Properties

Label 1305.2.a.l.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} -0.618034 q^{10} -4.23607 q^{11} +1.00000 q^{13} -0.145898 q^{14} +1.85410 q^{16} +1.47214 q^{17} -6.47214 q^{19} -1.61803 q^{20} +2.61803 q^{22} +4.47214 q^{23} +1.00000 q^{25} -0.618034 q^{26} -0.381966 q^{28} +1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} -0.909830 q^{34} +0.236068 q^{35} +4.00000 q^{38} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +6.85410 q^{44} -2.76393 q^{46} -8.23607 q^{47} -6.94427 q^{49} -0.618034 q^{50} -1.61803 q^{52} +6.47214 q^{53} -4.23607 q^{55} +0.527864 q^{56} -0.618034 q^{58} +6.00000 q^{59} +0.472136 q^{61} +4.94427 q^{62} -0.236068 q^{64} +1.00000 q^{65} -14.7082 q^{67} -2.38197 q^{68} -0.145898 q^{70} -2.47214 q^{71} -6.00000 q^{73} +10.4721 q^{76} -1.00000 q^{77} -6.00000 q^{79} +1.85410 q^{80} -3.70820 q^{82} -6.47214 q^{83} +1.47214 q^{85} +3.70820 q^{86} -9.47214 q^{88} -7.94427 q^{89} +0.236068 q^{91} -7.23607 q^{92} +5.09017 q^{94} -6.47214 q^{95} -11.4164 q^{97} +4.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} + 2 q^{13} - 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - q^{20} + 3 q^{22} + 2 q^{25} + q^{26} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32} - 13 q^{34} - 4 q^{35} + 8 q^{38} + 12 q^{41} - 12 q^{43} + 7 q^{44} - 10 q^{46} - 12 q^{47} + 4 q^{49} + q^{50} - q^{52} + 4 q^{53} - 4 q^{55} + 10 q^{56} + q^{58} + 12 q^{59} - 8 q^{61} - 8 q^{62} + 4 q^{64} + 2 q^{65} - 16 q^{67} - 7 q^{68} - 7 q^{70} + 4 q^{71} - 12 q^{73} + 12 q^{76} - 2 q^{77} - 12 q^{79} - 3 q^{80} + 6 q^{82} - 4 q^{83} - 6 q^{85} - 6 q^{86} - 10 q^{88} + 2 q^{89} - 4 q^{91} - 10 q^{92} - q^{94} - 4 q^{95} + 4 q^{97} + 22 q^{98}+O(q^{100}) \) 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 \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.236068 0.0892253 0.0446127 0.999004i \(-0.485795\pi\)
0.0446127 + 0.999004i \(0.485795\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) −0.618034 −0.195440
\(11\) −4.23607 −1.27722 −0.638611 0.769529i \(-0.720491\pi\)
−0.638611 + 0.769529i \(0.720491\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −0.145898 −0.0389929
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 1.47214 0.357045 0.178523 0.983936i \(-0.442868\pi\)
0.178523 + 0.983936i \(0.442868\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) −1.61803 −0.361803
\(21\) 0 0
\(22\) 2.61803 0.558167
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.618034 −0.121206
\(27\) 0 0
\(28\) −0.381966 −0.0721848
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −0.909830 −0.156035
\(35\) 0.236068 0.0399028
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 6.85410 1.03329
\(45\) 0 0
\(46\) −2.76393 −0.407520
\(47\) −8.23607 −1.20135 −0.600677 0.799492i \(-0.705102\pi\)
−0.600677 + 0.799492i \(0.705102\pi\)
\(48\) 0 0
\(49\) −6.94427 −0.992039
\(50\) −0.618034 −0.0874032
\(51\) 0 0
\(52\) −1.61803 −0.224381
\(53\) 6.47214 0.889016 0.444508 0.895775i \(-0.353378\pi\)
0.444508 + 0.895775i \(0.353378\pi\)
\(54\) 0 0
\(55\) −4.23607 −0.571191
\(56\) 0.527864 0.0705388
\(57\) 0 0
\(58\) −0.618034 −0.0811518
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 0.472136 0.0604508 0.0302254 0.999543i \(-0.490377\pi\)
0.0302254 + 0.999543i \(0.490377\pi\)
\(62\) 4.94427 0.627923
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −14.7082 −1.79689 −0.898447 0.439083i \(-0.855303\pi\)
−0.898447 + 0.439083i \(0.855303\pi\)
\(68\) −2.38197 −0.288856
\(69\) 0 0
\(70\) −0.145898 −0.0174382
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 10.4721 1.20124
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 1.85410 0.207295
\(81\) 0 0
\(82\) −3.70820 −0.409503
\(83\) −6.47214 −0.710409 −0.355205 0.934789i \(-0.615589\pi\)
−0.355205 + 0.934789i \(0.615589\pi\)
\(84\) 0 0
\(85\) 1.47214 0.159676
\(86\) 3.70820 0.399866
\(87\) 0 0
\(88\) −9.47214 −1.00973
\(89\) −7.94427 −0.842091 −0.421046 0.907039i \(-0.638337\pi\)
−0.421046 + 0.907039i \(0.638337\pi\)
\(90\) 0 0
\(91\) 0.236068 0.0247466
\(92\) −7.23607 −0.754412
\(93\) 0 0
\(94\) 5.09017 0.525011
\(95\) −6.47214 −0.664027
\(96\) 0 0
\(97\) −11.4164 −1.15916 −0.579580 0.814915i \(-0.696783\pi\)
