584\pi\)
0.167024 + 0.985953i \(0.446584\pi\)
\(422\) 0 0
\(423\) 30.6525i 1.49037i
\(424\) 0 0
\(425\) − 29.2705i − 1.41983i
\(426\) 0 0
\(427\) 18.7082i 0.905353i
\(428\) 0 0
\(429\) −0.798374 −0.0385459
\(430\) 0 0
\(431\) 6.76393 0.325807 0.162904 0.986642i \(-0.447914\pi\)
0.162904 + 0.986642i \(0.447914\pi\)
\(432\) 0 0
\(433\) 29.6180i 1.42335i 0.702508 + 0.711676i \(0.252064\pi\)
−0.702508 + 0.711676i \(0.747936\pi\)
\(434\) 0 0
\(435\) 5.12461i 0.245706i
\(436\) 0 0
\(437\) − 6.85410i − 0.327876i
\(438\) 0 0
\(439\) −0.270510 −0.0129107 −0.00645536 0.999979i \(-0.502055\pi\)
−0.00645536 + 0.999979i \(0.502055\pi\)
\(440\) 0 0
\(441\) −43.3607 −2.06479
\(442\) 0 0
\(443\) 6.85410i 0.325648i 0.986655 + 0.162824i \(0.0520603\pi\)
−0.986655 + 0.162824i \(0.947940\pi\)
\(444\) 0 0
\(445\) 21.7082 1.02907
\(446\) 0 0
\(447\) − 0.236068i − 0.0111656i
\(448\) 0 0
\(449\) −2.56231 −0.120923 −0.0604613 0.998171i \(-0.519257\pi\)
−0.0604613 + 0.998171i \(0.519257\pi\)
\(450\) 0 0
\(451\) −11.4377 −0.538580
\(452\) 0 0
\(453\) − 11.9098i − 0.559573i
\(454\) 0 0
\(455\) −4.14590 −0.194363
\(456\) 0 0
\(457\) − 25.4164i − 1.18893i −0.804122 0.594465i \(-0.797364\pi\)
0.804122 0.594465i \(-0.202636\pi\)
\(458\) 0 0
\(459\) −20.3262 −0.948748
\(460\) 0 0
\(461\) −25.3050 −1.17857 −0.589285 0.807926i \(-0.700590\pi\)
−0.589285 + 0.807926i \(0.700590\pi\)
\(462\) 0 0
\(463\) − 2.29180i − 0.106509i −0.998581 0.0532544i \(-0.983041\pi\)
0.998581 0.0532544i \(-0.0169594\pi\)
\(464\) 0 0
\(465\) − 12.2361i − 0.567434i
\(466\) 0 0
\(467\) 17.1246i 0.792433i 0.918157 + 0.396216i \(0.129677\pi\)
−0.918157 + 0.396216i \(0.870323\pi\)
\(468\) 0 0
\(469\) −3.70820 −0.171229
\(470\) 0 0
\(471\) 9.70820 0.447330
\(472\) 0 0
\(473\) 22.8754i 1.05181i
\(474\) 0 0
\(475\) −34.2705 −1.57244
\(476\) 0 0
\(477\) 5.23607i 0.239743i
\(478\) 0 0
\(479\) −19.5279 −0.892251 −0.446125 0.894970i \(-0.647196\pi\)
−0.446125 + 0.894970i \(0.647196\pi\)
\(480\) 0 0
\(481\) 1.41641 0.0645826
\(482\) 0 0
\(483\) − 3.00000i − 0.136505i
\(484\) 0 0
\(485\) − 35.8541i − 1.62805i
\(486\) 0 0
\(487\) 21.7082i 0.983693i 0.870682 + 0.491846i \(0.163678\pi\)
−0.870682 + 0.491846i \(0.836322\pi\)
\(488\) 0 0
\(489\) 4.34752 0.196602
\(490\) 0 0
\(491\) 26.8328 1.21095 0.605474 0.795865i \(-0.292984\pi\)
0.605474 + 0.795865i \(0.292984\pi\)
\(492\) 0 0
\(493\) 21.7082i 0.977688i
\(494\) 0 0
\(495\) 19.7984 0.889871
\(496\) 0 0
\(497\) 12.7082i 0.570041i
\(498\) 0 0
\(499\) 11.4164 0.511069 0.255534 0.966800i \(-0.417749\pi\)
0.255534 + 0.966800i \(0.417749\pi\)
\(500\) 0 0
