Properties

Label 475.2.a.h.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $3$
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] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.246980 q^{2} -0.801938 q^{3} -1.93900 q^{4} +0.198062 q^{6} +1.69202 q^{7} +0.972853 q^{8} -2.35690 q^{9} +O(q^{10})\) \(q-0.246980 q^{2} -0.801938 q^{3} -1.93900 q^{4} +0.198062 q^{6} +1.69202 q^{7} +0.972853 q^{8} -2.35690 q^{9} -0.911854 q^{11} +1.55496 q^{12} +1.55496 q^{13} -0.417895 q^{14} +3.63773 q^{16} +5.29590 q^{17} +0.582105 q^{18} -1.00000 q^{19} -1.35690 q^{21} +0.225209 q^{22} +4.24698 q^{23} -0.780167 q^{24} -0.384043 q^{26} +4.29590 q^{27} -3.28083 q^{28} +5.00969 q^{29} +1.82908 q^{31} -2.84415 q^{32} +0.731250 q^{33} -1.30798 q^{34} +4.57002 q^{36} +6.29590 q^{37} +0.246980 q^{38} -1.24698 q^{39} +4.18060 q^{41} +0.335126 q^{42} +7.31767 q^{43} +1.76809 q^{44} -1.04892 q^{46} -2.04892 q^{47} -2.91723 q^{48} -4.13706 q^{49} -4.24698 q^{51} -3.01507 q^{52} -2.70171 q^{53} -1.06100 q^{54} +1.64609 q^{56} +0.801938 q^{57} -1.23729 q^{58} +9.87800 q^{59} +0.542877 q^{61} -0.451747 q^{62} -3.98792 q^{63} -6.57301 q^{64} -0.180604 q^{66} -13.9976 q^{67} -10.2687 q^{68} -3.40581 q^{69} -12.8780 q^{71} -2.29291 q^{72} -2.80731 q^{73} -1.55496 q^{74} +1.93900 q^{76} -1.54288 q^{77} +0.307979 q^{78} +1.59419 q^{79} +3.62565 q^{81} -1.03252 q^{82} +12.2349 q^{83} +2.63102 q^{84} -1.80731 q^{86} -4.01746 q^{87} -0.887100 q^{88} +2.91723 q^{89} +2.63102 q^{91} -8.23490 q^{92} -1.46681 q^{93} +0.506041 q^{94} +2.28083 q^{96} +1.55496 q^{97} +1.02177 q^{98} +2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9} + q^{11} + 5 q^{12} + 5 q^{13} - 7 q^{14} + 18 q^{16} + 2 q^{17} - 4 q^{18} - 3 q^{19} - q^{22} + 8 q^{23} - q^{24} + 9 q^{26} - q^{27} - 21 q^{28} - 7 q^{29} - 5 q^{31} + 27 q^{32} + 10 q^{33} - 9 q^{34} - 11 q^{36} + 5 q^{37} - 4 q^{38} + q^{39} + q^{41} + 5 q^{43} - 15 q^{44} + 6 q^{46} + 3 q^{47} - 2 q^{48} - 7 q^{49} - 8 q^{51} + 16 q^{52} + 19 q^{53} - 13 q^{54} - 35 q^{56} - 2 q^{57} - 21 q^{58} + 10 q^{59} - 17 q^{61} - 23 q^{62} + 7 q^{63} + 49 q^{64} + 11 q^{66} - q^{67} - 23 q^{68} + 3 q^{69} - 19 q^{71} - 37 q^{72} - q^{73} - 5 q^{74} - 4 q^{76} + 14 q^{77} + 6 q^{78} + 18 q^{79} - q^{81} + 6 q^{82} + 13 q^{83} - 7 q^{84} + 2 q^{86} - 28 q^{87} - 46 q^{88} + 2 q^{89} - 7 q^{91} - q^{92} - q^{93} + 11 q^{94} + 18 q^{96} + 5 q^{97} + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.246980 −0.174641 −0.0873205 0.996180i \(-0.527830\pi\)
−0.0873205 + 0.996180i \(0.527830\pi\)
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) −1.93900 −0.969501
\(5\) 0 0
\(6\) 0.198062 0.0808586
\(7\) 1.69202 0.639524 0.319762 0.947498i \(-0.396397\pi\)
0.319762 + 0.947498i \(0.396397\pi\)
\(8\) 0.972853 0.343955
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) −0.911854 −0.274934 −0.137467 0.990506i \(-0.543896\pi\)
−0.137467 + 0.990506i \(0.543896\pi\)
\(12\) 1.55496 0.448878
\(13\) 1.55496 0.431268 0.215634 0.976474i \(-0.430818\pi\)
0.215634 + 0.976474i \(0.430818\pi\)
\(14\) −0.417895 −0.111687
\(15\) 0 0
\(16\) 3.63773 0.909432
\(17\) 5.29590 1.28444 0.642222 0.766519i \(-0.278013\pi\)
0.642222 + 0.766519i \(0.278013\pi\)
\(18\) 0.582105 0.137204
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.35690 −0.296099
\(22\) 0.225209 0.0480148
\(23\) 4.24698 0.885556 0.442778 0.896631i \(-0.353993\pi\)
0.442778 + 0.896631i \(0.353993\pi\)
\(24\) −0.780167 −0.159251
\(25\) 0 0
\(26\) −0.384043 −0.0753170
\(27\) 4.29590 0.826746
\(28\) −3.28083 −0.620019
\(29\) 5.00969 0.930276 0.465138 0.885238i \(-0.346005\pi\)
0.465138 + 0.885238i \(0.346005\pi\)
\(30\) 0 0
