Properties

Label 9075.2.a.x.1.2
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} -1.76393 q^{13} -1.85410 q^{14} +1.85410 q^{16} -1.61803 q^{17} +0.618034 q^{18} +5.85410 q^{19} +3.00000 q^{21} -3.47214 q^{23} +2.23607 q^{24} -1.09017 q^{26} -1.00000 q^{27} +4.85410 q^{28} +4.47214 q^{29} +2.85410 q^{31} +5.61803 q^{32} -1.00000 q^{34} -1.61803 q^{36} -0.236068 q^{37} +3.61803 q^{38} +1.76393 q^{39} -11.9443 q^{41} +1.85410 q^{42} -6.23607 q^{43} -2.14590 q^{46} -1.61803 q^{47} -1.85410 q^{48} +2.00000 q^{49} +1.61803 q^{51} +2.85410 q^{52} +9.61803 q^{53} -0.618034 q^{54} +6.70820 q^{56} -5.85410 q^{57} +2.76393 q^{58} +10.3262 q^{59} +7.85410 q^{61} +1.76393 q^{62} -3.00000 q^{63} -0.236068 q^{64} +9.56231 q^{67} +2.61803 q^{68} +3.47214 q^{69} -5.56231 q^{71} -2.23607 q^{72} +3.23607 q^{73} -0.145898 q^{74} -9.47214 q^{76} +1.09017 q^{78} +9.47214 q^{79} +1.00000 q^{81} -7.38197 q^{82} -0.708204 q^{83} -4.85410 q^{84} -3.85410 q^{86} -4.47214 q^{87} +0.527864 q^{89} +5.29180 q^{91} +5.61803 q^{92} -2.85410 q^{93} -1.00000 q^{94} -5.61803 q^{96} +14.0344 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9} + q^{12} - 8 q^{13} + 3 q^{14} - 3 q^{16} - q^{17} - q^{18} + 5 q^{19} + 6 q^{21} + 2 q^{23} + 9 q^{26} - 2 q^{27} + 3 q^{28} - q^{31} + 9 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.61803 0.467086
\(13\) −1.76393 −0.489227 −0.244613 0.969621i \(-0.578661\pi\)
−0.244613 + 0.969621i \(0.578661\pi\)
\(14\) −1.85410 −0.495530
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) −1.61803 −0.392431 −0.196215 0.980561i \(-0.562865\pi\)
−0.196215 + 0.980561i \(0.562865\pi\)
\(18\) 0.618034 0.145672
\(19\) 5.85410 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −3.47214 −0.723990 −0.361995 0.932180i \(-0.617904\pi\)
−0.361995 + 0.932180i \(0.617904\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −1.09017 −0.213800
\(27\) −1.00000 −0.192450
\(28\) 4.85410 0.917339
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.85410 0.512612 0.256306 0.966596i \(-0.417495\pi\)
0.256306 + 0.966596i \(0.417495\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) −1.61803 −0.269672
\(37\) −0.236068 −0.0388093 −0.0194047 0.999812i \(-0.506177\pi\)
−0.0194047 + 0.999812i \(0.506177\pi\)
\(38\) 3.61803 0.586923
\(39\) 1.76393 0.282455
\(40\) 0 0
\(41\) −11.9443 −1.86538 −0.932691 0.360677i \(-0.882546\pi\)
−0.932691 + 0.360677i \(0.882546\pi\)
\(42\) 1.85410 0.286094
\(43\) −6.23607 −0.950991 −0.475496 0.879718i \(-0.657731\pi\)
−0.475496 + 0.879718i \(0.657731\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.14590 −0.316395
\(47\) −1.61803 −0.236015 −0.118007 0.993013i \(-0.537651\pi\)
−0.118007 + 0.993013i \(0.537651\pi\)
\(48\) −1.85410 −0.267617
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 1.61803 0.226570
\(52\) 2.85410 0.395793
\(53\) 9.61803 1.32114 0.660569 0.750765i \(-0.270315\pi\)
0.660569 + 0.750765i \(0.270315\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) −5.85410 −0.775395
\(58\) 2.76393 0.362922
\(59\) 10.3262 1.34436 0.672181 0.740387i \(-0.265358\pi\)
0.672181 + 0.740387i \(0.265358\pi\)
\(60\) 0 0
\(61\) 7.85410 1.00561 0.502807 0.864398i \(-0.332301\pi\)
0.502807 + 0.864398i \(0.332301\pi\)
\(62\) 1.76393 0.224020
\(63\) −3.00000 −0.377964
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 9.56231 1.16822 0.584111 0.811674i \(-0.301443\pi\)
0.584111 + 0.811674i \(0.301443\pi\)
\(68\) 2.61803 0.317483
\(69\) 3.47214 0.417996
\(70\) 0 0
\(71\) −5.56231 −0.660124 −0.330062 0.943959i \(-0.607070\pi\)
−0.330062 + 0.943959i \(0.607070\pi\)
\(72\) −2.23607 −0.263523
\(73\) 3.23607 0.378753 0.189377 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) −0.145898 −0.0169603
\(75\) 0 0
\(76\) −9.47214 −1.08653
\(77\) 0 0
\(78\) 1.09017 0.123437
\(79\) 9.47214 1.06570 0.532849 0.846210i \(-0.321121\pi\)
0.532849 + 0.846210i \(0.321121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.38197 −0.815202
\(83\) −0.708204 −0.0777355 −0.0388677 0.999244i \(-0.512375\pi\)
−0.0388677 + 0.999244i \(0.512375\pi\)
\(84\) −4.85410 −0.529626
\(85\) 0 0
\(86\) −3.85410 −0.415599
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 0.527864 0.0559535 0.0279767 0.999609i \(-0.491094\pi\)
0.0279767 + 0.999609i \(0.491094\pi\)
\(90\) 0 0
\(91\) 5.29180 0.554731
\(92\) 5.61803 0.585721
\(93\) −2.85410 −0.295957
\(94\) −1.00000 −0.103142
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) 14.0344 1.42498 0.712491 0.701681i \(-0.247567\pi\)
0.712491 + 0.701681i \(0.247567\pi\)
\(98\) 1.23607 0.124862
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 1.00000 0.0990148