−0.579580 + 0.814915i \(0.696783\pi\)
\(98\) 4.29180 0.433537
\(99\) 0 0
\(100\) −1.61803 −0.161803
\(101\) −15.4721 −1.53954 −0.769768 0.638324i \(-0.779628\pi\)
−0.769768 + 0.638324i \(0.779628\pi\)
\(102\) 0 0
\(103\) 16.9443 1.66957 0.834784 0.550577i \(-0.185592\pi\)
0.834784 + 0.550577i \(0.185592\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −4.47214 −0.432338 −0.216169 0.976356i \(-0.569356\pi\)
−0.216169 + 0.976356i \(0.569356\pi\)
\(108\) 0 0
\(109\) −8.41641 −0.806146 −0.403073 0.915168i \(-0.632058\pi\)
−0.403073 + 0.915168i \(0.632058\pi\)
\(110\) 2.61803 0.249620
\(111\) 0 0
\(112\) 0.437694 0.0413582
\(113\) −7.47214 −0.702919 −0.351460 0.936203i \(-0.614315\pi\)
−0.351460 + 0.936203i \(0.614315\pi\)
\(114\) 0 0
\(115\) 4.47214 0.417029
\(116\) −1.61803 −0.150231
\(117\) 0 0
\(118\) −3.70820 −0.341368
\(119\) 0.347524 0.0318575
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) −0.291796 −0.0264180
\(123\) 0 0
\(124\) 12.9443 1.16243
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) −0.618034 −0.0542052
\(131\) −0.236068 −0.0206254 −0.0103127 0.999947i \(-0.503283\pi\)
−0.0103127 + 0.999947i \(0.503283\pi\)
\(132\) 0 0
\(133\) −1.52786 −0.132483
\(134\) 9.09017 0.785271
\(135\) 0 0
\(136\) 3.29180 0.282269
\(137\) −10.9443 −0.935032 −0.467516 0.883985i \(-0.654851\pi\)
−0.467516 + 0.883985i \(0.654851\pi\)
\(138\) 0 0
\(139\) 12.2361 1.03785 0.518925 0.854820i \(-0.326332\pi\)
0.518925 + 0.854820i \(0.326332\pi\)
\(140\) −0.381966 −0.0322820
\(141\) 0 0
\(142\) 1.52786 0.128216
\(143\) −4.23607 −0.354238
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 3.70820 0.306893
\(147\) 0 0
\(148\) 0 0
\(149\) 14.4721 1.18560 0.592802 0.805348i \(-0.298022\pi\)
0.592802 + 0.805348i \(0.298022\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −14.4721 −1.17385
\(153\) 0 0
\(154\) 0.618034 0.0498026
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) 3.70820 0.295009
\(159\) 0 0
\(160\) −5.61803 −0.444145
\(161\) 1.05573 0.0832030
\(162\) 0 0
\(163\) −5.05573 −0.395995 −0.197998 0.980203i \(-0.563444\pi\)
−0.197998 + 0.980203i \(0.563444\pi\)
\(164\) −9.70820 −0.758083
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −13.4164 −1.03819 −0.519096 0.854716i \(-0.673731\pi\)
−0.519096 + 0.854716i \(0.673731\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −0.909830 −0.0697808
\(171\) 0 0
\(172\) 9.70820 0.740244
\(173\) 23.8885 1.81621 0.908106 0.418740i \(-0.137528\pi\)
0.908106 + 0.418740i \(0.137528\pi\)
\(174\) 0 0
\(175\) 0.236068 0.0178451
\(176\) −7.85410 −0.592025
\(177\) 0 0
\(178\) 4.90983 0.368007
\(179\) 2.94427 0.220065 0.110033 0.993928i \(-0.464904\pi\)
0.110033 + 0.993928i \(0.464904\pi\)
\(180\) 0 0
\(181\) 4.41641 0.328269 0.164135 0.986438i \(-0.447517\pi\)
0.164135 + 0.986438i \(0.447517\pi\)
\(182\) −0.145898 −0.0108147
\(183\) 0 0
\(184\) 10.0000 0.737210
\(185\) 0 0
\(186\) 0 0
\(187\) −6.23607 −0.456026
\(188\) 13.3262 0.971916
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 8.94427 0.647185 0.323592 0.946197i \(-0.395109\pi\)
0.323592 + 0.946197i \(0.395109\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 7.05573 0.506572
\(195\) 0 0
\(196\) 11.2361 0.802576
\(197\) 7.05573 0.502700 0.251350 0.967896i \(-0.419126\pi\)
0.251350 + 0.967896i \(0.419126\pi\)
\(198\) 0 0
\(199\) 20.1246 1.42660 0.713298 0.700861i \(-0.247201\pi\)
0.713298 + 0.700861i \(0.247201\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 9.56231 0.672801
\(203\) 0.236068 0.0165687
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −10.4721 −0.729628
\(207\) 0 0
\(208\) 1.85410 0.128559
\(209\) 27.4164 1.89643
\(210\) 0 0
\(211\) 16.9443 1.16649 0.583246 0.812296i \(-0.301782\pi\)
0.583246 + 0.812296i \(0.301782\pi\)
\(212\) −10.4721 −0.719229
\(213\) 0 0
\(214\) 2.76393 0.188939
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) −1.88854 −0.128203
\(218\) 5.20163 0.352299
\(219\) 0 0
\(220\) 6.85410 0.462103
\(221\) 1.47214 0.0990266
\(222\) 0 0
\(223\) −9.18034 −0.614761 −0.307381 0.951587i \(-0.599452\pi\)
−0.307381 + 0.951587i \(0.599452\pi\)
\(224\) −1.32624 −0.0886130
\(225\) 0 0
\(226\) 4.61803 0.307187
\(227\) −22.4721 −1.49153 −0.745764 0.666210i \(-0.767915\pi\)