\(31\) 1.82908 0.328513 0.164257 0.986418i \(-0.447477\pi\)
0.164257 + 0.986418i \(0.447477\pi\)
\(32\) −2.84415 −0.502779
\(33\) 0.731250 0.127294
\(34\) −1.30798 −0.224316
\(35\) 0 0
\(36\) 4.57002 0.761671
\(37\) 6.29590 1.03504 0.517520 0.855671i \(-0.326855\pi\)
0.517520 + 0.855671i \(0.326855\pi\)
\(38\) 0.246980 0.0400654
\(39\) −1.24698 −0.199677
\(40\) 0 0
\(41\) 4.18060 0.652901 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(42\) 0.335126 0.0517110
\(43\) 7.31767 1.11593 0.557967 0.829863i \(-0.311582\pi\)
0.557967 + 0.829863i \(0.311582\pi\)
\(44\) 1.76809 0.266549
\(45\) 0 0
\(46\) −1.04892 −0.154654
\(47\) −2.04892 −0.298865 −0.149433 0.988772i \(-0.547745\pi\)
−0.149433 + 0.988772i \(0.547745\pi\)
\(48\) −2.91723 −0.421066
\(49\) −4.13706 −0.591009
\(50\) 0 0
\(51\) −4.24698 −0.594696
\(52\) −3.01507 −0.418114
\(53\) −2.70171 −0.371108 −0.185554 0.982634i \(-0.559408\pi\)
−0.185554 + 0.982634i \(0.559408\pi\)
\(54\) −1.06100 −0.144384
\(55\) 0 0
\(56\) 1.64609 0.219968
\(57\) 0.801938 0.106219
\(58\) −1.23729 −0.162464
\(59\) 9.87800 1.28601 0.643003 0.765864i \(-0.277688\pi\)
0.643003 + 0.765864i \(0.277688\pi\)
\(60\) 0 0
\(61\) 0.542877 0.0695082 0.0347541 0.999396i \(-0.488935\pi\)
0.0347541 + 0.999396i \(0.488935\pi\)
\(62\) −0.451747 −0.0573719
\(63\) −3.98792 −0.502430
\(64\) −6.57301 −0.821626
\(65\) 0 0
\(66\) −0.180604 −0.0222308
\(67\) −13.9976 −1.71008 −0.855040 0.518562i \(-0.826467\pi\)
−0.855040 + 0.518562i \(0.826467\pi\)
\(68\) −10.2687 −1.24527
\(69\) −3.40581 −0.410012
\(70\) 0 0
\(71\) −12.8780 −1.52834 −0.764169 0.645016i \(-0.776851\pi\)
−0.764169 + 0.645016i \(0.776851\pi\)
\(72\) −2.29291 −0.270222
\(73\) −2.80731 −0.328571 −0.164286 0.986413i \(-0.552532\pi\)
−0.164286 + 0.986413i \(0.552532\pi\)
\(74\) −1.55496 −0.180760
\(75\) 0 0
\(76\) 1.93900 0.222419
\(77\) −1.54288 −0.175827
\(78\) 0.307979 0.0348717
\(79\) 1.59419 0.179360 0.0896800 0.995971i \(-0.471416\pi\)
0.0896800 + 0.995971i \(0.471416\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) −1.03252 −0.114023
\(83\) 12.2349 1.34295 0.671477 0.741025i \(-0.265660\pi\)
0.671477 + 0.741025i \(0.265660\pi\)
\(84\) 2.63102 0.287068
\(85\) 0 0
\(86\) −1.80731 −0.194888
\(87\) −4.01746 −0.430717
\(88\) −0.887100 −0.0945652
\(89\) 2.91723 0.309226 0.154613 0.987975i \(-0.450587\pi\)
0.154613 + 0.987975i \(0.450587\pi\)
\(90\) 0 0
\(91\) 2.63102 0.275806
\(92\) −8.23490 −0.858547
\(93\) −1.46681 −0.152101
\(94\) 0.506041 0.0521941
\(95\) 0 0
\(96\) 2.28083 0.232786
\(97\) 1.55496 0.157882 0.0789410 0.996879i \(-0.474846\pi\)
0.0789410 + 0.996879i \(0.474846\pi\)
\(98\) 1.02177 0.103214
\(99\) 2.14914 0.215997
\(100\) 0 0
\(101\) −16.6015 −1.65191 −0.825955 0.563737i \(-0.809363\pi\)
−0.825955 + 0.563737i \(0.809363\pi\)
\(102\) 1.04892 0.103858
\(103\) −4.84548 −0.477439 −0.238720 0.971089i \(-0.576728\pi\)
−0.238720 + 0.971089i \(0.576728\pi\)
\(104\) 1.51275 0.148337
\(105\) 0 0
\(106\) 0.667267 0.0648107
\(107\) −4.46681 −0.431823 −0.215912 0.976413i \(-0.569272\pi\)
−0.215912 + 0.976413i \(0.569272\pi\)
\(108\) −8.32975 −0.801530
\(109\) 18.8267 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(110\) 0 0
\(111\) −5.04892 −0.479222
\(112\) 6.15511 0.581603
\(113\) 20.0368 1.88491 0.942453 0.334337i \(-0.108512\pi\)
0.942453 + 0.334337i \(0.108512\pi\)
\(114\) −0.198062 −0.0185502
\(115\) 0 0
\(116\) −9.71379 −0.901903
\(117\) −3.66487 −0.338818
\(118\) −2.43967 −0.224589
\(119\) 8.96077 0.821433
\(120\) 0 0
\(121\) −10.1685 −0.924411
\(122\) −0.134079 −0.0121390
\(123\) −3.35258 −0.302292
\(124\) −3.54660 −0.318494
\(125\) 0 0
\(126\) 0.984935 0.0877449
\(127\) −17.8702 −1.58573 −0.792863 0.609399i \(-0.791411\pi\)