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 3.94427 0.386768
\(105\) 0 0
\(106\) 5.94427 0.577359
\(107\) 4.23607 0.409516 0.204758 0.978813i \(-0.434359\pi\)
0.204758 + 0.978813i \(0.434359\pi\)
\(108\) 1.61803 0.155695
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0.236068 0.0224066
\(112\) −5.56231 −0.525589
\(113\) −0.708204 −0.0666222 −0.0333111 0.999445i \(-0.510605\pi\)
−0.0333111 + 0.999445i \(0.510605\pi\)
\(114\) −3.61803 −0.338860
\(115\) 0 0
\(116\) −7.23607 −0.671852
\(117\) −1.76393 −0.163076
\(118\) 6.38197 0.587508
\(119\) 4.85410 0.444975
\(120\) 0 0
\(121\) 0 0
\(122\) 4.85410 0.439470
\(123\) 11.9443 1.07698
\(124\) −4.61803 −0.414712
\(125\) 0 0
\(126\) −1.85410 −0.165177
\(127\) 3.70820 0.329050 0.164525 0.986373i \(-0.447391\pi\)
0.164525 + 0.986373i \(0.447391\pi\)
\(128\) −11.3820 −1.00603
\(129\) 6.23607 0.549055
\(130\) 0 0
\(131\) −7.14590 −0.624340 −0.312170 0.950026i \(-0.601056\pi\)
−0.312170 + 0.950026i \(0.601056\pi\)
\(132\) 0 0
\(133\) −17.5623 −1.52285
\(134\) 5.90983 0.510532
\(135\) 0 0
\(136\) 3.61803 0.310244
\(137\) −7.47214 −0.638388 −0.319194 0.947689i \(-0.603412\pi\)
−0.319194 + 0.947689i \(0.603412\pi\)
\(138\) 2.14590 0.182671
\(139\) −0.854102 −0.0724440 −0.0362220 0.999344i \(-0.511532\pi\)
−0.0362220 + 0.999344i \(0.511532\pi\)
\(140\) 0 0
\(141\) 1.61803 0.136263
\(142\) −3.43769 −0.288485
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −2.00000 −0.164957
\(148\) 0.381966 0.0313974
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −13.0902 −1.06175
\(153\) −1.61803 −0.130810
\(154\) 0 0
\(155\) 0 0
\(156\) −2.85410 −0.228511
\(157\) 3.70820 0.295947 0.147973 0.988991i \(-0.452725\pi\)
0.147973 + 0.988991i \(0.452725\pi\)
\(158\) 5.85410 0.465727
\(159\) −9.61803 −0.762760
\(160\) 0 0
\(161\) 10.4164 0.820928
\(162\) 0.618034 0.0485573
\(163\) −18.2705 −1.43106 −0.715528 0.698584i \(-0.753814\pi\)
−0.715528 + 0.698584i \(0.753814\pi\)
\(164\) 19.3262 1.50913
\(165\) 0 0
\(166\) −0.437694 −0.0339717
\(167\) −10.0344 −0.776488 −0.388244 0.921557i \(-0.626918\pi\)
−0.388244 + 0.921557i \(0.626918\pi\)
\(168\) −6.70820 −0.517549
\(169\) −9.88854 −0.760657
\(170\) 0 0
\(171\) 5.85410 0.447674
\(172\) 10.0902 0.769368
\(173\) −15.3820 −1.16947 −0.584735 0.811225i \(-0.698801\pi\)
−0.584735 + 0.811225i \(0.698801\pi\)
\(174\) −2.76393 −0.209533
\(175\) 0 0
\(176\) 0 0
\(177\) −10.3262 −0.776168
\(178\) 0.326238 0.0244526
\(179\) −2.23607 −0.167132 −0.0835658 0.996502i \(-0.526631\pi\)
−0.0835658 + 0.996502i \(0.526631\pi\)
\(180\) 0 0
\(181\) −17.4721 −1.29869 −0.649347 0.760492i \(-0.724958\pi\)
−0.649347 + 0.760492i \(0.724958\pi\)
\(182\) 3.27051 0.242426
\(183\) −7.85410 −0.580592
\(184\) 7.76393 0.572365
\(185\) 0 0
\(186\) −1.76393 −0.129338
\(187\) 0 0
\(188\) 2.61803 0.190940
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −7.47214 −0.540665 −0.270332 0.962767i \(-0.587134\pi\)
−0.270332 + 0.962767i \(0.587134\pi\)
\(192\) 0.236068 0.0170367
\(193\) 18.5623 1.33614 0.668072 0.744097i \(-0.267120\pi\)
0.668072 + 0.744097i \(0.267120\pi\)
\(194\) 8.67376 0.622740
\(195\) 0 0
\(196\) −3.23607 −0.231148
\(197\) −24.3820 −1.73714 −0.868572 0.495564i \(-0.834961\pi\)
−0.868572 + 0.495564i \(0.834961\pi\)
\(198\) 0 0
\(199\) −16.7082 −1.18441 −0.592207 0.805786i \(-0.701743\pi\)
−0.592207 + 0.805786i \(0.701743\pi\)
\(200\) 0 0
\(201\) −9.56231 −0.674473
\(202\) −1.85410 −0.130454
\(203\) −13.4164 −0.941647
\(204\) −2.61803 −0.183299
\(205\) 0 0
\(206\) 3.70820 0.258363
\(207\) −3.47214 −0.241330
\(208\) −3.27051 −0.226769
\(209\) 0 0
\(210\) 0 0
\(211\) −22.2705 −1.53317 −0.766583 0.642146i \(-0.778044\pi\)
−0.766583 + 0.642146i \(0.778044\pi\)
\(212\) −15.5623 −1.06882
\(213\) 5.56231 0.381123
\(214\) 2.61803 0.178965
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) −8.56231 −0.581247
\(218\) 0 0
\(219\) −3.23607 −0.218673
\(220\) 0 0
\(221\) 2.85410 0.191988
\(222\) 0.145898 0.00979203
\(223\) −0.708204 −0.0474248 −0.0237124 0.999719i \(-0.507549\pi\)
−0.0237124 + 0.999719i \(0.507549\pi\)
\(224\) −16.8541 −1.12611
\(225\) 0 0
\(226\) −0.437694 −0.0291150
\(227\) 24.8885 1.65191 0.825955 0.563736i \(-0.190636\pi\)
0.825955 + 0.563736i \(0.190636\pi\)
\(228\) 9.47214 0.627308
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) −24.3262 −1.59366 −0.796832 0.604200i \(-0.793493\pi\)
−0.796832 + 0.604200i \(0.793493\pi\)
\(234\) −1.09017 −0.0712666
\(235\) 0 0
\(236\) −16.7082 −1.08761
\(237\) −9.47214 −0.615281
\(238\) 3.00000 0.194461
\(239\) 2.56231 0.165742 0.0828709 0.996560i \(-0.473591\pi\)
0.0828709 + 0.996560i \(0.473591\pi\)
\(240\) 0 0
\(241\) −23.1246 −1.48959 −0.744794 0.667295i \(-0.767452\pi\)