\(501\) −2.29180 −0.102390
\(502\) 0 0
\(503\) 16.8541i 0.751487i 0.926724 + 0.375744i \(0.122613\pi\)
−0.926724 + 0.375744i \(0.877387\pi\)
\(504\) 0 0
\(505\) −23.4164 −1.04202
\(506\) 0 0
\(507\) − 7.94427i − 0.352818i
\(508\) 0 0
\(509\) 24.6525 1.09270 0.546351 0.837556i \(-0.316017\pi\)
0.546351 + 0.837556i \(0.316017\pi\)
\(510\) 0 0
\(511\) 36.5410 1.61648
\(512\) 0 0
\(513\) 23.7984i 1.05072i
\(514\) 0 0
\(515\) 9.27051i 0.408507i
\(516\) 0 0
\(517\) 39.5967i 1.74146i
\(518\) 0 0
\(519\) 2.20163 0.0966407
\(520\) 0 0
\(521\) 20.0689 0.879234 0.439617 0.898185i \(-0.355114\pi\)
0.439617 + 0.898185i \(0.355114\pi\)
\(522\) 0 0
\(523\) − 40.3607i − 1.76485i −0.470454 0.882425i \(-0.655910\pi\)
0.470454 0.882425i \(-0.344090\pi\)
\(524\) 0 0
\(525\) −15.0000 −0.654654
\(526\) 0 0
\(527\) − 51.8328i − 2.25787i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −15.7082 −0.681678
\(532\) 0 0
\(533\) 1.29180i 0.0559539i
\(534\) 0 0
\(535\) 3.81966i 0.165138i
\(536\) 0 0
\(537\) − 10.1803i − 0.439314i
\(538\) 0 0
\(539\) −56.0132 −2.41266
\(540\) 0 0
\(541\) −27.4164 −1.17872 −0.589362 0.807869i \(-0.700621\pi\)
−0.589362 + 0.807869i \(0.700621\pi\)
\(542\) 0 0
\(543\) 2.81966i 0.121003i
\(544\) 0 0
\(545\) 28.0902 1.20325
\(546\) 0 0
\(547\) 1.14590i 0.0489951i 0.999700 + 0.0244975i \(0.00779859\pi\)
−0.999700 + 0.0244975i \(0.992201\pi\)
\(548\) 0 0
\(549\) −10.0902 −0.430638
\(550\) 0 0
\(551\) 25.4164 1.08278
\(552\) 0 0
\(553\) − 27.7082i − 1.17827i
\(554\) 0 0
\(555\) 5.12461 0.217528
\(556\) 0 0
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 0 0
\(559\) 2.58359 0.109274
\(560\) 0 0
\(561\) −12.2361 −0.516607
\(562\) 0 0
\(563\) − 2.29180i − 0.0965877i −0.998833 0.0482938i \(-0.984622\pi\)
0.998833 0.0482938i \(-0.0153784\pi\)
\(564\) 0 0
\(565\) − 30.0000i − 1.26211i
\(566\) 0 0
\(567\) − 27.7082i − 1.16364i
\(568\) 0 0
\(569\) −28.3607 −1.18894 −0.594471 0.804117i \(-0.702638\pi\)
−0.594471 + 0.804117i \(0.702638\pi\)
\(570\) 0 0
\(571\) 25.2705 1.05754 0.528769 0.848766i \(-0.322654\pi\)
0.528769 + 0.848766i \(0.322654\pi\)
\(572\) 0 0
\(573\) − 0.875388i − 0.0365699i
\(574\) 0 0
\(575\) 5.00000i 0.208514i
\(576\) 0 0
\(577\) − 26.2918i − 1.09454i −0.836956 0.547271i \(-0.815667\pi\)
0.836956 0.547271i \(-0.184333\pi\)
\(578\) 0 0
\(579\) 1.41641 0.0588639
\(580\) 0 0
\(581\) −27.7082 −1.14953
\(582\) 0 0
\(583\) 6.76393i 0.280133i
\(584\) 0 0
\(585\) − 2.23607i − 0.0924500i
\(586\) 0 0
\(587\) 2.14590i 0.0885707i 0.999019 + 0.0442853i \(0.0141011\pi\)
−0.999019 + 0.0442853i \(0.985899\pi\)
\(588\) 0 0
\(589\) −60.6869 −2.50056