−0.792863 + 0.609399i \(0.791411\pi\)
\(128\) 7.31170 0.646269
\(129\) −5.86831 −0.516676
\(130\) 0 0
\(131\) 7.44265 0.650267 0.325134 0.945668i \(-0.394591\pi\)
0.325134 + 0.945668i \(0.394591\pi\)
\(132\) −1.41789 −0.123412
\(133\) −1.69202 −0.146717
\(134\) 3.45712 0.298650
\(135\) 0 0
\(136\) 5.15213 0.441791
\(137\) 5.68664 0.485843 0.242921 0.970046i \(-0.421894\pi\)
0.242921 + 0.970046i \(0.421894\pi\)
\(138\) 0.841166 0.0716048
\(139\) 3.61596 0.306701 0.153351 0.988172i \(-0.450994\pi\)
0.153351 + 0.988172i \(0.450994\pi\)
\(140\) 0 0
\(141\) 1.64310 0.138374
\(142\) 3.18060 0.266910
\(143\) −1.41789 −0.118570
\(144\) −8.57374 −0.714479
\(145\) 0 0
\(146\) 0.693349 0.0573820
\(147\) 3.31767 0.273637
\(148\) −12.2078 −1.00347
\(149\) 3.29052 0.269570 0.134785 0.990875i \(-0.456966\pi\)
0.134785 + 0.990875i \(0.456966\pi\)
\(150\) 0 0
\(151\) −10.2131 −0.831133 −0.415566 0.909563i \(-0.636417\pi\)
−0.415566 + 0.909563i \(0.636417\pi\)
\(152\) −0.972853 −0.0789088
\(153\) −12.4819 −1.00910
\(154\) 0.381059 0.0307066
\(155\) 0 0
\(156\) 2.41789 0.193587
\(157\) 14.3448 1.14484 0.572420 0.819960i \(-0.306005\pi\)
0.572420 + 0.819960i \(0.306005\pi\)
\(158\) −0.393732 −0.0313236
\(159\) 2.16660 0.171823
\(160\) 0 0
\(161\) 7.18598 0.566335
\(162\) −0.895461 −0.0703540
\(163\) −19.5308 −1.52977 −0.764885 0.644167i \(-0.777204\pi\)
−0.764885 + 0.644167i \(0.777204\pi\)
\(164\) −8.10620 −0.632988
\(165\) 0 0
\(166\) −3.02177 −0.234535
\(167\) 11.8823 0.919481 0.459741 0.888053i \(-0.347942\pi\)
0.459741 + 0.888053i \(0.347942\pi\)
\(168\) −1.32006 −0.101845
\(169\) −10.5821 −0.814008
\(170\) 0 0
\(171\) 2.35690 0.180236
\(172\) −14.1890 −1.08190
\(173\) 15.4722 1.17633 0.588164 0.808741i \(-0.299851\pi\)
0.588164 + 0.808741i \(0.299851\pi\)
\(174\) 0.992230 0.0752208
\(175\) 0 0
\(176\) −3.31708 −0.250034
\(177\) −7.92154 −0.595420
\(178\) −0.720497 −0.0540035
\(179\) 2.16421 0.161761 0.0808803 0.996724i \(-0.474227\pi\)
0.0808803 + 0.996724i \(0.474227\pi\)
\(180\) 0 0
\(181\) 16.8974 1.25597 0.627986 0.778225i \(-0.283879\pi\)
0.627986 + 0.778225i \(0.283879\pi\)
\(182\) −0.649809 −0.0481670
\(183\) −0.435353 −0.0321822
\(184\) 4.13169 0.304592
\(185\) 0 0
\(186\) 0.362273 0.0265631
\(187\) −4.82908 −0.353138
\(188\) 3.97285 0.289750
\(189\) 7.26875 0.528724
\(190\) 0 0
\(191\) 5.92394 0.428641 0.214320 0.976763i \(-0.431246\pi\)
0.214320 + 0.976763i \(0.431246\pi\)
\(192\) 5.27114 0.380412
\(193\) −4.43535 −0.319264 −0.159632 0.987177i \(-0.551031\pi\)
−0.159632 + 0.987177i \(0.551031\pi\)
\(194\) −0.384043 −0.0275727
\(195\) 0 0
\(196\) 8.02177 0.572984
\(197\) 16.4722 1.17359 0.586797 0.809734i \(-0.300388\pi\)
0.586797 + 0.809734i \(0.300388\pi\)
\(198\) −0.530795 −0.0377220
\(199\) −24.4131 −1.73060 −0.865300 0.501255i \(-0.832872\pi\)
−0.865300 + 0.501255i \(0.832872\pi\)
\(200\) 0 0
\(201\) 11.2252 0.791765
\(202\) 4.10023 0.288491
\(203\) 8.47650 0.594934
\(204\) 8.23490 0.576558
\(205\) 0 0
\(206\) 1.19673 0.0833804
\(207\) −10.0097 −0.695721
\(208\) 5.65651 0.392209
\(209\) 0.911854 0.0630743
\(210\) 0 0
\(211\) −4.34050 −0.298813 −0.149406 0.988776i \(-0.547736\pi\)
−0.149406 + 0.988776i \(0.547736\pi\)
\(212\) 5.23862 0.359790
\(213\) 10.3274 0.707619
\(214\) 1.10321 0.0754140
\(215\) 0 0
\(216\) 4.17928 0.284364
\(217\) 3.09485 0.210092
\(218\) −4.64981 −0.314925
\(219\) 2.25129 0.152128
\(220\) 0 0
\(221\) 8.23490 0.553939
\(222\) 1.24698 0.0836918
\(223\) 26.2379 1.75702 0.878509 0.477725i \(-0.158539\pi\)
0.878509 + 0.477725i \(0.158539\pi\)
\(224\) −4.81236 −0.321540
\(225\) 0 0
\(226\) −4.94869 −0.329182
\(227\) 14.6853 0.974699 0.487349 0.873207i \(-0.337964\pi\)