−0.744794 + 0.667295i \(0.767452\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −12.7082 −0.813559
\(245\) 0 0
\(246\) 7.38197 0.470657
\(247\) −10.3262 −0.657043
\(248\) −6.38197 −0.405255
\(249\) 0.708204 0.0448806
\(250\) 0 0
\(251\) −7.79837 −0.492229 −0.246114 0.969241i \(-0.579154\pi\)
−0.246114 + 0.969241i \(0.579154\pi\)
\(252\) 4.85410 0.305780
\(253\) 0 0
\(254\) 2.29180 0.143800
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 11.6738 0.728189 0.364095 0.931362i \(-0.381378\pi\)
0.364095 + 0.931362i \(0.381378\pi\)
\(258\) 3.85410 0.239946
\(259\) 0.708204 0.0440057
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) −4.41641 −0.272847
\(263\) 16.3262 1.00672 0.503359 0.864077i \(-0.332097\pi\)
0.503359 + 0.864077i \(0.332097\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.8541 −0.665508
\(267\) −0.527864 −0.0323048
\(268\) −15.4721 −0.945111
\(269\) 15.5279 0.946751 0.473375 0.880861i \(-0.343035\pi\)
0.473375 + 0.880861i \(0.343035\pi\)
\(270\) 0 0
\(271\) −27.2705 −1.65657 −0.828283 0.560310i \(-0.810682\pi\)
−0.828283 + 0.560310i \(0.810682\pi\)
\(272\) −3.00000 −0.181902
\(273\) −5.29180 −0.320274
\(274\) −4.61803 −0.278986
\(275\) 0 0
\(276\) −5.61803 −0.338166
\(277\) −30.5623 −1.83631 −0.918155 0.396220i \(-0.870322\pi\)
−0.918155 + 0.396220i \(0.870322\pi\)
\(278\) −0.527864 −0.0316592
\(279\) 2.85410 0.170871
\(280\) 0 0
\(281\) −0.763932 −0.0455724 −0.0227862 0.999740i \(-0.507254\pi\)
−0.0227862 + 0.999740i \(0.507254\pi\)
\(282\) 1.00000 0.0595491
\(283\) −0.180340 −0.0107201 −0.00536005 0.999986i \(-0.501706\pi\)
−0.00536005 + 0.999986i \(0.501706\pi\)
\(284\) 9.00000 0.534052
\(285\) 0 0
\(286\) 0 0
\(287\) 35.8328 2.11514
\(288\) 5.61803 0.331046
\(289\) −14.3820 −0.845998
\(290\) 0 0
\(291\) −14.0344 −0.822714
\(292\) −5.23607 −0.306418
\(293\) −0.0557281 −0.00325567 −0.00162783 0.999999i \(-0.500518\pi\)
−0.00162783 + 0.999999i \(0.500518\pi\)
\(294\) −1.23607 −0.0720889
\(295\) 0 0
\(296\) 0.527864 0.0306815
\(297\) 0 0
\(298\) 9.27051 0.537026
\(299\) 6.12461 0.354195
\(300\) 0 0
\(301\) 18.7082 1.07832
\(302\) 1.23607 0.0711277
\(303\) 3.00000 0.172345
\(304\) 10.8541 0.622525
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) −0.562306 −0.0320925 −0.0160462 0.999871i \(-0.505108\pi\)
−0.0160462 + 0.999871i \(0.505108\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 2.52786 0.143342 0.0716710 0.997428i \(-0.477167\pi\)
0.0716710 + 0.997428i \(0.477167\pi\)
\(312\) −3.94427 −0.223300
\(313\) −25.8328 −1.46016 −0.730079 0.683363i \(-0.760517\pi\)
−0.730079 + 0.683363i \(0.760517\pi\)
\(314\) 2.29180 0.129334
\(315\) 0 0
\(316\) −15.3262 −0.862168
\(317\) 19.3607 1.08740 0.543702 0.839278i \(-0.317022\pi\)
0.543702 + 0.839278i \(0.317022\pi\)
\(318\) −5.94427 −0.333338
\(319\) 0 0
\(320\) 0 0
\(321\) −4.23607 −0.236434
\(322\) 6.43769 0.358759
\(323\) −9.47214 −0.527044
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) −11.2918 −0.625395
\(327\) 0 0
\(328\) 26.7082 1.47471
\(329\) 4.85410 0.267615
\(330\) 0 0
\(331\) 26.5967 1.46189 0.730945 0.682437i \(-0.239080\pi\)
0.730945 + 0.682437i \(0.239080\pi\)
\(332\) 1.14590 0.0628893
\(333\) −0.236068 −0.0129364
\(334\) −6.20163 −0.339338
\(335\) 0 0
\(336\) 5.56231 0.303449
\(337\) 0.291796 0.0158951 0.00794757 0.999968i \(-0.497470\pi\)
0.00794757 + 0.999968i \(0.497470\pi\)
\(338\) −6.11146 −0.332419
\(339\) 0.708204 0.0384644
\(340\) 0 0
\(341\) 0 0
\(342\) 3.61803 0.195641
\(343\) 15.0000 0.809924
\(344\) 13.9443 0.751825
\(345\) 0 0
\(346\) −9.50658 −0.511077
\(347\) 20.9443 1.12435 0.562174 0.827019i \(-0.309965\pi\)
0.562174 + 0.827019i \(0.309965\pi\)
\(348\) 7.23607 0.387894
\(349\) −10.1246 −0.541958 −0.270979 0.962585i \(-0.587347\pi\)
−0.270979 + 0.962585i \(0.587347\pi\)
\(350\) 0 0
\(351\) 1.76393 0.0941517
\(352\) 0 0
\(353\) 10.4721 0.557376 0.278688 0.960382i \(-0.410101\pi\)
0.278688 + 0.960382i \(0.410101\pi\)
\(354\) −6.38197 −0.339198
\(355\) 0 0
\(356\) −0.854102 −0.0452673
\(357\) −4.85410 −0.256906
\(358\) −1.38197 −0.0730392
\(359\) −12.7639 −0.673655 −0.336827 0.941566i \(-0.609354\pi\)
−0.336827 + 0.941566i \(0.609354\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) −10.7984 −0.567550
\(363\) 0 0
\(364\) −8.56231 −0.448787
\(365\) 0 0
\(366\) −4.85410 −0.253728
\(367\) −5.56231 −0.290350 −0.145175 0.989406i \(-0.546375\pi\)
−0.145175 + 0.989406i \(0.546375\pi\)
\(368\) −6.43769 −0.335588
\(369\) −11.9443 −0.621794
\(370\) 0 0
\(371\) −28.8541 −1.49803
\(372\) 4.61803 0.239434
\(373\) 4.41641 0.228673 0.114336 0.993442i \(-0.463526\pi\)