\(590\) 0 0
\(591\) −0.270510 −0.0111273
\(592\) 0 0
\(593\) − 33.4164i − 1.37225i −0.727485 0.686124i \(-0.759311\pi\)
0.727485 0.686124i \(-0.240689\pi\)
\(594\) 0 0
\(595\) −63.5410 −2.60493
\(596\) 0 0
\(597\) 9.52786i 0.389950i
\(598\) 0 0
\(599\) 21.3820 0.873643 0.436822 0.899548i \(-0.356104\pi\)
0.436822 + 0.899548i \(0.356104\pi\)
\(600\) 0 0
\(601\) −3.14590 −0.128324 −0.0641619 0.997940i \(-0.520437\pi\)
−0.0641619 + 0.997940i \(0.520437\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 0 0
\(605\) 0.978714 0.0397904
\(606\) 0 0
\(607\) − 38.9443i − 1.58070i −0.612656 0.790350i \(-0.709899\pi\)
0.612656 0.790350i \(-0.290101\pi\)
\(608\) 0 0
\(609\) 11.1246 0.450792
\(610\) 0 0
\(611\) 4.47214 0.180923
\(612\) 0 0
\(613\) 5.23607i 0.211483i 0.994394 + 0.105741i \(0.0337216\pi\)
−0.994394 + 0.105741i \(0.966278\pi\)
\(614\) 0 0
\(615\) 4.67376i 0.188464i
\(616\) 0 0
\(617\) 17.5623i 0.707032i 0.935429 + 0.353516i \(0.115014\pi\)
−0.935429 + 0.353516i \(0.884986\pi\)
\(618\) 0 0
\(619\) −21.1459 −0.849925 −0.424963 0.905211i \(-0.639713\pi\)
−0.424963 + 0.905211i \(0.639713\pi\)
\(620\) 0 0
\(621\) 3.47214 0.139332
\(622\) 0 0
\(623\) − 47.1246i − 1.88801i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 14.3262i 0.572135i
\(628\) 0 0
\(629\) 21.7082 0.865563
\(630\) 0 0
\(631\) −3.41641 −0.136005 −0.0680025 0.997685i \(-0.521663\pi\)
−0.0680025 + 0.997685i \(0.521663\pi\)
\(632\) 0 0
\(633\) 3.52786i 0.140220i
\(634\) 0 0
\(635\) − 8.29180i − 0.329050i
\(636\) 0 0
\(637\) 6.32624i 0.250655i
\(638\) 0 0
\(639\) −6.85410 −0.271144
\(640\) 0 0
\(641\) 30.6525 1.21070 0.605350 0.795959i \(-0.293033\pi\)
0.605350 + 0.795959i \(0.293033\pi\)
\(642\) 0 0
\(643\) 35.0132i 1.38078i 0.723435 + 0.690392i \(0.242562\pi\)
−0.723435 + 0.690392i \(0.757438\pi\)
\(644\) 0 0
\(645\) 9.34752 0.368058
\(646\) 0 0
\(647\) 0.291796i 0.0114717i 0.999984 + 0.00573584i \(0.00182579\pi\)
−0.999984 + 0.00573584i \(0.998174\pi\)
\(648\) 0 0
\(649\) −20.2918 −0.796523
\(650\) 0 0
\(651\) −26.5623 −1.04106
\(652\) 0 0
\(653\) 11.9787i 0.468763i 0.972145 + 0.234382i \(0.0753065\pi\)
−0.972145 + 0.234382i \(0.924693\pi\)
\(654\) 0 0
\(655\) −18.2918 −0.714720
\(656\) 0 0
\(657\) 19.7082i 0.768890i
\(658\) 0 0
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) 34.3951 1.33782 0.668908 0.743346i \(-0.266762\pi\)
0.668908 + 0.743346i \(0.266762\pi\)
\(662\) 0 0
\(663\) 1.38197i 0.0536711i
\(664\) 0 0
\(665\) 74.3951i 2.88492i
\(666\) 0 0
\(667\) − 3.70820i − 0.143582i
\(668\) 0 0
\(669\) −13.8197 −0.534299
\(670\) 0 0
\(671\) −13.0344 −0.503189