−0.745764 + 0.666210i \(0.767915\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −2.76393 −0.182248
\(231\) 0 0
\(232\) 2.23607 0.146805
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −8.23607 −0.537262
\(236\) −9.70820 −0.631950
\(237\) 0 0
\(238\) −0.214782 −0.0139222
\(239\) 3.05573 0.197659 0.0988293 0.995104i \(-0.468490\pi\)
0.0988293 + 0.995104i \(0.468490\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −4.29180 −0.275887
\(243\) 0 0
\(244\) −0.763932 −0.0489057
\(245\) −6.94427 −0.443653
\(246\) 0 0
\(247\) −6.47214 −0.411812
\(248\) −17.8885 −1.13592
\(249\) 0 0
\(250\) −0.618034 −0.0390879
\(251\) 30.5967 1.93125 0.965625 0.259940i \(-0.0837028\pi\)
0.965625 + 0.259940i \(0.0837028\pi\)
\(252\) 0 0
\(253\) −18.9443 −1.19102
\(254\) 7.41641 0.465347
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −22.4721 −1.40177 −0.700887 0.713273i \(-0.747212\pi\)
−0.700887 + 0.713273i \(0.747212\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.61803 −0.100346
\(261\) 0 0
\(262\) 0.145898 0.00901361
\(263\) 7.05573 0.435075 0.217537 0.976052i \(-0.430198\pi\)
0.217537 + 0.976052i \(0.430198\pi\)
\(264\) 0 0
\(265\) 6.47214 0.397580
\(266\) 0.944272 0.0578970
\(267\) 0 0
\(268\) 23.7984 1.45372
\(269\) 0.527864 0.0321844 0.0160922 0.999871i \(-0.494877\pi\)
0.0160922 + 0.999871i \(0.494877\pi\)
\(270\) 0 0
\(271\) −12.4721 −0.757628 −0.378814 0.925473i \(-0.623668\pi\)
−0.378814 + 0.925473i \(0.623668\pi\)
\(272\) 2.72949 0.165500
\(273\) 0 0
\(274\) 6.76393 0.408624
\(275\) −4.23607 −0.255445
\(276\) 0 0
\(277\) 2.05573 0.123517 0.0617584 0.998091i \(-0.480329\pi\)
0.0617584 + 0.998091i \(0.480329\pi\)
\(278\) −7.56231 −0.453557
\(279\) 0 0
\(280\) 0.527864 0.0315459
\(281\) 13.5279 0.807005 0.403502 0.914979i \(-0.367793\pi\)
0.403502 + 0.914979i \(0.367793\pi\)
\(282\) 0 0
\(283\) −12.9443 −0.769457 −0.384729 0.923030i \(-0.625705\pi\)
−0.384729 + 0.923030i \(0.625705\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 2.61803 0.154808
\(287\) 1.41641 0.0836079
\(288\) 0 0
\(289\) −14.8328 −0.872519
\(290\) −0.618034 −0.0362922
\(291\) 0 0
\(292\) 9.70820 0.568130
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) −8.94427 −0.518128
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) −1.41641 −0.0816404
\(302\) 7.41641 0.426766
\(303\) 0 0
\(304\) −12.0000 −0.688247
\(305\) 0.472136 0.0270344
\(306\) 0 0
\(307\) 26.8328 1.53143 0.765715 0.643180i \(-0.222385\pi\)
0.765715 + 0.643180i \(0.222385\pi\)
\(308\) 1.61803 0.0921960
\(309\) 0 0
\(310\) 4.94427 0.280816
\(311\) 20.1246 1.14116 0.570581 0.821241i \(-0.306718\pi\)
0.570581 + 0.821241i \(0.306718\pi\)
\(312\) 0 0
\(313\) 1.58359 0.0895099 0.0447550 0.998998i \(-0.485749\pi\)
0.0447550 + 0.998998i \(0.485749\pi\)
\(314\) −7.05573 −0.398178
\(315\) 0 0
\(316\) 9.70820 0.546129
\(317\) 14.8885 0.836224 0.418112 0.908395i \(-0.362692\pi\)
0.418112 + 0.908395i \(0.362692\pi\)
\(318\) 0 0
\(319\) −4.23607 −0.237174
\(320\) −0.236068 −0.0131966
\(321\) 0 0
\(322\) −0.652476 −0.0363611
\(323\) −9.52786 −0.530145
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 3.12461 0.173056
\(327\) 0 0
\(328\) 13.4164 0.740797
\(329\) −1.94427 −0.107191
\(330\) 0 0
\(331\) 25.8885 1.42296 0.711482 0.702705i \(-0.248025\pi\)
0.711482 + 0.702705i \(0.248025\pi\)
\(332\) 10.4721 0.574733
\(333\) 0 0
\(334\) 8.29180 0.453707
\(335\) −14.7082 −0.803595
\(336\) 0 0
\(337\) 5.05573 0.275403 0.137702 0.990474i \(-0.456029\pi\)
0.137702 + 0.990474i \(0.456029\pi\)
\(338\) 7.41641 0.403399
\(339\) 0 0
\(340\) −2.38197 −0.129180
\(341\) 33.8885 1.83517
\(342\) 0 0
\(343\) −3.29180 −0.177740
\(344\) −13.4164 −0.723364
\(345\) 0 0
\(346\) −14.7639 −0.793714
\(347\) 5.88854 0.316114 0.158057 0.987430i \(-0.449477\pi\)
0.158057 + 0.987430i \(0.449477\pi\)
\(348\) 0 0
\(349\) −5.05573 −0.270627 −0.135313 0.990803i \(-0.543204\pi\)
−0.135313 + 0.990803i \(0.543204\pi\)
\(350\) −0.145898 −0.00779858
\(351\) 0 0
\(352\) 23.7984 1.26846
\(353\) −33.8885 −1.80371 −0.901853 0.432044i \(-0.857793\pi\)
−0.901853 + 0.432044i \(0.857793\pi\)
\(354\) 0 0
\(355\) −2.47214 −0.131207
\(356\) 12.8541 0.681266
\(357\) 0 0
\(358\) −1.81966 −0.0961720