0.114336 + 0.993442i \(0.463526\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.61803 0.186586
\(377\) −7.88854 −0.406281
\(378\) 1.85410 0.0953647
\(379\) 1.58359 0.0813437 0.0406718 0.999173i \(-0.487050\pi\)
0.0406718 + 0.999173i \(0.487050\pi\)
\(380\) 0 0
\(381\) −3.70820 −0.189977
\(382\) −4.61803 −0.236279
\(383\) −26.8885 −1.37394 −0.686970 0.726686i \(-0.741060\pi\)
−0.686970 + 0.726686i \(0.741060\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) 11.4721 0.583916
\(387\) −6.23607 −0.316997
\(388\) −22.7082 −1.15283
\(389\) 24.2705 1.23056 0.615282 0.788307i \(-0.289042\pi\)
0.615282 + 0.788307i \(0.289042\pi\)
\(390\) 0 0
\(391\) 5.61803 0.284116
\(392\) −4.47214 −0.225877
\(393\) 7.14590 0.360463
\(394\) −15.0689 −0.759159
\(395\) 0 0
\(396\) 0 0
\(397\) 38.7082 1.94271 0.971355 0.237635i \(-0.0763722\pi\)
0.971355 + 0.237635i \(0.0763722\pi\)
\(398\) −10.3262 −0.517608
\(399\) 17.5623 0.879215
\(400\) 0 0
\(401\) −26.0902 −1.30288 −0.651440 0.758700i \(-0.725835\pi\)
−0.651440 + 0.758700i \(0.725835\pi\)
\(402\) −5.90983 −0.294756
\(403\) −5.03444 −0.250783
\(404\) 4.85410 0.241501
\(405\) 0 0
\(406\) −8.29180 −0.411515
\(407\) 0 0
\(408\) −3.61803 −0.179119
\(409\) −11.0557 −0.546671 −0.273335 0.961919i \(-0.588127\pi\)
−0.273335 + 0.961919i \(0.588127\pi\)
\(410\) 0 0
\(411\) 7.47214 0.368573
\(412\) −9.70820 −0.478289
\(413\) −30.9787 −1.52436
\(414\) −2.14590 −0.105465
\(415\) 0 0
\(416\) −9.90983 −0.485869
\(417\) 0.854102 0.0418256
\(418\) 0 0
\(419\) 16.5066 0.806399 0.403200 0.915112i \(-0.367898\pi\)
0.403200 + 0.915112i \(0.367898\pi\)
\(420\) 0 0
\(421\) −37.2705 −1.81645 −0.908227 0.418478i \(-0.862564\pi\)
−0.908227 + 0.418478i \(0.862564\pi\)
\(422\) −13.7639 −0.670018
\(423\) −1.61803 −0.0786715
\(424\) −21.5066 −1.04445
\(425\) 0 0
\(426\) 3.43769 0.166557
\(427\) −23.5623 −1.14026
\(428\) −6.85410 −0.331306
\(429\) 0 0
\(430\) 0 0
\(431\) −39.5066 −1.90296 −0.951482 0.307703i \(-0.900440\pi\)
−0.951482 + 0.307703i \(0.900440\pi\)
\(432\) −1.85410 −0.0892055
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −5.29180 −0.254014
\(435\) 0 0
\(436\) 0 0
\(437\) −20.3262 −0.972336
\(438\) −2.00000 −0.0955637
\(439\) 3.29180 0.157109 0.0785544 0.996910i \(-0.474970\pi\)
0.0785544 + 0.996910i \(0.474970\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 1.76393 0.0839017
\(443\) 41.1246 1.95389 0.976945 0.213493i \(-0.0684840\pi\)
0.976945 + 0.213493i \(0.0684840\pi\)
\(444\) −0.381966 −0.0181273
\(445\) 0 0
\(446\) −0.437694 −0.0207254
\(447\) −15.0000 −0.709476
\(448\) 0.708204 0.0334595
\(449\) −24.4721 −1.15491 −0.577456 0.816422i \(-0.695954\pi\)
−0.577456 + 0.816422i \(0.695954\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.14590 0.0538985
\(453\) −2.00000 −0.0939682
\(454\) 15.3820 0.721911
\(455\) 0 0
\(456\) 13.0902 0.613003
\(457\) −8.20163 −0.383656 −0.191828 0.981429i \(-0.561442\pi\)
−0.191828 + 0.981429i \(0.561442\pi\)
\(458\) 6.18034 0.288788
\(459\) 1.61803 0.0755234
\(460\) 0 0
\(461\) −21.0902 −0.982267 −0.491134 0.871084i \(-0.663417\pi\)
−0.491134 + 0.871084i \(0.663417\pi\)
\(462\) 0 0
\(463\) 15.7984 0.734213 0.367106 0.930179i \(-0.380349\pi\)
0.367106 + 0.930179i \(0.380349\pi\)
\(464\) 8.29180 0.384937
\(465\) 0 0
\(466\) −15.0344 −0.696457
\(467\) 9.76393 0.451821 0.225910 0.974148i \(-0.427464\pi\)
0.225910 + 0.974148i \(0.427464\pi\)
\(468\) 2.85410 0.131931
\(469\) −28.6869 −1.32464
\(470\) 0 0
\(471\) −3.70820 −0.170865
\(472\) −23.0902 −1.06281
\(473\) 0 0
\(474\) −5.85410 −0.268888
\(475\) 0 0
\(476\) −7.85410 −0.359992
\(477\) 9.61803 0.440380
\(478\) 1.58359 0.0724318
\(479\) −28.0902 −1.28347 −0.641736 0.766925i \(-0.721786\pi\)
−0.641736 + 0.766925i \(0.721786\pi\)
\(480\) 0 0
\(481\) 0.416408 0.0189866
\(482\) −14.2918 −0.650973
\(483\) −10.4164 −0.473963
\(484\) 0 0
\(485\) 0 0
\(486\) −0.618034 −0.0280346
\(487\) −16.8197 −0.762172 −0.381086 0.924540i \(-0.624450\pi\)
−0.381086 + 0.924540i \(0.624450\pi\)
\(488\) −17.5623 −0.795008
\(489\) 18.2705 0.826221
\(490\) 0 0
\(491\) 25.2148 1.13793 0.568964 0.822363i \(-0.307345\pi\)
0.568964 + 0.822363i \(0.307345\pi\)
\(492\) −19.3262 −0.871294
\(493\) −7.23607 −0.325896
\(494\) −6.38197 −0.287138
\(495\) 0 0
\(496\) 5.29180 0.237609
\(497\) 16.6869 0.748511
\(498\) 0.437694 0.0196135
\(499\) −17.5623 −0.786197 −0.393098 0.919496i \(-0.628597\pi\)
−0.393098 + 0.919496i \(0.628597\pi\)
\(500\) 0 0
\(501\) 10.0344 0.448306
\(502\) −4.81966 −0.215112
\(503\) −28.0689 −1.25153 −0.625765 0.780012i \(-0.715213\pi\)
−0.625765 + 0.780012i \(0.715213\pi\)
\(504\) 6.70820 0.298807
\(505\) 0 0
\(506\) 0 0
\(507\) 9.88854 0.439166
\(508\) −6.00000 −0.266207