\(672\) 0 0
\(673\) 14.2918i 0.550908i 0.961314 + 0.275454i \(0.0888282\pi\)
−0.961314 + 0.275454i \(0.911172\pi\)
\(674\) 0 0
\(675\) − 17.3607i − 0.668213i
\(676\) 0 0
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) −77.8328 −2.98695
\(680\) 0 0
\(681\) −14.6525 −0.561484
\(682\) 0 0
\(683\) 35.9787i 1.37669i 0.725385 + 0.688344i \(0.241662\pi\)
−0.725385 + 0.688344i \(0.758338\pi\)
\(684\) 0 0
\(685\) − 15.9787i − 0.610515i
\(686\) 0 0
\(687\) − 6.18034i − 0.235795i
\(688\) 0 0
\(689\) 0.763932 0.0291035
\(690\) 0 0
\(691\) −19.1246 −0.727535 −0.363767 0.931490i \(-0.618510\pi\)
−0.363767 + 0.931490i \(0.618510\pi\)
\(692\) 0 0
\(693\) − 42.9787i − 1.63263i
\(694\) 0 0
\(695\) 39.5967 1.50199
\(696\) 0 0
\(697\) 19.7984i 0.749917i
\(698\) 0 0
\(699\) 11.8197 0.447061
\(700\) 0 0
\(701\) 36.9230 1.39456 0.697281 0.716798i \(-0.254393\pi\)
0.697281 + 0.716798i \(0.254393\pi\)
\(702\) 0 0
\(703\) − 25.4164i − 0.958598i
\(704\) 0 0
\(705\) 16.1803 0.609387
\(706\) 0 0
\(707\) 50.8328i 1.91176i
\(708\) 0 0
\(709\) 11.5623 0.434232 0.217116 0.976146i \(-0.430335\pi\)
0.217116 + 0.976146i \(0.430335\pi\)
\(710\) 0 0
\(711\) 14.9443 0.560454
\(712\) 0 0
\(713\) 8.85410i 0.331589i
\(714\) 0 0
\(715\) − 2.88854i − 0.108025i
\(716\) 0 0
\(717\) 0.944272i 0.0352645i
\(718\) 0 0
\(719\) −32.4508 −1.21021 −0.605106 0.796145i \(-0.706869\pi\)
−0.605106 + 0.796145i \(0.706869\pi\)
\(720\) 0 0
\(721\) 20.1246 0.749480
\(722\) 0 0
\(723\) 1.23607i 0.0459699i
\(724\) 0 0
\(725\) −18.5410 −0.688596
\(726\) 0 0
\(727\) 1.74265i 0.0646312i 0.999478 + 0.0323156i \(0.0102882\pi\)
−0.999478 + 0.0323156i \(0.989712\pi\)
\(728\) 0 0
\(729\) 8.50658 0.315058
\(730\) 0 0
\(731\) 39.5967 1.46454
\(732\) 0 0
\(733\) − 35.8885i − 1.32557i −0.748808 0.662787i \(-0.769374\pi\)
0.748808 0.662787i \(-0.230626\pi\)
\(734\) 0 0
\(735\) 22.8885i 0.844257i
\(736\) 0 0
\(737\) − 2.58359i − 0.0951678i
\(738\) 0 0
\(739\) −32.5410 −1.19704 −0.598520 0.801108i \(-0.704244\pi\)
−0.598520 + 0.801108i \(0.704244\pi\)
\(740\) 0 0
\(741\) 1.61803 0.0594400
\(742\) 0 0
\(743\) − 9.97871i − 0.366084i −0.983105 0.183042i \(-0.941406\pi\)
0.983105 0.183042i \(-0.0585944\pi\)
\(744\) 0 0
\(745\) 0.854102 0.0312919
\(746\) 0 0
\(747\) − 14.9443i − 0.546782i
\(748\) 0 0
\(749\) 8.29180 0.302976
\(750\) 0 0
\(751\) 41.1246 1.50066 0.750329 0.661064i \(-0.229895\pi\)
0.750329 + 0.661064i \(0.229895\pi\)
\(752\) 0 0
\(753\) 4.81966i 0.175638i
\(754\) 0 0
\(755\) 43.0902 1.56821
\(756\) 0 0
\(757\) − 47.0132i − 1.70872i −0.519680 0.854361i \(-0.673949\pi\)
0.519680 0.854361i \(-0.326051\pi\)