0.487349 + 0.873207i \(0.337964\pi\)
\(228\) −1.55496 −0.102980
\(229\) −21.6407 −1.43006 −0.715029 0.699095i \(-0.753587\pi\)
−0.715029 + 0.699095i \(0.753587\pi\)
\(230\) 0 0
\(231\) 1.23729 0.0814078
\(232\) 4.87369 0.319973
\(233\) 27.1183 1.77658 0.888289 0.459286i \(-0.151895\pi\)
0.888289 + 0.459286i \(0.151895\pi\)
\(234\) 0.905149 0.0591715
\(235\) 0 0
\(236\) −19.1535 −1.24678
\(237\) −1.27844 −0.0830435
\(238\) −2.21313 −0.143456
\(239\) −11.5308 −0.745865 −0.372933 0.927858i \(-0.621648\pi\)
−0.372933 + 0.927858i \(0.621648\pi\)
\(240\) 0 0
\(241\) 11.8194 0.761354 0.380677 0.924708i \(-0.375691\pi\)
0.380677 + 0.924708i \(0.375691\pi\)
\(242\) 2.51142 0.161440
\(243\) −15.7952 −1.01326
\(244\) −1.05264 −0.0673883
\(245\) 0 0
\(246\) 0.828020 0.0527926
\(247\) −1.55496 −0.0989396
\(248\) 1.77943 0.112994
\(249\) −9.81163 −0.621787
\(250\) 0 0
\(251\) −9.66487 −0.610041 −0.305021 0.952346i \(-0.598663\pi\)
−0.305021 + 0.952346i \(0.598663\pi\)
\(252\) 7.73258 0.487107
\(253\) −3.87263 −0.243470
\(254\) 4.41358 0.276933
\(255\) 0 0
\(256\) 11.3402 0.708761
\(257\) 14.1860 0.884897 0.442449 0.896794i \(-0.354110\pi\)
0.442449 + 0.896794i \(0.354110\pi\)
\(258\) 1.44935 0.0902328
\(259\) 10.6528 0.661932
\(260\) 0 0
\(261\) −11.8073 −0.730854
\(262\) −1.83818 −0.113563
\(263\) −21.7942 −1.34389 −0.671943 0.740603i \(-0.734540\pi\)
−0.671943 + 0.740603i \(0.734540\pi\)
\(264\) 0.711399 0.0437836
\(265\) 0 0
\(266\) 0.417895 0.0256228
\(267\) −2.33944 −0.143171
\(268\) 27.1414 1.65792
\(269\) −24.7265 −1.50760 −0.753800 0.657104i \(-0.771781\pi\)
−0.753800 + 0.657104i \(0.771781\pi\)
\(270\) 0 0
\(271\) −13.2295 −0.803636 −0.401818 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(272\) 19.2650 1.16811
\(273\) −2.10992 −0.127698
\(274\) −1.40449 −0.0848481
\(275\) 0 0
\(276\) 6.60388 0.397507
\(277\) −0.560335 −0.0336673 −0.0168336 0.999858i \(-0.505359\pi\)
−0.0168336 + 0.999858i \(0.505359\pi\)
\(278\) −0.893068 −0.0535626
\(279\) −4.31096 −0.258091
\(280\) 0 0
\(281\) 27.6039 1.64671 0.823355 0.567527i \(-0.192100\pi\)
0.823355 + 0.567527i \(0.192100\pi\)
\(282\) −0.405813 −0.0241658
\(283\) −15.9608 −0.948769 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(284\) 24.9705 1.48172
\(285\) 0 0
\(286\) 0.350191 0.0207072
\(287\) 7.07367 0.417546
\(288\) 6.70337 0.395000
\(289\) 11.0465 0.649796
\(290\) 0 0
\(291\) −1.24698 −0.0730992
\(292\) 5.44339 0.318550
\(293\) 25.3327 1.47995 0.739977 0.672632i \(-0.234836\pi\)
0.739977 + 0.672632i \(0.234836\pi\)
\(294\) −0.819396 −0.0477882
\(295\) 0 0
\(296\) 6.12498 0.356007
\(297\) −3.91723 −0.227301
\(298\) −0.812691 −0.0470779
\(299\) 6.60388 0.381912
\(300\) 0 0
\(301\) 12.3817 0.713666
\(302\) 2.52243 0.145150
\(303\) 13.3134 0.764832
\(304\) −3.63773 −0.208638
\(305\) 0 0
\(306\) 3.08277 0.176230
\(307\) −11.5574 −0.659613 −0.329806 0.944049i \(-0.606983\pi\)
−0.329806 + 0.944049i \(0.606983\pi\)
\(308\) 2.99164 0.170464
\(309\) 3.88577 0.221054
\(310\) 0 0
\(311\) −18.3424 −1.04010 −0.520052 0.854135i \(-0.674087\pi\)
−0.520052 + 0.854135i \(0.674087\pi\)
\(312\) −1.21313 −0.0686798
\(313\) −18.9119 −1.06896 −0.534481 0.845181i \(-0.679493\pi\)
−0.534481 + 0.845181i \(0.679493\pi\)
\(314\) −3.54288 −0.199936
\(315\) 0 0
\(316\) −3.09113 −0.173890
\(317\) 20.2784 1.13895 0.569475 0.822008i \(-0.307146\pi\)
0.569475 + 0.822008i \(0.307146\pi\)
\(318\) −0.535107 −0.0300073
\(319\) −4.56810 −0.255765
\(320\) 0 0
\(321\) 3.58211 0.199934
\(322\) −1.77479 −0.0989052
\(323\) −5.29590 −0.294672
\(324\) −7.03013 −0.390563
\(325\) 0 0
\(326\) 4.82371 0.267160
\(327\) −15.0978 −0.834912
\(328\) 4.06711 0.224569
\(329\) −3.46681 −0.191132
\(330\) 0 0