\(509\) 23.6180 1.04685 0.523425 0.852071i \(-0.324654\pi\)
0.523425 + 0.852071i \(0.324654\pi\)
\(510\) 0 0
\(511\) −9.70820 −0.429466
\(512\) 18.7082 0.826794
\(513\) −5.85410 −0.258465
\(514\) 7.21478 0.318230
\(515\) 0 0
\(516\) −10.0902 −0.444195
\(517\) 0 0
\(518\) 0.437694 0.0192312
\(519\) 15.3820 0.675193
\(520\) 0 0
\(521\) −14.8328 −0.649837 −0.324919 0.945742i \(-0.605337\pi\)
−0.324919 + 0.945742i \(0.605337\pi\)
\(522\) 2.76393 0.120974
\(523\) 11.9787 0.523793 0.261896 0.965096i \(-0.415652\pi\)
0.261896 + 0.965096i \(0.415652\pi\)
\(524\) 11.5623 0.505102
\(525\) 0 0
\(526\) 10.0902 0.439952
\(527\) −4.61803 −0.201165
\(528\) 0 0
\(529\) −10.9443 −0.475838
\(530\) 0 0
\(531\) 10.3262 0.448121
\(532\) 28.4164 1.23201
\(533\) 21.0689 0.912595
\(534\) −0.326238 −0.0141177
\(535\) 0 0
\(536\) −21.3820 −0.923560
\(537\) 2.23607 0.0964935
\(538\) 9.59675 0.413745
\(539\) 0 0
\(540\) 0 0
\(541\) 19.5623 0.841049 0.420525 0.907281i \(-0.361846\pi\)
0.420525 + 0.907281i \(0.361846\pi\)
\(542\) −16.8541 −0.723946
\(543\) 17.4721 0.749801
\(544\) −9.09017 −0.389738
\(545\) 0 0
\(546\) −3.27051 −0.139965
\(547\) −21.6180 −0.924320 −0.462160 0.886796i \(-0.652925\pi\)
−0.462160 + 0.886796i \(0.652925\pi\)
\(548\) 12.0902 0.516466
\(549\) 7.85410 0.335205
\(550\) 0 0
\(551\) 26.1803 1.11532
\(552\) −7.76393 −0.330455
\(553\) −28.4164 −1.20839
\(554\) −18.8885 −0.802497
\(555\) 0 0
\(556\) 1.38197 0.0586084
\(557\) −14.9098 −0.631750 −0.315875 0.948801i \(-0.602298\pi\)
−0.315875 + 0.948801i \(0.602298\pi\)
\(558\) 1.76393 0.0746732
\(559\) 11.0000 0.465250
\(560\) 0 0
\(561\) 0 0
\(562\) −0.472136 −0.0199159
\(563\) 8.88854 0.374607 0.187304 0.982302i \(-0.440025\pi\)
0.187304 + 0.982302i \(0.440025\pi\)
\(564\) −2.61803 −0.110239
\(565\) 0 0
\(566\) −0.111456 −0.00468485
\(567\) −3.00000 −0.125988
\(568\) 12.4377 0.521874
\(569\) −24.0689 −1.00902 −0.504510 0.863406i \(-0.668327\pi\)
−0.504510 + 0.863406i \(0.668327\pi\)
\(570\) 0 0
\(571\) 34.6869 1.45160 0.725801 0.687905i \(-0.241469\pi\)
0.725801 + 0.687905i \(0.241469\pi\)
\(572\) 0 0
\(573\) 7.47214 0.312153
\(574\) 22.1459 0.924352
\(575\) 0 0
\(576\) −0.236068 −0.00983617
\(577\) −10.7639 −0.448108 −0.224054 0.974577i \(-0.571929\pi\)
−0.224054 + 0.974577i \(0.571929\pi\)
\(578\) −8.88854 −0.369715
\(579\) −18.5623 −0.771423
\(580\) 0 0
\(581\) 2.12461 0.0881437
\(582\) −8.67376 −0.359539
\(583\) 0 0
\(584\) −7.23607 −0.299431
\(585\) 0 0
\(586\) −0.0344419 −0.00142278
\(587\) 38.3050 1.58101 0.790507 0.612453i \(-0.209817\pi\)
0.790507 + 0.612453i \(0.209817\pi\)
\(588\) 3.23607 0.133453
\(589\) 16.7082 0.688450
\(590\) 0 0
\(591\) 24.3820 1.00294
\(592\) −0.437694 −0.0179891
\(593\) −22.2148 −0.912252 −0.456126 0.889915i \(-0.650763\pi\)
−0.456126 + 0.889915i \(0.650763\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24.2705 −0.994159
\(597\) 16.7082 0.683821
\(598\) 3.78522 0.154789
\(599\) −8.29180 −0.338794 −0.169397 0.985548i \(-0.554182\pi\)
−0.169397 + 0.985548i \(0.554182\pi\)
\(600\) 0 0
\(601\) 33.8328 1.38007 0.690035 0.723776i \(-0.257595\pi\)
0.690035 + 0.723776i \(0.257595\pi\)
\(602\) 11.5623 0.471244
\(603\) 9.56231 0.389407
\(604\) −3.23607 −0.131674
\(605\) 0 0
\(606\) 1.85410 0.0753177
\(607\) −13.3262 −0.540895 −0.270448 0.962735i \(-0.587172\pi\)
−0.270448 + 0.962735i \(0.587172\pi\)
\(608\) 32.8885 1.33381
\(609\) 13.4164 0.543660
\(610\) 0 0
\(611\) 2.85410 0.115465
\(612\) 2.61803 0.105828
\(613\) −34.6525 −1.39960 −0.699800 0.714339i \(-0.746728\pi\)
−0.699800 + 0.714339i \(0.746728\pi\)
\(614\) −0.347524 −0.0140249
\(615\) 0 0
\(616\) 0 0
\(617\) −19.5836 −0.788406 −0.394203 0.919023i \(-0.628979\pi\)
−0.394203 + 0.919023i \(0.628979\pi\)
\(618\) −3.70820 −0.149166
\(619\) −8.81966 −0.354492 −0.177246 0.984167i \(-0.556719\pi\)
−0.177246 + 0.984167i \(0.556719\pi\)
\(620\) 0 0
\(621\) 3.47214 0.139332
\(622\) 1.56231 0.0626428
\(623\) −1.58359 −0.0634453
\(624\) 3.27051 0.130925
\(625\) 0 0
\(626\) −15.9656 −0.638112
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) 0.381966 0.0152300
\(630\) 0 0
\(631\) 46.2705 1.84200 0.921000 0.389563i \(-0.127374\pi\)
0.921000 + 0.389563i \(0.127374\pi\)
\(632\) −21.1803 −0.842509
\(633\) 22.2705 0.885173
\(634\) 11.9656 0.475213
\(635\) 0 0
\(636\) 15.5623 0.617086
\(637\) −3.52786 −0.139779
\(638\) 0 0
\(639\) −5.56231 −0.220041
\(640\) 0 0
\(641\) 35.7426 1.41175 0.705875 0.708337i \(-0.250554\pi\)
0.705875 + 0.708337i \(0.250554\pi\)
\(642\) −2.61803 −0.103326
\(643\) 38.5623 1.52075 0.760374 0.649485i \(-0.225015\pi\)
0.760374 + 0.649485i \(0.225015\pi\)
\(644\) −16.8541 −0.664145
\(645\) 0 0