\(758\) 0 0
\(759\) 2.09017 0.0758684
\(760\) 0 0
\(761\) −11.5066 −0.417113 −0.208557 0.978010i \(-0.566877\pi\)
−0.208557 + 0.978010i \(0.566877\pi\)
\(762\) 0 0
\(763\) − 60.9787i − 2.20758i
\(764\) 0 0
\(765\) − 34.2705i − 1.23905i
\(766\) 0 0
\(767\) 2.29180i 0.0827520i
\(768\) 0 0
\(769\) 34.2918 1.23659 0.618297 0.785945i \(-0.287823\pi\)
0.618297 + 0.785945i \(0.287823\pi\)
\(770\) 0 0
\(771\) 9.52786 0.343138
\(772\) 0 0
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 44.2705 1.59024
\(776\) 0 0
\(777\) − 11.1246i − 0.399093i
\(778\) 0 0
\(779\) 23.1803 0.830522
\(780\) 0 0
\(781\) −8.85410 −0.316825
\(782\) 0 0
\(783\) 12.8754i 0.460129i
\(784\) 0 0
\(785\) 35.1246i 1.25365i
\(786\) 0 0
\(787\) − 31.4164i − 1.11987i −0.828535 0.559937i \(-0.810825\pi\)
0.828535 0.559937i \(-0.189175\pi\)
\(788\) 0 0
\(789\) 9.43769 0.335991
\(790\) 0 0
\(791\) −65.1246 −2.31556
\(792\) 0 0
\(793\) 1.47214i 0.0522771i
\(794\) 0 0
\(795\) 2.76393 0.0980266
\(796\) 0 0
\(797\) 18.5836i 0.658265i 0.944284 + 0.329132i \(0.106756\pi\)
−0.944284 + 0.329132i \(0.893244\pi\)
\(798\) 0 0
\(799\) 68.5410 2.42481
\(800\) 0 0
\(801\) 25.4164 0.898045
\(802\) 0 0
\(803\) 25.4590i 0.898428i
\(804\) 0 0
\(805\) 10.8541 0.382557
\(806\) 0 0
\(807\) − 6.47214i − 0.227830i
\(808\) 0 0
\(809\) −27.1591 −0.954861 −0.477431 0.878669i \(-0.658432\pi\)
−0.477431 + 0.878669i \(0.658432\pi\)
\(810\) 0 0
\(811\) 7.70820 0.270672 0.135336 0.990800i \(-0.456789\pi\)
0.135336 + 0.990800i \(0.456789\pi\)
\(812\) 0 0
\(813\) 11.8328i 0.414995i
\(814\) 0 0
\(815\) 15.7295i 0.550980i
\(816\) 0 0
\(817\) − 46.3607i − 1.62195i
\(818\) 0 0
\(819\) −4.85410 −0.169616
\(820\) 0 0
\(821\) −2.94427 −0.102756 −0.0513779 0.998679i \(-0.516361\pi\)
−0.0513779 + 0.998679i \(0.516361\pi\)
\(822\) 0 0
\(823\) 20.8328i 0.726186i 0.931753 + 0.363093i \(0.118279\pi\)
−0.931753 + 0.363093i \(0.881721\pi\)
\(824\) 0 0
\(825\) − 10.4508i − 0.363852i
\(826\) 0 0
\(827\) − 48.2492i − 1.67779i −0.544293 0.838895i \(-0.683202\pi\)
0.544293 0.838895i \(-0.316798\pi\)
\(828\) 0 0
\(829\) −32.5410 −1.13020 −0.565098 0.825024i \(-0.691162\pi\)
−0.565098 + 0.825024i \(0.691162\pi\)
\(830\) 0 0
\(831\) 10.1803 0.353152
\(832\) 0 0
\(833\) 96.9574i 3.35938i
\(834\) 0 0
\(835\) − 8.29180i − 0.286949i
\(836\) 0 0
\(837\) − 30.7426i − 1.06262i
\(838\) 0 0
\(839\) −6.76393 −0.233517 −0.116758 0.993160i \(-0.537250\pi\)
−0.116758 + 0.993160i \(0.537250\pi\)
\(840\) 0 0
\(841\) −15.2492 −0.525835
\(842\) 0 0
\(843\) 0.403252i 0.0138887i
\(844\) 0 0
\(845\) 28.7426 0.988777
\(846\) 0 0