\(331\) −4.77479 −0.262446 −0.131223 0.991353i \(-0.541890\pi\)
−0.131223 + 0.991353i \(0.541890\pi\)
\(332\) −23.7235 −1.30200
\(333\) −14.8388 −0.813160
\(334\) −2.93469 −0.160579
\(335\) 0 0
\(336\) −4.93602 −0.269282
\(337\) 24.3967 1.32897 0.664487 0.747300i \(-0.268650\pi\)
0.664487 + 0.747300i \(0.268650\pi\)
\(338\) 2.61356 0.142159
\(339\) −16.0683 −0.872710
\(340\) 0 0
\(341\) −1.66786 −0.0903196
\(342\) −0.582105 −0.0314766
\(343\) −18.8442 −1.01749
\(344\) 7.11901 0.383832
\(345\) 0 0
\(346\) −3.82132 −0.205435
\(347\) −9.99761 −0.536700 −0.268350 0.963322i \(-0.586478\pi\)
−0.268350 + 0.963322i \(0.586478\pi\)
\(348\) 7.78986 0.417580
\(349\) 21.9584 1.17541 0.587703 0.809077i \(-0.300033\pi\)
0.587703 + 0.809077i \(0.300033\pi\)
\(350\) 0 0
\(351\) 6.67994 0.356549
\(352\) 2.59345 0.138231
\(353\) −36.8786 −1.96285 −0.981425 0.191848i \(-0.938552\pi\)
−0.981425 + 0.191848i \(0.938552\pi\)
\(354\) 1.95646 0.103985
\(355\) 0 0
\(356\) −5.65651 −0.299795
\(357\) −7.18598 −0.380322
\(358\) −0.534516 −0.0282500
\(359\) 16.5187 0.871824 0.435912 0.899989i \(-0.356426\pi\)
0.435912 + 0.899989i \(0.356426\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −4.17331 −0.219344
\(363\) 8.15452 0.428001
\(364\) −5.10156 −0.267394
\(365\) 0 0
\(366\) 0.107523 0.00562034
\(367\) −6.83148 −0.356600 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(368\) 15.4494 0.805353
\(369\) −9.85325 −0.512940
\(370\) 0 0
\(371\) −4.57135 −0.237333
\(372\) 2.84415 0.147462
\(373\) −4.76271 −0.246604 −0.123302 0.992369i \(-0.539348\pi\)
−0.123302 + 0.992369i \(0.539348\pi\)
\(374\) 1.19269 0.0616723
\(375\) 0 0
\(376\) −1.99330 −0.102796
\(377\) 7.78986 0.401198
\(378\) −1.79523 −0.0923368
\(379\) 14.3773 0.738514 0.369257 0.929327i \(-0.379612\pi\)
0.369257 + 0.929327i \(0.379612\pi\)
\(380\) 0 0
\(381\) 14.3308 0.734190
\(382\) −1.46309 −0.0748583
\(383\) −30.6708 −1.56721 −0.783603 0.621261i \(-0.786621\pi\)
−0.783603 + 0.621261i \(0.786621\pi\)
\(384\) −5.86353 −0.299222
\(385\) 0 0
\(386\) 1.09544 0.0557565
\(387\) −17.2470 −0.876713
\(388\) −3.01507 −0.153067
\(389\) −12.9215 −0.655148 −0.327574 0.944825i \(-0.606231\pi\)
−0.327574 + 0.944825i \(0.606231\pi\)
\(390\) 0 0
\(391\) 22.4916 1.13745
\(392\) −4.02475 −0.203281
\(393\) −5.96854 −0.301073
\(394\) −4.06829 −0.204958
\(395\) 0 0
\(396\) −4.16719 −0.209409
\(397\) −29.8471 −1.49798 −0.748992 0.662579i \(-0.769462\pi\)
−0.748992 + 0.662579i \(0.769462\pi\)
\(398\) 6.02954 0.302234
\(399\) 1.35690 0.0679298
\(400\) 0 0
\(401\) −28.7101 −1.43371 −0.716856 0.697221i \(-0.754420\pi\)
−0.716856 + 0.697221i \(0.754420\pi\)
\(402\) −2.77240 −0.138275
\(403\) 2.84415 0.141677
\(404\) 32.1903 1.60153
\(405\) 0 0
\(406\) −2.09352 −0.103900
\(407\) −5.74094 −0.284568
\(408\) −4.13169 −0.204549
\(409\) −13.6203 −0.673479 −0.336739 0.941598i \(-0.609324\pi\)
−0.336739 + 0.941598i \(0.609324\pi\)
\(410\) 0 0
\(411\) −4.56033 −0.224945
\(412\) 9.39539 0.462878
\(413\) 16.7138 0.822432
\(414\) 2.47219 0.121501
\(415\) 0 0
\(416\) −4.42253 −0.216833
\(417\) −2.89977 −0.142002
\(418\) −0.225209 −0.0110153
\(419\) −2.13946 −0.104519 −0.0522596 0.998634i \(-0.516642\pi\)
−0.0522596 + 0.998634i \(0.516642\pi\)
\(420\) 0 0
\(421\) −15.0562 −0.733795 −0.366897 0.930261i \(-0.619580\pi\)
−0.366897 + 0.930261i \(0.619580\pi\)
\(422\) 1.07202 0.0521849
\(423\) 4.82908 0.234798
\(424\) −2.62837 −0.127645
\(425\) 0 0
\(426\) −2.55065 −0.123579
\(427\) 0.918559 0.0444522
\(428\) 8.66115 0.418653
\(429\) 1.13706 0.0548979
\(430\) 0 0
\(431\) 4.37435 0.210705 0.105353 0.994435i \(-0.466403\pi\)
0.105353 + 0.994435i \(0.466403\pi\)
\(432\) 15.6273 0.751869