\(359\) 35.7771 1.88824 0.944121 0.329598i \(-0.106913\pi\)
0.944121 + 0.329598i \(0.106913\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) −2.72949 −0.143459
\(363\) 0 0
\(364\) −0.381966 −0.0200205
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −25.4164 −1.32673 −0.663363 0.748298i \(-0.730871\pi\)
−0.663363 + 0.748298i \(0.730871\pi\)
\(368\) 8.29180 0.432240
\(369\) 0 0
\(370\) 0 0
\(371\) 1.52786 0.0793227
\(372\) 0 0
\(373\) 24.8328 1.28579 0.642897 0.765952i \(-0.277732\pi\)
0.642897 + 0.765952i \(0.277732\pi\)
\(374\) 3.85410 0.199291
\(375\) 0 0
\(376\) −18.4164 −0.949754
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −29.4164 −1.51102 −0.755510 0.655137i \(-0.772611\pi\)
−0.755510 + 0.655137i \(0.772611\pi\)
\(380\) 10.4721 0.537209
\(381\) 0 0
\(382\) −5.52786 −0.282830
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −12.3607 −0.629142
\(387\) 0 0
\(388\) 18.4721 0.937781
\(389\) 10.5279 0.533784 0.266892 0.963726i \(-0.414003\pi\)
0.266892 + 0.963726i \(0.414003\pi\)
\(390\) 0 0
\(391\) 6.58359 0.332947
\(392\) −15.5279 −0.784276
\(393\) 0 0
\(394\) −4.36068 −0.219688
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 27.8885 1.39969 0.699843 0.714297i \(-0.253253\pi\)
0.699843 + 0.714297i \(0.253253\pi\)
\(398\) −12.4377 −0.623445
\(399\) 0 0
\(400\) 1.85410 0.0927051
\(401\) −34.3607 −1.71589 −0.857945 0.513741i \(-0.828259\pi\)
−0.857945 + 0.513741i \(0.828259\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 25.0344 1.24551
\(405\) 0 0
\(406\) −0.145898 −0.00724080
\(407\) 0 0
\(408\) 0 0
\(409\) 0.583592 0.0288568 0.0144284 0.999896i \(-0.495407\pi\)
0.0144284 + 0.999896i \(0.495407\pi\)
\(410\) −3.70820 −0.183135
\(411\) 0 0
\(412\) −27.4164 −1.35071
\(413\) 1.41641 0.0696969
\(414\) 0 0
\(415\) −6.47214 −0.317705
\(416\) −5.61803 −0.275447
\(417\) 0 0
\(418\) −16.9443 −0.828771
\(419\) 10.5836 0.517042 0.258521 0.966006i \(-0.416765\pi\)
0.258521 + 0.966006i \(0.416765\pi\)
\(420\) 0 0
\(421\) 9.41641 0.458928 0.229464 0.973317i \(-0.426303\pi\)
0.229464 + 0.973317i \(0.426303\pi\)
\(422\) −10.4721 −0.509776
\(423\) 0 0
\(424\) 14.4721 0.702829
\(425\) 1.47214 0.0714091
\(426\) 0 0
\(427\) 0.111456 0.00539374
\(428\) 7.23607 0.349769
\(429\) 0 0
\(430\) 3.70820 0.178825
\(431\) −3.52786 −0.169931 −0.0849656 0.996384i \(-0.527078\pi\)
−0.0849656 + 0.996384i \(0.527078\pi\)
\(432\) 0 0
\(433\) 23.3050 1.11996 0.559982 0.828505i \(-0.310808\pi\)
0.559982 + 0.828505i \(0.310808\pi\)
\(434\) 1.16718 0.0560266
\(435\) 0 0
\(436\) 13.6180 0.652186
\(437\) −28.9443 −1.38459
\(438\) 0 0
\(439\) −1.76393 −0.0841879 −0.0420939 0.999114i \(-0.513403\pi\)
−0.0420939 + 0.999114i \(0.513403\pi\)
\(440\) −9.47214 −0.451566
\(441\) 0 0
\(442\) −0.909830 −0.0432762
\(443\) −10.2361 −0.486330 −0.243165 0.969985i \(-0.578186\pi\)
−0.243165 + 0.969985i \(0.578186\pi\)
\(444\) 0 0
\(445\) −7.94427 −0.376595
\(446\) 5.67376 0.268660
\(447\) 0 0
\(448\) −0.0557281 −0.00263290
\(449\) −24.0557 −1.13526 −0.567630 0.823284i \(-0.692140\pi\)
−0.567630 + 0.823284i \(0.692140\pi\)
\(450\) 0 0
\(451\) −25.4164 −1.19681
\(452\) 12.0902 0.568674
\(453\) 0 0
\(454\) 13.8885 0.651822
\(455\) 0.236068 0.0110670
\(456\) 0 0
\(457\) 21.3607 0.999210 0.499605 0.866253i \(-0.333478\pi\)
0.499605 + 0.866253i \(0.333478\pi\)
\(458\) −2.47214 −0.115515
\(459\) 0 0
\(460\) −7.23607 −0.337383
\(461\) −29.7771 −1.38686 −0.693429 0.720525i \(-0.743901\pi\)
−0.693429 + 0.720525i \(0.743901\pi\)
\(462\) 0 0
\(463\) −12.2361 −0.568658 −0.284329 0.958727i \(-0.591771\pi\)
−0.284329 + 0.958727i \(0.591771\pi\)
\(464\) 1.85410 0.0860745
\(465\) 0 0
\(466\) 11.1246 0.515338
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −3.47214 −0.160328
\(470\) 5.09017 0.234792
\(471\) 0 0
\(472\) 13.4164 0.617540
\(473\) 25.4164 1.16865
\(474\) 0 0
\(475\) −6.47214 −0.296962
\(476\) −0.562306 −0.0257732
\(477\) 0 0
\(478\) −1.88854 −0.0863800
\(479\) −1.88854 −0.0862898 −0.0431449 0.999069i \(-0.513738\pi\)
−0.0431449 + 0.999069i \(0.513738\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.09017 0.140753
\(483\) 0 0
\(484\) −11.2361 −0.510730
\(485\) −11.4164 −0.518392
\(486\) 0 0