\(847\) − 2.12461i − 0.0730025i
\(848\) 0 0
\(849\) 1.88854 0.0648147
\(850\) 0 0
\(851\) −3.70820 −0.127116
\(852\) 0 0
\(853\) 51.2148i 1.75356i 0.480891 + 0.876780i \(0.340313\pi\)
−0.480891 + 0.876780i \(0.659687\pi\)
\(854\) 0 0
\(855\) −40.1246 −1.37223
\(856\) 0 0
\(857\) − 36.0000i − 1.22974i −0.788630 0.614868i \(-0.789209\pi\)
0.788630 0.614868i \(-0.210791\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 10.1459 0.345771
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 7.96556i 0.270837i
\(866\) 0 0
\(867\) 10.6738i 0.362500i
\(868\) 0 0
\(869\) 19.3050 0.654876
\(870\) 0 0
\(871\) −0.291796 −0.00988713
\(872\) 0 0
\(873\) − 41.9787i − 1.42076i
\(874\) 0 0
\(875\) − 54.2705i − 1.83468i
\(876\) 0 0
\(877\) − 34.6869i − 1.17129i −0.810566 0.585647i \(-0.800840\pi\)
0.810566 0.585647i \(-0.199160\pi\)
\(878\) 0 0
\(879\) 17.1246 0.577599
\(880\) 0 0
\(881\) −38.2918 −1.29008 −0.645042 0.764147i \(-0.723160\pi\)
−0.645042 + 0.764147i \(0.723160\pi\)
\(882\) 0 0
\(883\) − 17.7295i − 0.596645i −0.954465 0.298322i \(-0.903573\pi\)
0.954465 0.298322i \(-0.0964271\pi\)
\(884\) 0 0
\(885\) 8.29180i 0.278726i
\(886\) 0 0
\(887\) 36.5410i 1.22693i 0.789723 + 0.613464i \(0.210224\pi\)
−0.789723 + 0.613464i \(0.789776\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 19.3050 0.646740
\(892\) 0 0
\(893\) − 80.2492i − 2.68544i
\(894\) 0 0
\(895\) 36.8328 1.23119
\(896\) 0 0
\(897\) − 0.236068i − 0.00788208i
\(898\) 0 0
\(899\) −32.8328 −1.09504
\(900\) 0 0
\(901\) 11.7082 0.390057
\(902\) 0 0
\(903\) − 20.2918i − 0.675269i
\(904\) 0 0
\(905\) −10.2016 −0.339114
\(906\) 0 0
\(907\) − 30.5410i − 1.01410i −0.861917 0.507049i \(-0.830736\pi\)
0.861917 0.507049i \(-0.169264\pi\)
\(908\) 0 0
\(909\) −27.4164 −0.909345
\(910\) 0 0
\(911\) −22.3607 −0.740842 −0.370421 0.928864i \(-0.620787\pi\)
−0.370421 + 0.928864i \(0.620787\pi\)
\(912\) 0 0
\(913\) − 19.3050i − 0.638901i
\(914\) 0 0
\(915\) 5.32624i 0.176080i
\(916\) 0 0
\(917\) 39.7082i 1.31128i
\(918\) 0 0
\(919\) −14.5836 −0.481068 −0.240534 0.970641i \(-0.577323\pi\)
−0.240534 + 0.970641i \(0.577323\pi\)
\(920\) 0 0
\(921\) −6.27051 −0.206620
\(922\) 0 0
\(923\) 1.00000i 0.0329154i
\(924\) 0 0
\(925\) 18.5410i 0.609625i
\(926\) 0 0
\(927\) 10.8541i 0.356495i
\(928\) 0 0
\(929\) −19.3050 −0.633375 −0.316687 0.948530i \(-0.602571\pi\)
−0.316687 + 0.948530i \(0.602571\pi\)
\(930\) 0 0
\(931\) 113.520 3.72046
\(932\) 0 0
\(933\) − 8.36068i − 0.273716i
\(934\) 0 0
\(935\) − 44.2705i − 1.44780i
\(936\) 0 0
\(937\) − 8.67376i − 0.283359i −0.989913 0.141680i \(-0.954750\pi\)