\(433\) −33.1564 −1.59340 −0.796698 0.604377i \(-0.793422\pi\)
−0.796698 + 0.604377i \(0.793422\pi\)
\(434\) −0.764365 −0.0366907
\(435\) 0 0
\(436\) −36.5050 −1.74827
\(437\) −4.24698 −0.203161
\(438\) −0.556023 −0.0265678
\(439\) 32.2368 1.53858 0.769290 0.638900i \(-0.220610\pi\)
0.769290 + 0.638900i \(0.220610\pi\)
\(440\) 0 0
\(441\) 9.75063 0.464316
\(442\) −2.03385 −0.0967405
\(443\) 21.7362 1.03272 0.516358 0.856373i \(-0.327287\pi\)
0.516358 + 0.856373i \(0.327287\pi\)
\(444\) 9.78986 0.464606
\(445\) 0 0
\(446\) −6.48022 −0.306847
\(447\) −2.63879 −0.124811
\(448\) −11.1217 −0.525450
\(449\) −24.9584 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(450\) 0 0
\(451\) −3.81210 −0.179505
\(452\) −38.8514 −1.82742
\(453\) 8.19029 0.384814
\(454\) −3.62697 −0.170222
\(455\) 0 0
\(456\) 0.780167 0.0365347
\(457\) 13.1347 0.614414 0.307207 0.951643i \(-0.400606\pi\)
0.307207 + 0.951643i \(0.400606\pi\)
\(458\) 5.34481 0.249747
\(459\) 22.7506 1.06191
\(460\) 0 0
\(461\) −35.3726 −1.64746 −0.823732 0.566979i \(-0.808112\pi\)
−0.823732 + 0.566979i \(0.808112\pi\)
\(462\) −0.305586 −0.0142171
\(463\) 14.6963 0.682997 0.341498 0.939882i \(-0.389066\pi\)
0.341498 + 0.939882i \(0.389066\pi\)
\(464\) 18.2239 0.846022
\(465\) 0 0
\(466\) −6.69766 −0.310263
\(467\) −33.2121 −1.53687 −0.768435 0.639927i \(-0.778964\pi\)
−0.768435 + 0.639927i \(0.778964\pi\)
\(468\) 7.10620 0.328484
\(469\) −23.6843 −1.09364
\(470\) 0 0
\(471\) −11.5036 −0.530060
\(472\) 9.60984 0.442329
\(473\) −6.67264 −0.306809
\(474\) 0.315748 0.0145028
\(475\) 0 0
\(476\) −17.3749 −0.796379
\(477\) 6.36765 0.291555
\(478\) 2.84787 0.130259
\(479\) 24.6219 1.12500 0.562502 0.826796i \(-0.309839\pi\)
0.562502 + 0.826796i \(0.309839\pi\)
\(480\) 0 0
\(481\) 9.78986 0.446379
\(482\) −2.91915 −0.132964
\(483\) −5.76271 −0.262212
\(484\) 19.7168 0.896217
\(485\) 0 0
\(486\) 3.90110 0.176958
\(487\) −29.5646 −1.33970 −0.669851 0.742496i \(-0.733642\pi\)
−0.669851 + 0.742496i \(0.733642\pi\)
\(488\) 0.528139 0.0239077
\(489\) 15.6625 0.708282
\(490\) 0 0
\(491\) 36.2978 1.63810 0.819049 0.573724i \(-0.194502\pi\)
0.819049 + 0.573724i \(0.194502\pi\)
\(492\) 6.50066 0.293073
\(493\) 26.5308 1.19489
\(494\) 0.384043 0.0172789
\(495\) 0 0
\(496\) 6.65371 0.298760
\(497\) −21.7899 −0.977409
\(498\) 2.42327 0.108589
\(499\) 7.84415 0.351152 0.175576 0.984466i \(-0.443821\pi\)
0.175576 + 0.984466i \(0.443821\pi\)
\(500\) 0 0
\(501\) −9.52888 −0.425719
\(502\) 2.38703 0.106538
\(503\) −20.4166 −0.910330 −0.455165 0.890407i \(-0.650420\pi\)
−0.455165 + 0.890407i \(0.650420\pi\)
\(504\) −3.87966 −0.172814
\(505\) 0 0
\(506\) 0.956459 0.0425198
\(507\) 8.48619 0.376885
\(508\) 34.6504 1.53736
\(509\) −14.7530 −0.653916 −0.326958 0.945039i \(-0.606024\pi\)
−0.326958 + 0.945039i \(0.606024\pi\)
\(510\) 0 0
\(511\) −4.75004 −0.210129
\(512\) −17.4242 −0.770048
\(513\) −4.29590 −0.189668
\(514\) −3.50365 −0.154539
\(515\) 0 0
\(516\) 11.3787 0.500918
\(517\) 1.86831 0.0821683
\(518\) −2.63102 −0.115600
\(519\) −12.4077 −0.544639
\(520\) 0 0
\(521\) 37.1487 1.62751 0.813756 0.581206i \(-0.197419\pi\)
0.813756 + 0.581206i \(0.197419\pi\)
\(522\) 2.91617 0.127637
\(523\) −16.3623 −0.715472 −0.357736 0.933823i \(-0.616451\pi\)
−0.357736 + 0.933823i \(0.616451\pi\)
\(524\) −14.4313 −0.630434
\(525\) 0 0
\(526\) 5.38271 0.234698
\(527\) 9.68664 0.421957
\(528\) 2.66009 0.115765
\(529\) −4.96316 −0.215790
\(530\) 0 0
\(531\) −23.2814 −1.01033
\(532\) 3.28083 0.142242
\(533\) 6.50066 0.281575
\(534\) 0.577793 0.0250036
\(535\) 0 0
\(536\) −13.6176 −0.588191
\(537\) −1.73556 −0.0748950
\(538\) 6.10693 0.263289
\(539\) 3.77240 0.162489
\(540\) 0 0