\(646\) −5.85410 −0.230327
\(647\) −27.7984 −1.09287 −0.546433 0.837503i \(-0.684015\pi\)
−0.546433 + 0.837503i \(0.684015\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) 8.56231 0.335583
\(652\) 29.5623 1.15775
\(653\) −51.0344 −1.99713 −0.998566 0.0535342i \(-0.982951\pi\)
−0.998566 + 0.0535342i \(0.982951\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −22.1459 −0.864652
\(657\) 3.23607 0.126251
\(658\) 3.00000 0.116952
\(659\) −10.6525 −0.414962 −0.207481 0.978239i \(-0.566526\pi\)
−0.207481 + 0.978239i \(0.566526\pi\)
\(660\) 0 0
\(661\) −9.90983 −0.385448 −0.192724 0.981253i \(-0.561732\pi\)
−0.192724 + 0.981253i \(0.561732\pi\)
\(662\) 16.4377 0.638869
\(663\) −2.85410 −0.110844
\(664\) 1.58359 0.0614553
\(665\) 0 0
\(666\) −0.145898 −0.00565343
\(667\) −15.5279 −0.601241
\(668\) 16.2361 0.628192
\(669\) 0.708204 0.0273807
\(670\) 0 0
\(671\) 0 0
\(672\) 16.8541 0.650161
\(673\) −12.4164 −0.478617 −0.239309 0.970944i \(-0.576921\pi\)
−0.239309 + 0.970944i \(0.576921\pi\)
\(674\) 0.180340 0.00694643
\(675\) 0 0
\(676\) 16.0000 0.615385
\(677\) −13.5279 −0.519918 −0.259959 0.965620i \(-0.583709\pi\)
−0.259959 + 0.965620i \(0.583709\pi\)
\(678\) 0.437694 0.0168095
\(679\) −42.1033 −1.61578
\(680\) 0 0
\(681\) −24.8885 −0.953731
\(682\) 0 0
\(683\) 3.11146 0.119057 0.0595283 0.998227i \(-0.481040\pi\)
0.0595283 + 0.998227i \(0.481040\pi\)
\(684\) −9.47214 −0.362176
\(685\) 0 0
\(686\) 9.27051 0.353950
\(687\) −10.0000 −0.381524
\(688\) −11.5623 −0.440809
\(689\) −16.9656 −0.646336
\(690\) 0 0
\(691\) −26.2918 −1.00019 −0.500094 0.865971i \(-0.666701\pi\)
−0.500094 + 0.865971i \(0.666701\pi\)
\(692\) 24.8885 0.946120
\(693\) 0 0
\(694\) 12.9443 0.491358
\(695\) 0 0
\(696\) 10.0000 0.379049
\(697\) 19.3262 0.732033
\(698\) −6.25735 −0.236844
\(699\) 24.3262 0.920103
\(700\) 0 0
\(701\) −10.6869 −0.403639 −0.201820 0.979423i \(-0.564685\pi\)
−0.201820 + 0.979423i \(0.564685\pi\)
\(702\) 1.09017 0.0411458
\(703\) −1.38197 −0.0521218
\(704\) 0 0
\(705\) 0 0
\(706\) 6.47214 0.243582
\(707\) 9.00000 0.338480
\(708\) 16.7082 0.627933
\(709\) 48.7426 1.83057 0.915284 0.402809i \(-0.131966\pi\)
0.915284 + 0.402809i \(0.131966\pi\)
\(710\) 0 0
\(711\) 9.47214 0.355233
\(712\) −1.18034 −0.0442351
\(713\) −9.90983 −0.371126
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 3.61803 0.135212
\(717\) −2.56231 −0.0956911
\(718\) −7.88854 −0.294398
\(719\) −1.58359 −0.0590580 −0.0295290 0.999564i \(-0.509401\pi\)
−0.0295290 + 0.999564i \(0.509401\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 9.43769 0.351235
\(723\) 23.1246 0.860014
\(724\) 28.2705 1.05067
\(725\) 0 0
\(726\) 0 0
\(727\) −38.8541 −1.44102 −0.720509 0.693445i \(-0.756092\pi\)
−0.720509 + 0.693445i \(0.756092\pi\)
\(728\) −11.8328 −0.438553
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.0902 0.373198
\(732\) 12.7082 0.469709
\(733\) 37.7082 1.39278 0.696392 0.717661i \(-0.254787\pi\)
0.696392 + 0.717661i \(0.254787\pi\)
\(734\) −3.43769 −0.126888
\(735\) 0 0
\(736\) −19.5066 −0.719022
\(737\) 0 0
\(738\) −7.38197 −0.271734
\(739\) −25.0000 −0.919640 −0.459820 0.888012i \(-0.652086\pi\)
−0.459820 + 0.888012i \(0.652086\pi\)
\(740\) 0 0
\(741\) 10.3262 0.379344
\(742\) −17.8328 −0.654663
\(743\) −35.1803 −1.29064 −0.645321 0.763912i \(-0.723276\pi\)
−0.645321 + 0.763912i \(0.723276\pi\)
\(744\) 6.38197 0.233974
\(745\) 0 0
\(746\) 2.72949 0.0999337
\(747\) −0.708204 −0.0259118
\(748\) 0 0
\(749\) −12.7082 −0.464348
\(750\) 0 0
\(751\) 11.9230 0.435076 0.217538 0.976052i \(-0.430197\pi\)
0.217538 + 0.976052i \(0.430197\pi\)
\(752\) −3.00000 −0.109399
\(753\) 7.79837 0.284189
\(754\) −4.87539 −0.177551
\(755\) 0 0
\(756\) −4.85410 −0.176542
\(757\) −1.94427 −0.0706658 −0.0353329 0.999376i \(-0.511249\pi\)
−0.0353329 + 0.999376i \(0.511249\pi\)
\(758\) 0.978714 0.0355485
\(759\) 0 0
\(760\) 0 0
\(761\) −30.8885 −1.11971 −0.559854 0.828591i \(-0.689143\pi\)
−0.559854 + 0.828591i \(0.689143\pi\)
\(762\) −2.29180 −0.0830230
\(763\) 0 0
\(764\) 12.0902 0.437407
\(765\) 0 0
\(766\) −16.6180 −0.600434
\(767\) −18.2148 −0.657698
\(768\) 6.56231 0.236797
\(769\) 12.6869 0.457502 0.228751 0.973485i \(-0.426536\pi\)
0.228751 + 0.973485i \(0.426536\pi\)
\(770\) 0 0
\(771\) −11.6738 −0.420420
\(772\) −30.0344 −1.08096
\(773\) 31.2492 1.12396 0.561978 0.827152i \(-0.310040\pi\)
0.561978 + 0.827152i \(0.310040\pi\)
\(774\) −3.85410 −0.138533
\(775\) 0 0
\(776\) −31.3820 −1.12655
\(777\) −0.708204 −0.0254067
\(778\) 15.0000 0.537776
\(779\) −69.9230 −2.50525
\(780\) 0 0
\(781\) 0 0
\(782\) 3.47214 0.124163
\(783\) −4.47214 −0.159821
\(784\) 3.70820 0.132436
\(785\) 0 0