\(487\) −24.9443 −1.13033 −0.565166 0.824977i \(-0.691188\pi\)
−0.565166 + 0.824977i \(0.691188\pi\)
\(488\) 1.05573 0.0477906
\(489\) 0 0
\(490\) 4.29180 0.193884
\(491\) 38.8328 1.75250 0.876250 0.481856i \(-0.160037\pi\)
0.876250 + 0.481856i \(0.160037\pi\)
\(492\) 0 0
\(493\) 1.47214 0.0663017
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −14.8328 −0.666013
\(497\) −0.583592 −0.0261777
\(498\) 0 0
\(499\) 14.1246 0.632304 0.316152 0.948708i \(-0.397609\pi\)
0.316152 + 0.948708i \(0.397609\pi\)
\(500\) −1.61803 −0.0723607
\(501\) 0 0
\(502\) −18.9098 −0.843987
\(503\) −26.1246 −1.16484 −0.582419 0.812888i \(-0.697894\pi\)
−0.582419 + 0.812888i \(0.697894\pi\)
\(504\) 0 0
\(505\) −15.4721 −0.688501
\(506\) 11.7082 0.520493
\(507\) 0 0
\(508\) 19.4164 0.861464
\(509\) 6.36068 0.281932 0.140966 0.990014i \(-0.454979\pi\)
0.140966 + 0.990014i \(0.454979\pi\)
\(510\) 0 0
\(511\) −1.41641 −0.0626582
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) 13.8885 0.612597
\(515\) 16.9443 0.746654
\(516\) 0 0
\(517\) 34.8885 1.53440
\(518\) 0 0
\(519\) 0 0
\(520\) 2.23607 0.0980581
\(521\) 3.52786 0.154559 0.0772793 0.997009i \(-0.475377\pi\)
0.0772793 + 0.997009i \(0.475377\pi\)
\(522\) 0 0
\(523\) 33.5410 1.46665 0.733323 0.679880i \(-0.237968\pi\)
0.733323 + 0.679880i \(0.237968\pi\)
\(524\) 0.381966 0.0166863
\(525\) 0 0
\(526\) −4.36068 −0.190135
\(527\) −11.7771 −0.513018
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) −4.00000 −0.173749
\(531\) 0 0
\(532\) 2.47214 0.107181
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) −4.47214 −0.193347
\(536\) −32.8885 −1.42057
\(537\) 0 0
\(538\) −0.326238 −0.0140651
\(539\) 29.4164 1.26705
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 7.70820 0.331096
\(543\) 0 0
\(544\) −8.27051 −0.354595
\(545\) −8.41641 −0.360519
\(546\) 0 0
\(547\) −17.2918 −0.739344 −0.369672 0.929162i \(-0.620530\pi\)
−0.369672 + 0.929162i \(0.620530\pi\)
\(548\) 17.7082 0.756457
\(549\) 0 0
\(550\) 2.61803 0.111633
\(551\) −6.47214 −0.275722
\(552\) 0 0
\(553\) −1.41641 −0.0602318
\(554\) −1.27051 −0.0539788
\(555\) 0 0
\(556\) −19.7984 −0.839638
\(557\) −4.58359 −0.194213 −0.0971065 0.995274i \(-0.530959\pi\)
−0.0971065 + 0.995274i \(0.530959\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0.437694 0.0184960
\(561\) 0 0
\(562\) −8.36068 −0.352674
\(563\) −3.18034 −0.134035 −0.0670177 0.997752i \(-0.521348\pi\)
−0.0670177 + 0.997752i \(0.521348\pi\)
\(564\) 0 0
\(565\) −7.47214 −0.314355
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) −5.52786 −0.231944
\(569\) −14.0557 −0.589247 −0.294623 0.955613i \(-0.595194\pi\)
−0.294623 + 0.955613i \(0.595194\pi\)
\(570\) 0 0
\(571\) −22.8328 −0.955524 −0.477762 0.878489i \(-0.658552\pi\)
−0.477762 + 0.878489i \(0.658552\pi\)
\(572\) 6.85410 0.286584
\(573\) 0 0
\(574\) −0.875388 −0.0365380
\(575\) 4.47214 0.186501
\(576\) 0 0
\(577\) −46.9443 −1.95432 −0.977158 0.212515i \(-0.931835\pi\)
−0.977158 + 0.212515i \(0.931835\pi\)
\(578\) 9.16718 0.381305
\(579\) 0 0
\(580\) −1.61803 −0.0671852
\(581\) −1.52786 −0.0633865
\(582\) 0 0
\(583\) −27.4164 −1.13547
\(584\) −13.4164 −0.555175
\(585\) 0 0
\(586\) −5.56231 −0.229777
\(587\) 10.9443 0.451718 0.225859 0.974160i \(-0.427481\pi\)
0.225859 + 0.974160i \(0.427481\pi\)
\(588\) 0 0
\(589\) 51.7771 2.13344
\(590\) −3.70820 −0.152664
\(591\) 0 0
\(592\) 0 0
\(593\) −3.52786 −0.144872 −0.0724360 0.997373i \(-0.523077\pi\)
−0.0724360 + 0.997373i \(0.523077\pi\)
\(594\) 0 0
\(595\) 0.347524 0.0142471
\(596\) −23.4164 −0.959173
\(597\) 0 0
\(598\) −2.76393 −0.113026
\(599\) 29.0689 1.18772 0.593861 0.804568i \(-0.297603\pi\)
0.593861 + 0.804568i \(0.297603\pi\)
\(600\) 0 0
\(601\) −18.8328 −0.768207 −0.384103 0.923290i \(-0.625489\pi\)
−0.384103 + 0.923290i \(0.625489\pi\)
\(602\) 0.875388 0.0356782
\(603\) 0 0
\(604\) 19.4164 0.790042
\(605\) 6.94427 0.282325
\(606\) 0 0
\(607\) 3.41641 0.138668 0.0693339 0.997594i \(-0.477913\pi\)
0.0693339 + 0.997594i \(0.477913\pi\)
\(608\) 36.3607 1.47462
\(609\) 0 0
\(610\) −0.291796 −0.0118145
\(611\) −8.23607 −0.333196
\(612\) 0 0
\(613\) −23.0000 −0.928961 −0.464481 0.885583i \(-0.653759\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(614\) −16.5836 −0.669259