\(541\) −2.08947 −0.0898335 −0.0449168 0.998991i \(-0.514302\pi\)
−0.0449168 + 0.998991i \(0.514302\pi\)
\(542\) 3.26742 0.140348
\(543\) −13.5506 −0.581514
\(544\) −15.0623 −0.645792
\(545\) 0 0
\(546\) 0.521106 0.0223013
\(547\) −1.86054 −0.0795511 −0.0397756 0.999209i \(-0.512664\pi\)
−0.0397756 + 0.999209i \(0.512664\pi\)
\(548\) −11.0264 −0.471025
\(549\) −1.27950 −0.0546079
\(550\) 0 0
\(551\) −5.00969 −0.213420
\(552\) −3.31336 −0.141026
\(553\) 2.69740 0.114705
\(554\) 0.138391 0.00587968
\(555\) 0 0
\(556\) −7.01134 −0.297347
\(557\) 9.80061 0.415265 0.207633 0.978207i \(-0.433424\pi\)
0.207633 + 0.978207i \(0.433424\pi\)
\(558\) 1.06472 0.0450732
\(559\) 11.3787 0.481266
\(560\) 0 0
\(561\) 3.87263 0.163502
\(562\) −6.81759 −0.287583
\(563\) −38.0170 −1.60222 −0.801112 0.598514i \(-0.795758\pi\)
−0.801112 + 0.598514i \(0.795758\pi\)
\(564\) −3.18598 −0.134154
\(565\) 0 0
\(566\) 3.94198 0.165694
\(567\) 6.13467 0.257632
\(568\) −12.5284 −0.525680
\(569\) −7.32975 −0.307279 −0.153640 0.988127i \(-0.549099\pi\)
−0.153640 + 0.988127i \(0.549099\pi\)
\(570\) 0 0
\(571\) 37.8775 1.58513 0.792563 0.609791i \(-0.208746\pi\)
0.792563 + 0.609791i \(0.208746\pi\)
\(572\) 2.74930 0.114954
\(573\) −4.75063 −0.198460
\(574\) −1.74705 −0.0729206
\(575\) 0 0
\(576\) 15.4919 0.645496
\(577\) −28.6993 −1.19477 −0.597384 0.801955i \(-0.703793\pi\)
−0.597384 + 0.801955i \(0.703793\pi\)
\(578\) −2.72827 −0.113481
\(579\) 3.55688 0.147819
\(580\) 0 0
\(581\) 20.7017 0.858852
\(582\) 0.307979 0.0127661
\(583\) 2.46357 0.102030
\(584\) −2.73110 −0.113014
\(585\) 0 0
\(586\) −6.25667 −0.258461
\(587\) −3.72348 −0.153684 −0.0768422 0.997043i \(-0.524484\pi\)
−0.0768422 + 0.997043i \(0.524484\pi\)
\(588\) −6.43296 −0.265291
\(589\) −1.82908 −0.0753661
\(590\) 0 0
\(591\) −13.2097 −0.543373
\(592\) 22.9028 0.941297
\(593\) −27.7399 −1.13914 −0.569570 0.821943i \(-0.692890\pi\)
−0.569570 + 0.821943i \(0.692890\pi\)
\(594\) 0.967476 0.0396960
\(595\) 0 0
\(596\) −6.38032 −0.261348
\(597\) 19.5778 0.801266
\(598\) −1.63102 −0.0666975
\(599\) −23.5579 −0.962551 −0.481276 0.876569i \(-0.659826\pi\)
−0.481276 + 0.876569i \(0.659826\pi\)
\(600\) 0 0
\(601\) −7.36898 −0.300587 −0.150293 0.988641i \(-0.548022\pi\)
−0.150293 + 0.988641i \(0.548022\pi\)
\(602\) −3.05802 −0.124635
\(603\) 32.9909 1.34349
\(604\) 19.8033 0.805783
\(605\) 0 0
\(606\) −3.28813 −0.133571
\(607\) 28.4198 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(608\) 2.84415 0.115346
\(609\) −6.79763 −0.275454
\(610\) 0 0
\(611\) −3.18598 −0.128891
\(612\) 24.2024 0.978323
\(613\) 6.48129 0.261777 0.130888 0.991397i \(-0.458217\pi\)
0.130888 + 0.991397i \(0.458217\pi\)
\(614\) 2.85443 0.115195
\(615\) 0 0
\(616\) −1.50099 −0.0604767
\(617\) −24.4650 −0.984924 −0.492462 0.870334i \(-0.663903\pi\)
−0.492462 + 0.870334i \(0.663903\pi\)
\(618\) −0.959706 −0.0386051
\(619\) −22.2457 −0.894128 −0.447064 0.894502i \(-0.647530\pi\)
−0.447064 + 0.894502i \(0.647530\pi\)
\(620\) 0 0
\(621\) 18.2446 0.732130
\(622\) 4.53020 0.181645
\(623\) 4.93602 0.197757
\(624\) −4.53617 −0.181592
\(625\) 0 0
\(626\) 4.67084 0.186684
\(627\) −0.731250 −0.0292033
\(628\) −27.8146 −1.10992
\(629\) 33.3424 1.32945
\(630\) 0 0
\(631\) −15.5888 −0.620581 −0.310290 0.950642i \(-0.600426\pi\)
−0.310290 + 0.950642i \(0.600426\pi\)
\(632\) 1.55091 0.0616919
\(633\) 3.48081 0.138350
\(634\) −5.00836 −0.198907
\(635\) 0 0
\(636\) −4.20105 −0.166582
\(637\) −6.43296 −0.254883
\(638\) 1.12823 0.0446670
\(639\) 30.3521 1.20071
\(640\) 0 0
\(641\) 8.17496 0.322892 0.161446 0.986882i \(-0.448384\pi\)
0.161446 + 0.986882i \(0.448384\pi\)
\(642\) −0.884707 −0.0349166
\(643\) 11.1836 0.441038 0.220519 0.975383i \(-0.429225\pi\)