\(786\) 4.41641 0.157528
\(787\) 10.2918 0.366863 0.183431 0.983033i \(-0.441279\pi\)
0.183431 + 0.983033i \(0.441279\pi\)
\(788\) 39.4508 1.40538
\(789\) −16.3262 −0.581229
\(790\) 0 0
\(791\) 2.12461 0.0755425
\(792\) 0 0
\(793\) −13.8541 −0.491974
\(794\) 23.9230 0.848995
\(795\) 0 0
\(796\) 27.0344 0.958210
\(797\) −10.7639 −0.381278 −0.190639 0.981660i \(-0.561056\pi\)
−0.190639 + 0.981660i \(0.561056\pi\)
\(798\) 10.8541 0.384231
\(799\) 2.61803 0.0926194
\(800\) 0 0
\(801\) 0.527864 0.0186512
\(802\) −16.1246 −0.569380
\(803\) 0 0
\(804\) 15.4721 0.545660
\(805\) 0 0
\(806\) −3.11146 −0.109596
\(807\) −15.5279 −0.546607
\(808\) 6.70820 0.235994
\(809\) 9.67376 0.340111 0.170056 0.985434i \(-0.445605\pi\)
0.170056 + 0.985434i \(0.445605\pi\)
\(810\) 0 0
\(811\) 3.25735 0.114381 0.0571906 0.998363i \(-0.481786\pi\)
0.0571906 + 0.998363i \(0.481786\pi\)
\(812\) 21.7082 0.761809
\(813\) 27.2705 0.956419
\(814\) 0 0
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) −36.5066 −1.27720
\(818\) −6.83282 −0.238904
\(819\) 5.29180 0.184910
\(820\) 0 0
\(821\) 40.4164 1.41054 0.705271 0.708938i \(-0.250825\pi\)
0.705271 + 0.708938i \(0.250825\pi\)
\(822\) 4.61803 0.161072
\(823\) 35.4721 1.23648 0.618240 0.785989i \(-0.287846\pi\)
0.618240 + 0.785989i \(0.287846\pi\)
\(824\) −13.4164 −0.467383
\(825\) 0 0
\(826\) −19.1459 −0.666171
\(827\) −53.1246 −1.84732 −0.923662 0.383208i \(-0.874819\pi\)
−0.923662 + 0.383208i \(0.874819\pi\)
\(828\) 5.61803 0.195240
\(829\) 17.6869 0.614292 0.307146 0.951662i \(-0.400626\pi\)
0.307146 + 0.951662i \(0.400626\pi\)
\(830\) 0 0
\(831\) 30.5623 1.06019
\(832\) 0.416408 0.0144363
\(833\) −3.23607 −0.112123
\(834\) 0.527864 0.0182784
\(835\) 0 0
\(836\) 0 0
\(837\) −2.85410 −0.0986522
\(838\) 10.2016 0.352409
\(839\) −36.7082 −1.26731 −0.633654 0.773617i \(-0.718446\pi\)
−0.633654 + 0.773617i \(0.718446\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −23.0344 −0.793819
\(843\) 0.763932 0.0263112
\(844\) 36.0344 1.24036
\(845\) 0 0
\(846\) −1.00000 −0.0343807
\(847\) 0 0
\(848\) 17.8328 0.612381
\(849\) 0.180340 0.00618925
\(850\) 0 0
\(851\) 0.819660 0.0280976
\(852\) −9.00000 −0.308335
\(853\) 9.94427 0.340485 0.170243 0.985402i \(-0.445545\pi\)
0.170243 + 0.985402i \(0.445545\pi\)
\(854\) −14.5623 −0.498312
\(855\) 0 0
\(856\) −9.47214 −0.323751
\(857\) −47.7214 −1.63013 −0.815065 0.579369i \(-0.803299\pi\)
−0.815065 + 0.579369i \(0.803299\pi\)
\(858\) 0 0
\(859\) 7.11146 0.242640 0.121320 0.992613i \(-0.461287\pi\)
0.121320 + 0.992613i \(0.461287\pi\)
\(860\) 0 0
\(861\) −35.8328 −1.22118
\(862\) −24.4164 −0.831626
\(863\) −11.8885 −0.404691 −0.202345 0.979314i \(-0.564856\pi\)
−0.202345 + 0.979314i \(0.564856\pi\)
\(864\) −5.61803 −0.191129
\(865\) 0 0
\(866\) 3.70820 0.126010
\(867\) 14.3820 0.488437
\(868\) 13.8541 0.470239
\(869\) 0 0
\(870\) 0 0
\(871\) −16.8673 −0.571525
\(872\) 0 0
\(873\) 14.0344 0.474994
\(874\) −12.5623 −0.424926
\(875\) 0 0
\(876\) 5.23607 0.176910
\(877\) −6.41641 −0.216667 −0.108333 0.994115i \(-0.534551\pi\)
−0.108333 + 0.994115i \(0.534551\pi\)
\(878\) 2.03444 0.0686591
\(879\) 0.0557281 0.00187966
\(880\) 0 0
\(881\) 13.9098 0.468634 0.234317 0.972160i \(-0.424715\pi\)
0.234317 + 0.972160i \(0.424715\pi\)
\(882\) 1.23607 0.0416206
\(883\) −10.5836 −0.356166 −0.178083 0.984015i \(-0.556990\pi\)
−0.178083 + 0.984015i \(0.556990\pi\)
\(884\) −4.61803 −0.155321
\(885\) 0 0
\(886\) 25.4164 0.853881
\(887\) −3.00000 −0.100730 −0.0503651 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) −0.527864 −0.0177140
\(889\) −11.1246 −0.373108
\(890\) 0 0
\(891\) 0 0
\(892\) 1.14590 0.0383675
\(893\) −9.47214 −0.316973
\(894\) −9.27051 −0.310052
\(895\) 0 0
\(896\) 34.1459 1.14073
\(897\) −6.12461 −0.204495
\(898\) −15.1246 −0.504715
\(899\) 12.7639 0.425701
\(900\) 0 0
\(901\) −15.5623 −0.518456
\(902\) 0 0
\(903\) −18.7082 −0.622570
\(904\) 1.58359 0.0526695
\(905\) 0 0
\(906\) −1.23607 −0.0410656
\(907\) 42.9787 1.42708 0.713542 0.700612i \(-0.247090\pi\)
0.713542 + 0.700612i \(0.247090\pi\)
\(908\) −40.2705 −1.33642
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 18.0557 0.598213 0.299106 0.954220i \(-0.403311\pi\)
0.299106 + 0.954220i \(0.403311\pi\)
\(912\) −10.8541 −0.359415
\(913\) 0 0
\(914\) −5.06888 −0.167664
\(915\) 0 0
\(916\) −16.1803 −0.534613
\(917\) 21.4377 0.707935
\(918\) 1.00000 0.0330049
\(919\) 46.9574 1.54898 0.774491 0.632585i \(-0.218006\pi\)
0.774491 + 0.632585i \(0.218006\pi\)
\(920\) 0 0
\(921\) 0.562306 0.0185286
\(922\) −13.0344 −0.429266
\(923\) 9.81153 0.322950
\(924\) 0 0
\(925\) 0 0
\(926\) 9.76393 0.320863
\(927\) 6.00000 0.197066