0.989913 0.141680i \(-0.0452503\pi\)
\(938\) 0 0
\(939\) −11.8328 −0.386149
\(940\) 0 0
\(941\) −24.2148 −0.789379 −0.394690 0.918814i \(-0.629148\pi\)
−0.394690 + 0.918814i \(0.629148\pi\)
\(942\) 0 0
\(943\) − 3.38197i − 0.110132i
\(944\) 0 0
\(945\) −37.6869 −1.22596
\(946\) 0 0
\(947\) 47.3951i 1.54013i 0.637963 + 0.770067i \(0.279777\pi\)
−0.637963 + 0.770067i \(0.720223\pi\)
\(948\) 0 0
\(949\) 2.87539 0.0933391
\(950\) 0 0
\(951\) 16.2361 0.526491
\(952\) 0 0
\(953\) 36.3951i 1.17895i 0.807785 + 0.589477i \(0.200666\pi\)
−0.807785 + 0.589477i \(0.799334\pi\)
\(954\) 0 0
\(955\) 3.16718 0.102488
\(956\) 0 0
\(957\) 7.75078i 0.250547i
\(958\) 0 0
\(959\) −34.6869 −1.12010
\(960\) 0 0
\(961\) 47.3951 1.52887
\(962\) 0 0
\(963\) 4.47214i 0.144113i
\(964\) 0 0
\(965\) 5.12461i 0.164967i
\(966\) 0 0
\(967\) − 24.7639i − 0.796354i −0.917309 0.398177i \(-0.869643\pi\)
0.917309 0.398177i \(-0.130357\pi\)
\(968\) 0 0
\(969\) 24.7984 0.796639
\(970\) 0 0
\(971\) −25.8541 −0.829698 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(972\) 0 0
\(973\) − 85.9574i − 2.75567i
\(974\) 0 0
\(975\) −1.18034 −0.0378011
\(976\) 0 0
\(977\) − 32.2705i − 1.03243i −0.856461 0.516213i \(-0.827341\pi\)
0.856461 0.516213i \(-0.172659\pi\)
\(978\) 0 0
\(979\) 32.8328 1.04934
\(980\) 0 0
\(981\) 32.8885 1.05005
\(982\) 0 0
\(983\) 26.3951i 0.841874i 0.907090 + 0.420937i \(0.138299\pi\)
−0.907090 + 0.420937i \(0.861701\pi\)
\(984\) 0 0
\(985\) − 0.978714i − 0.0311844i
\(986\) 0 0
\(987\) − 35.1246i − 1.11803i
\(988\) 0 0
\(989\) −6.76393 −0.215081
\(990\) 0 0
\(991\) −51.2705 −1.62866 −0.814331 0.580401i \(-0.802896\pi\)
−0.814331 + 0.580401i \(0.802896\pi\)
\(992\) 0 0
\(993\) 9.34752i 0.296635i
\(994\) 0 0
\(995\) −34.4721 −1.09284
\(996\) 0 0
\(997\) 26.7214i 0.846274i 0.906066 + 0.423137i \(0.139071\pi\)
−0.906066 + 0.423137i \(0.860929\pi\)
\(998\) 0 0
\(999\) 12.8754 0.407359
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 1840.2.e.c.369.2 4
4.3 odd 2 230.2.b.a.139.2 4
5.2 odd 4 9200.2.a.by.1.1 2
5.3 odd 4 9200.2.a.bo.1.2 2
5.4 even 2 inner 1840.2.e.c.369.3 4
12.11 even 2 2070.2.d.c.829.3 4
20.3 even 4 1150.2.a.l.1.1 2
20.7 even 4 1150.2.a.n.1.2 2
20.19 odd 2 230.2.b.a.139.3 yes 4
60.59 even 2 2070.2.d.c.829.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.2 4 4.3 odd 2
230.2.b.a.139.3 yes 4 20.19 odd 2
1150.2.a.l.1.1 2 20.3 even 4
1150.2.a.n.1.2 2 20.7 even 4
1840.2.e.c.369.2 4 1.1 even 1 trivial
1840.2.e.c.369.3 4 5.4 even 2 inner
2070.2.d.c.829.1 4 60.59 even 2
2070.2.d.c.829.3 4 12.11 even 2
9200.2.a.bo.1.2 2 5.3 odd 4
9200.2.a.by.1.1 2 5.2 odd 4