\(615\) 0 0
\(616\) −2.23607 −0.0900937
\(617\) 9.05573 0.364570 0.182285 0.983246i \(-0.441651\pi\)
0.182285 + 0.983246i \(0.441651\pi\)
\(618\) 0 0
\(619\) −27.3050 −1.09748 −0.548739 0.835994i \(-0.684892\pi\)
−0.548739 + 0.835994i \(0.684892\pi\)
\(620\) 12.9443 0.519854
\(621\) 0 0
\(622\) −12.4377 −0.498706
\(623\) −1.87539 −0.0751358
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.978714 −0.0391173
\(627\) 0 0
\(628\) −18.4721 −0.737118
\(629\) 0 0
\(630\) 0 0
\(631\) −7.87539 −0.313514 −0.156757 0.987637i \(-0.550104\pi\)
−0.156757 + 0.987637i \(0.550104\pi\)
\(632\) −13.4164 −0.533676
\(633\) 0 0
\(634\) −9.20163 −0.365443
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) −6.94427 −0.275142
\(638\) 2.61803 0.103649
\(639\) 0 0
\(640\) 11.3820 0.449912
\(641\) 13.9443 0.550766 0.275383 0.961335i \(-0.411195\pi\)
0.275383 + 0.961335i \(0.411195\pi\)
\(642\) 0 0
\(643\) −21.5410 −0.849495 −0.424747 0.905312i \(-0.639637\pi\)
−0.424747 + 0.905312i \(0.639637\pi\)
\(644\) −1.70820 −0.0673127
\(645\) 0 0
\(646\) 5.88854 0.231682
\(647\) 29.0557 1.14230 0.571149 0.820846i \(-0.306498\pi\)
0.571149 + 0.820846i \(0.306498\pi\)
\(648\) 0 0
\(649\) −25.4164 −0.997681
\(650\) −0.618034 −0.0242413
\(651\) 0 0
\(652\) 8.18034 0.320367
\(653\) 18.8885 0.739166 0.369583 0.929198i \(-0.379501\pi\)
0.369583 + 0.929198i \(0.379501\pi\)
\(654\) 0 0
\(655\) −0.236068 −0.00922394
\(656\) 11.1246 0.434343
\(657\) 0 0
\(658\) 1.20163 0.0468443
\(659\) −48.4853 −1.88872 −0.944359 0.328915i \(-0.893317\pi\)
−0.944359 + 0.328915i \(0.893317\pi\)
\(660\) 0 0
\(661\) −26.4164 −1.02748 −0.513740 0.857946i \(-0.671740\pi\)
−0.513740 + 0.857946i \(0.671740\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) −14.4721 −0.561628
\(665\) −1.52786 −0.0592480
\(666\) 0 0
\(667\) 4.47214 0.173162
\(668\) 21.7082 0.839916
\(669\) 0 0
\(670\) 9.09017 0.351184
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 1.58359 0.0610430 0.0305215 0.999534i \(-0.490283\pi\)
0.0305215 + 0.999534i \(0.490283\pi\)
\(674\) −3.12461 −0.120356
\(675\) 0 0
\(676\) 19.4164 0.746785
\(677\) −0.0557281 −0.00214180 −0.00107090 0.999999i \(-0.500341\pi\)
−0.00107090 + 0.999999i \(0.500341\pi\)
\(678\) 0 0
\(679\) −2.69505 −0.103426
\(680\) 3.29180 0.126235
\(681\) 0 0
\(682\) −20.9443 −0.801998
\(683\) −36.9443 −1.41363 −0.706817 0.707397i \(-0.749869\pi\)
−0.706817 + 0.707397i \(0.749869\pi\)
\(684\) 0 0
\(685\) −10.9443 −0.418159
\(686\) 2.03444 0.0776754
\(687\) 0 0
\(688\) −11.1246 −0.424122
\(689\) 6.47214 0.246569
\(690\) 0 0
\(691\) −15.2918 −0.581727 −0.290864 0.956765i \(-0.593943\pi\)
−0.290864 + 0.956765i \(0.593943\pi\)
\(692\) −38.6525 −1.46935
\(693\) 0 0
\(694\) −3.63932 −0.138147
\(695\) 12.2361 0.464141
\(696\) 0 0
\(697\) 8.83282 0.334567
\(698\) 3.12461 0.118268
\(699\) 0 0
\(700\) −0.381966 −0.0144370
\(701\) 25.5279 0.964174 0.482087 0.876123i \(-0.339879\pi\)
0.482087 + 0.876123i \(0.339879\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 20.9443 0.788248
\(707\) −3.65248 −0.137365
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 1.52786 0.0573397
\(711\) 0 0
\(712\) −17.7639 −0.665731
\(713\) −35.7771 −1.33986
\(714\) 0 0
\(715\) −4.23607 −0.158420
\(716\) −4.76393 −0.178036
\(717\) 0 0
\(718\) −22.1115 −0.825192
\(719\) 11.8885 0.443368 0.221684 0.975119i \(-0.428845\pi\)
0.221684 + 0.975119i \(0.428845\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −14.1459 −0.526456
\(723\) 0 0
\(724\) −7.14590 −0.265575
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 32.2492 1.19606 0.598029 0.801475i \(-0.295951\pi\)
0.598029 + 0.801475i \(0.295951\pi\)
\(728\) 0.527864 0.0195639
\(729\) 0 0
\(730\) 3.70820 0.137247
\(731\) −8.83282 −0.326693
\(732\) 0 0
\(733\) −12.4721 −0.460669 −0.230334 0.973112i \(-0.573982\pi\)
−0.230334 + 0.973112i \(0.573982\pi\)
\(734\) 15.7082 0.579800
\(735\) 0 0
\(736\) −25.1246 −0.926105
\(737\) 62.3050 2.29503
\(738\) 0 0
\(739\) 16.5836 0.610037 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.944272 −0.0346653
\(743\) −44.2361 −1.62286 −0.811432 0.584447i \(-0.801312\pi\)
−0.811432 + 0.584447i \(0.801312\pi\)
\(744\) 0 0
\(745\) 14.4721 0.530218