\(928\) 25.1246 0.824756
\(929\) −32.8885 −1.07904 −0.539519 0.841973i \(-0.681394\pi\)
−0.539519 + 0.841973i \(0.681394\pi\)
\(930\) 0 0
\(931\) 11.7082 0.383721
\(932\) 39.3607 1.28930
\(933\) −2.52786 −0.0827586
\(934\) 6.03444 0.197453
\(935\) 0 0
\(936\) 3.94427 0.128923
\(937\) 16.5967 0.542192 0.271096 0.962552i \(-0.412614\pi\)
0.271096 + 0.962552i \(0.412614\pi\)
\(938\) −17.7295 −0.578888
\(939\) 25.8328 0.843022
\(940\) 0 0
\(941\) −44.6312 −1.45494 −0.727468 0.686142i \(-0.759303\pi\)
−0.727468 + 0.686142i \(0.759303\pi\)
\(942\) −2.29180 −0.0746708
\(943\) 41.4721 1.35052
\(944\) 19.1459 0.623146
\(945\) 0 0
\(946\) 0 0
\(947\) −18.3262 −0.595523 −0.297761 0.954640i \(-0.596240\pi\)
−0.297761 + 0.954640i \(0.596240\pi\)
\(948\) 15.3262 0.497773
\(949\) −5.70820 −0.185296
\(950\) 0 0
\(951\) −19.3607 −0.627813
\(952\) −10.8541 −0.351783
\(953\) −37.8197 −1.22510 −0.612549 0.790432i \(-0.709856\pi\)
−0.612549 + 0.790432i \(0.709856\pi\)
\(954\) 5.94427 0.192453
\(955\) 0 0
\(956\) −4.14590 −0.134088
\(957\) 0 0
\(958\) −17.3607 −0.560898
\(959\) 22.4164 0.723864
\(960\) 0 0
\(961\) −22.8541 −0.737229
\(962\) 0.257354 0.00829743
\(963\) 4.23607 0.136505
\(964\) 37.4164 1.20510
\(965\) 0 0
\(966\) −6.43769 −0.207129
\(967\) 34.6869 1.11546 0.557728 0.830024i \(-0.311673\pi\)
0.557728 + 0.830024i \(0.311673\pi\)
\(968\) 0 0
\(969\) 9.47214 0.304289
\(970\) 0 0
\(971\) 37.7771 1.21232 0.606162 0.795341i \(-0.292708\pi\)
0.606162 + 0.795341i \(0.292708\pi\)
\(972\) 1.61803 0.0518985
\(973\) 2.56231 0.0821438
\(974\) −10.3951 −0.333081
\(975\) 0 0
\(976\) 14.5623 0.466128
\(977\) 4.63932 0.148425 0.0742125 0.997242i \(-0.476356\pi\)
0.0742125 + 0.997242i \(0.476356\pi\)
\(978\) 11.2918 0.361072
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 15.5836 0.497292
\(983\) 27.3050 0.870893 0.435446 0.900215i \(-0.356591\pi\)
0.435446 + 0.900215i \(0.356591\pi\)
\(984\) −26.7082 −0.851426
\(985\) 0 0
\(986\) −4.47214 −0.142422
\(987\) −4.85410 −0.154508
\(988\) 16.7082 0.531559
\(989\) 21.6525 0.688509
\(990\) 0 0
\(991\) 21.2705 0.675680 0.337840 0.941204i \(-0.390304\pi\)
0.337840 + 0.941204i \(0.390304\pi\)
\(992\) 16.0344 0.509094
\(993\) −26.5967 −0.844022
\(994\) 10.3131 0.327111
\(995\) 0 0
\(996\) −1.14590 −0.0363092
\(997\) 12.9787 0.411040 0.205520 0.978653i \(-0.434111\pi\)
0.205520 + 0.978653i \(0.434111\pi\)
\(998\) −10.8541 −0.343581
\(999\) 0.236068 0.00746886
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 9075.2.a.x.1.2 2
5.4 even 2 363.2.a.h.1.1 2
11.3 even 5 825.2.n.f.526.1 4
11.4 even 5 825.2.n.f.676.1 4
11.10 odd 2 9075.2.a.bv.1.1 2
15.14 odd 2 1089.2.a.m.1.2 2
20.19 odd 2 5808.2.a.bl.1.1 2
55.3 odd 20 825.2.bx.b.724.1 8
55.4 even 10 33.2.e.a.16.1 4
55.9 even 10 363.2.e.h.202.1 4
55.14 even 10 33.2.e.a.31.1 yes 4
55.19 odd 10 363.2.e.j.130.1 4
55.24 odd 10 363.2.e.c.202.1 4
55.29 odd 10 363.2.e.j.148.1 4
55.37 odd 20 825.2.bx.b.49.1 8
55.39 odd 10 363.2.e.c.124.1 4
55.47 odd 20 825.2.bx.b.724.2 8
55.48 odd 20 825.2.bx.b.49.2 8
55.49 even 10 363.2.e.h.124.1 4
55.54 odd 2 363.2.a.e.1.2 2
165.14 odd 10 99.2.f.b.64.1 4
165.59 odd 10 99.2.f.b.82.1 4
165.164 even 2 1089.2.a.s.1.1 2
220.59 odd 10 528.2.y.f.49.1 4
220.179 odd 10 528.2.y.f.97.1 4
220.219 even 2 5808.2.a.bm.1.1 2
495.4 even 30 891.2.n.d.379.1 8
495.14 odd 30 891.2.n.a.460.1 8
495.59 odd 30 891.2.n.a.379.1 8
495.124 even 30 891.2.n.d.757.1 8
495.169 even 30 891.2.n.d.676.1 8
495.344 odd 30 891.2.n.a.757.1 8
495.389 odd 30 891.2.n.a.676.1 8
495.454 even 30 891.2.n.d.460.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 4 55.4 even 10
33.2.e.a.31.1 yes 4 55.14 even 10
99.2.f.b.64.1 4 165.14 odd 10
99.2.f.b.82.1 4 165.59 odd 10
363.2.a.e.1.2 2 55.54 odd 2
363.2.a.h.1.1 2 5.4 even 2
363.2.e.c.124.1 4 55.39 odd 10
363.2.e.c.202.1 4 55.24 odd 10
363.2.e.h.124.1 4 55.49 even 10
363.2.e.h.202.1 4 55.9 even 10
363.2.e.j.130.1 4 55.19 odd 10
363.2.e.j.148.1 4 55.29 odd 10
528.2.y.f.49.1 4 220.59 odd 10
528.2.y.f.97.1 4 220.179 odd 10
825.2.n.f.526.1 4 11.3 even 5
825.2.n.f.676.1 4 11.4 even 5
825.2.bx.b.49.1 8 55.37 odd 20
825.2.bx.b.49.2 8 55.48 odd 20
825.2.bx.b.724.1 8 55.3 odd 20
825.2.bx.b.724.2 8 55.47 odd 20
891.2.n.a.379.1 8 495.59 odd 30
891.2.n.a.460.1 8 495.14 odd 30
891.2.n.a.676.1 8 495.389 odd 30
891.2.n.a.757.1 8 495.344 odd 30
891.2.n.d.379.1 8 495.4 even 30
891.2.n.d.460.1 8 495.454 even 30
891.2.n.d.676.1 8 495.169 even 30
891.2.n.d.757.1 8 495.124 even 30
1089.2.a.m.1.2 2 15.14 odd 2
1089.2.a.s.1.1 2 165.164 even 2
5808.2.a.bl.1.1 2 20.19 odd 2
5808.2.a.bm.1.1 2 220.219 even 2
9075.2.a.x.1.2 2 1.1 even 1 trivial
9075.2.a.bv.1.1 2 11.10 odd 2