\(746\) −15.3475 −0.561913
\(747\) 0 0
\(748\) 10.0902 0.368933
\(749\) −1.05573 −0.0385755
\(750\) 0 0
\(751\) −45.8885 −1.67450 −0.837248 0.546823i \(-0.815837\pi\)
−0.837248 + 0.546823i \(0.815837\pi\)
\(752\) −15.2705 −0.556858
\(753\) 0 0
\(754\) −0.618034 −0.0225075
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 24.8328 0.902564 0.451282 0.892381i \(-0.350967\pi\)
0.451282 + 0.892381i \(0.350967\pi\)
\(758\) 18.1803 0.660340
\(759\) 0 0
\(760\) −14.4721 −0.524960
\(761\) 35.5279 1.28788 0.643942 0.765074i \(-0.277298\pi\)
0.643942 + 0.765074i \(0.277298\pi\)
\(762\) 0 0
\(763\) −1.98684 −0.0719286
\(764\) −14.4721 −0.523584
\(765\) 0 0
\(766\) −11.1246 −0.401949
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 45.3050 1.63374 0.816869 0.576823i \(-0.195708\pi\)
0.816869 + 0.576823i \(0.195708\pi\)
\(770\) 0.618034 0.0222724
\(771\) 0 0
\(772\) −32.3607 −1.16469
\(773\) −32.8328 −1.18091 −0.590457 0.807069i \(-0.701053\pi\)
−0.590457 + 0.807069i \(0.701053\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −25.5279 −0.916397
\(777\) 0 0
\(778\) −6.50658 −0.233272
\(779\) −38.8328 −1.39133
\(780\) 0 0
\(781\) 10.4721 0.374722
\(782\) −4.06888 −0.145503
\(783\) 0 0
\(784\) −12.8754 −0.459835
\(785\) 11.4164 0.407469
\(786\) 0 0
\(787\) −43.7771 −1.56048 −0.780242 0.625477i \(-0.784904\pi\)
−0.780242 + 0.625477i \(0.784904\pi\)
\(788\) −11.4164 −0.406693
\(789\) 0 0
\(790\) 3.70820 0.131932
\(791\) −1.76393 −0.0627182
\(792\) 0 0
\(793\) 0.472136 0.0167660
\(794\) −17.2361 −0.611685
\(795\) 0 0
\(796\) −32.5623 −1.15414
\(797\) −46.7214 −1.65496 −0.827478 0.561499i \(-0.810225\pi\)
−0.827478 + 0.561499i \(0.810225\pi\)
\(798\) 0 0
\(799\) −12.1246 −0.428938
\(800\) −5.61803 −0.198627
\(801\) 0 0
\(802\) 21.2361 0.749872
\(803\) 25.4164 0.896926
\(804\) 0 0
\(805\) 1.05573 0.0372095
\(806\) 4.94427 0.174155
\(807\) 0 0
\(808\) −34.5967 −1.21711
\(809\) 2.05573 0.0722756 0.0361378 0.999347i \(-0.488494\pi\)
0.0361378 + 0.999347i \(0.488494\pi\)
\(810\) 0 0
\(811\) 2.81966 0.0990117 0.0495058 0.998774i \(-0.484235\pi\)
0.0495058 + 0.998774i \(0.484235\pi\)
\(812\) −0.381966 −0.0134044
\(813\) 0 0
\(814\) 0 0
\(815\) −5.05573 −0.177094
\(816\) 0 0
\(817\) 38.8328 1.35859
\(818\) −0.360680 −0.0126109
\(819\) 0 0
\(820\) −9.70820 −0.339025
\(821\) −48.9443 −1.70817 −0.854083 0.520136i \(-0.825881\pi\)
−0.854083 + 0.520136i \(0.825881\pi\)
\(822\) 0 0
\(823\) −2.00000 −0.0697156 −0.0348578 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) 37.8885 1.31991
\(825\) 0 0
\(826\) −0.875388 −0.0304587
\(827\) 41.8885 1.45661 0.728304 0.685254i \(-0.240309\pi\)
0.728304 + 0.685254i \(0.240309\pi\)
\(828\) 0 0
\(829\) −19.8885 −0.690758 −0.345379 0.938463i \(-0.612250\pi\)
−0.345379 + 0.938463i \(0.612250\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −0.236068 −0.00818418
\(833\) −10.2229 −0.354203
\(834\) 0 0
\(835\) −13.4164 −0.464294
\(836\) −44.3607 −1.53425
\(837\) 0 0
\(838\) −6.54102 −0.225956
\(839\) −18.7082 −0.645879 −0.322939 0.946420i \(-0.604671\pi\)
−0.322939 + 0.946420i \(0.604671\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −5.81966 −0.200559
\(843\) 0 0
\(844\) −27.4164 −0.943712
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 1.63932 0.0563277
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −0.909830 −0.0312069
\(851\) 0 0
\(852\) 0 0
\(853\) −11.3050 −0.387074 −0.193537 0.981093i \(-0.561996\pi\)
−0.193537 + 0.981093i \(0.561996\pi\)
\(854\) −0.0688837 −0.00235715
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 40.7214 1.39102 0.695508 0.718519i \(-0.255180\pi\)
0.695508 + 0.718519i \(0.255180\pi\)
\(858\) 0 0
\(859\) −16.8328 −0.574328 −0.287164 0.957881i \(-0.592713\pi\)
−0.287164 + 0.957881i \(0.592713\pi\)
\(860\) 9.70820 0.331047
\(861\) 0 0
\(862\) 2.18034 0.0742627
\(863\) 54.4721 1.85425 0.927127 0.374748i \(-0.122271\pi\)
0.927127 + 0.374748i \(0.122271\pi\)
\(864\) 0 0
\(865\) 23.8885 0.812235
\(866\) −14.4033 −0.489442
\(867\) 0 0
\(868\) 3.05573 0.103718
\(869\) 25.4164 0.862193
\(870\) 0 0
\(871\) −14.7082 −0.498368
\(872\) −18.8197 −0.637314
\(873\) 0 0
\(874\) 17.8885 0.605089
\(875\) 0.236068 0.00798055
\(876\) 0 0
\(877\) 15.8885 0.536518 0.268259 0.963347i \(-0.413552\pi\)
0.268259 + 0.963347i \(0.413552\pi\)
\(878\) 1.09017 0.0367915
\(879\) 0 0
\(880\) −7.85410 −0.264762
\(881\) 50.8885 1.71448 0.857239 0.514918i \(-0.172178\pi\)
0.857239 + 0.514918i \(0.172178\pi\)
\(882\) 0 0
\(883\) 38.8328 1.30683 0.653414 0.757001i \(-0.273336\pi\)
0.653414 + 0.757001i \(0.273336\pi\)
\(884\) −2.38197 −0.0801142
\(885\) 0 0
\(886\) 6.32624 0.212534
\(887\) 17.1803 0.576859 0.288430 0.957501i \(-0.406867\pi\)
0.288430 + 0.957501i \(0.406867\pi\)
\(888\) 0 0
\(889\) −2.83282 −0.0950096
\(890\) 4.90983 0.164578
\(891\) 0 0
\(892\) 14.8541 0.497352
\(893\) 53.3050 1.78378
\(894\) 0 0
\(895\) 2.94427 0.0984162
\(896\) 2.68692 0.0897636
\(897\) 0 0
\(898\) 14.8673 0.496127
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 9.52786 0.317419
\(902\) 15.7082 0.523026
\(903\) 0 0
\(904\) −16.7082 −0.555707
\(905\) 4.41641 0.146806
\(906\) 0 0
\(907\) 28.8328 0.957378 0.478689 0.877985i \(-0.341112\pi\)
0.478689 + 0.877985i \(0.341112\pi\)
\(908\) 36.3607 1.20667
\(909\) 0 0
\(910\) −0.145898 −0.00483647
\(911\) 26.2361 0.869240 0.434620 0.900614i \(-0.356883\pi\)
0.434620 + 0.900614i \(0.356883\pi\)
\(912\) 0 0
\(913\) 27.4164 0.907351
\(914\) −13.2016 −0.436671
\(915\) 0 0
\(916\) −6.47214 −0.213845
\(917\) −0.0557281 −0.00184030
\(918\) 0 0
\(919\) −45.5410 −1.50226 −0.751130 0.660155i \(-0.770491\pi\)
−0.751130 + 0.660155i \(0.770491\pi\)
\(920\) 10.0000 0.329690
\(921\) 0 0
\(922\) 18.4033 0.606079
\(923\) −2.47214 −0.0813713
\(924\) 0 0
\(925\) 0 0
\(926\) 7.56231 0.248513
\(927\) 0 0
\(928\) −5.61803 −0.184421
\(929\) −26.9443 −0.884013 −0.442006 0.897012i \(-0.645733\pi\)
−0.442006 + 0.897012i \(0.645733\pi\)
\(930\) 0 0
\(931\) 44.9443 1.47299
\(932\) 29.1246 0.954008
\(933\) 0 0
\(934\) 14.8328 0.485345
\(935\) −6.23607 −0.203941
\(936\) 0 0
\(937\) 36.5279 1.19331 0.596657 0.802497i \(-0.296495\pi\)
0.596657 + 0.802497i \(0.296495\pi\)
\(938\) 2.14590 0.0700661
\(939\) 0 0
\(940\) 13.3262 0.434654
\(941\) 13.4164 0.437362 0.218681 0.975796i \(-0.429825\pi\)
0.218681 + 0.975796i \(0.429825\pi\)
\(942\) 0 0
\(943\) 26.8328 0.873797
\(944\) 11.1246 0.362075
\(945\) 0 0
\(946\) −15.7082 −0.510718
\(947\) −9.54102 −0.310041 −0.155021 0.987911i \(-0.549544\pi\)
−0.155021 + 0.987911i \(0.549544\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0.777088 0.0251856
\(953\) −46.3607 −1.50177 −0.750885 0.660433i \(-0.770373\pi\)
−0.750885 + 0.660433i \(0.770373\pi\)
\(954\) 0 0
\(955\) 8.94427 0.289430
\(956\) −4.94427 −0.159909
\(957\) 0 0
\(958\) 1.16718 0.0377100
\(959\) −2.58359 −0.0834285
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 8.09017 0.260567
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −15.4164 −0.495758 −0.247879 0.968791i \(-0.579734\pi\)
−0.247879 + 0.968791i \(0.579734\pi\)
\(968\) 15.5279 0.499084
\(969\) 0 0
\(970\) 7.05573 0.226546
\(971\) −45.8885 −1.47263 −0.736317 0.676637i \(-0.763437\pi\)
−0.736317 + 0.676637i \(0.763437\pi\)
\(972\) 0 0
\(973\) 2.88854 0.0926025
\(974\) 15.4164 0.493974
\(975\) 0 0
\(976\) 0.875388 0.0280205
\(977\) 31.0557 0.993561 0.496780 0.867876i \(-0.334515\pi\)
0.496780 + 0.867876i \(0.334515\pi\)
\(978\) 0 0
\(979\) 33.6525 1.07554
\(980\) 11.2361 0.358923
\(981\) 0 0
\(982\) −24.0000 −0.765871
\(983\) 16.9443 0.540438 0.270219 0.962799i \(-0.412904\pi\)
0.270219 + 0.962799i \(0.412904\pi\)
\(984\) 0 0
\(985\) 7.05573 0.224814
\(986\) −0.909830 −0.0289749
\(987\) 0 0
\(988\) 10.4721 0.333163
\(989\) −26.8328 −0.853234
\(990\) 0 0
\(991\) 10.7082 0.340157 0.170079 0.985430i \(-0.445598\pi\)
0.170079 + 0.985430i \(0.445598\pi\)
\(992\) 44.9443 1.42698
\(993\) 0 0
\(994\) 0.360680 0.0114401
\(995\) 20.1246 0.637993
\(996\) 0 0
\(997\) −4.94427 −0.156587 −0.0782933 0.996930i \(-0.524947\pi\)
−0.0782933 + 0.996930i \(0.524947\pi\)
\(998\) −8.72949 −0.276327
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1305.2.a.l.1.1 yes 2
3.2 odd 2 1305.2.a.h.1.2 2
5.4 even 2 6525.2.a.r.1.2 2
15.14 odd 2 6525.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.h.1.2 2 3.2 odd 2
1305.2.a.l.1.1 yes 2 1.1 even 1 trivial
6525.2.a.r.1.2 2 5.4 even 2
6525.2.a.bb.1.1 2 15.14 odd 2