0.220519 + 0.975383i \(0.429225\pi\)
\(644\) −13.9336 −0.549062
\(645\) 0 0
\(646\) 1.30798 0.0514617
\(647\) 16.2121 0.637362 0.318681 0.947862i \(-0.396760\pi\)
0.318681 + 0.947862i \(0.396760\pi\)
\(648\) 3.52722 0.138562
\(649\) −9.00730 −0.353567
\(650\) 0 0
\(651\) −2.48188 −0.0972725
\(652\) 37.8702 1.48311
\(653\) 24.7952 0.970312 0.485156 0.874427i \(-0.338763\pi\)
0.485156 + 0.874427i \(0.338763\pi\)
\(654\) 3.72886 0.145810
\(655\) 0 0
\(656\) 15.2079 0.593769
\(657\) 6.61655 0.258136
\(658\) 0.856232 0.0333794
\(659\) −20.2868 −0.790262 −0.395131 0.918625i \(-0.629301\pi\)
−0.395131 + 0.918625i \(0.629301\pi\)
\(660\) 0 0
\(661\) 20.4222 0.794332 0.397166 0.917747i \(-0.369994\pi\)
0.397166 + 0.917747i \(0.369994\pi\)
\(662\) 1.17928 0.0458339
\(663\) −6.60388 −0.256473
\(664\) 11.9028 0.461917
\(665\) 0 0
\(666\) 3.66487 0.142011
\(667\) 21.2760 0.823812
\(668\) −23.0398 −0.891437
\(669\) −21.0411 −0.813498
\(670\) 0 0
\(671\) −0.495024 −0.0191102
\(672\) 3.85922 0.148872
\(673\) 28.7254 1.10728 0.553641 0.832755i \(-0.313238\pi\)
0.553641 + 0.832755i \(0.313238\pi\)
\(674\) −6.02549 −0.232093
\(675\) 0 0
\(676\) 20.5187 0.789181
\(677\) 44.0062 1.69130 0.845648 0.533740i \(-0.179214\pi\)
0.845648 + 0.533740i \(0.179214\pi\)
\(678\) 3.96854 0.152411
\(679\) 2.63102 0.100969
\(680\) 0 0
\(681\) −11.7767 −0.451284
\(682\) 0.411927 0.0157735
\(683\) 8.48427 0.324642 0.162321 0.986738i \(-0.448102\pi\)
0.162321 + 0.986738i \(0.448102\pi\)
\(684\) −4.57002 −0.174739
\(685\) 0 0
\(686\) 4.65412 0.177695
\(687\) 17.3545 0.662116
\(688\) 26.6197 1.01487
\(689\) −4.20105 −0.160047
\(690\) 0 0
\(691\) 20.2586 0.770673 0.385336 0.922776i \(-0.374085\pi\)
0.385336 + 0.922776i \(0.374085\pi\)
\(692\) −30.0006 −1.14045
\(693\) 3.63640 0.138135
\(694\) 2.46921 0.0937297
\(695\) 0 0
\(696\) −3.90840 −0.148147
\(697\) 22.1400 0.838614
\(698\) −5.42327 −0.205274
\(699\) −21.7472 −0.822553
\(700\) 0 0
\(701\) −23.4101 −0.884188 −0.442094 0.896969i \(-0.645764\pi\)
−0.442094 + 0.896969i \(0.645764\pi\)
\(702\) −1.64981 −0.0622680
\(703\) −6.29590 −0.237454
\(704\) 5.99362 0.225893
\(705\) 0 0
\(706\) 9.10826 0.342794
\(707\) −28.0901 −1.05644
\(708\) 15.3599 0.577260
\(709\) 30.3472 1.13971 0.569857 0.821744i \(-0.306999\pi\)
0.569857 + 0.821744i \(0.306999\pi\)
\(710\) 0 0
\(711\) −3.75733 −0.140911
\(712\) 2.83804 0.106360
\(713\) 7.76809 0.290917
\(714\) 1.77479 0.0664199
\(715\) 0 0
\(716\) −4.19641 −0.156827
\(717\) 9.24698 0.345335
\(718\) −4.07979 −0.152256
\(719\) −38.3230 −1.42921 −0.714604 0.699529i \(-0.753393\pi\)
−0.714604 + 0.699529i \(0.753393\pi\)
\(720\) 0 0
\(721\) −8.19865 −0.305334
\(722\) −0.246980 −0.00919163
\(723\) −9.47842 −0.352506
\(724\) −32.7640 −1.21767
\(725\) 0 0
\(726\) −2.01400 −0.0747466
\(727\) 2.69069 0.0997923 0.0498961 0.998754i \(-0.484111\pi\)
0.0498961 + 0.998754i \(0.484111\pi\)
\(728\) 2.55960 0.0948650
\(729\) 1.78986 0.0662910
\(730\) 0 0
\(731\) 38.7536 1.43335
\(732\) 0.844150 0.0312007
\(733\) 18.9651 0.700491 0.350246 0.936658i \(-0.386098\pi\)
0.350246 + 0.936658i \(0.386098\pi\)
\(734\) 1.68724 0.0622770
\(735\) 0 0
\(736\) −12.0790 −0.445240
\(737\) 12.7638 0.470160
\(738\) 2.43355 0.0895803
\(739\) 29.8278 1.09723 0.548616 0.836075i \(-0.315155\pi\)
0.548616 + 0.836075i \(0.315155\pi\)
\(740\) 0 0
\(741\) 1.24698 0.0458089
\(742\) 1.12903 0.0414480
\(743\) 8.78448 0.322271 0.161136 0.986932i \(-0.448484\pi\)
0.161136 + 0.986932i \(0.448484\pi\)
\(744\) −1.42699 −0.0523161
\(745\) 0 0
\(746\) 1.17629 0.0430671
\(747\) −28.8364 −1.05507
\(748\) 9.36360 0.342367
\(749\) −7.55794 −0.276161
\(750\) 0 0
\(751\) −18.1142 −0.660998 −0.330499 0.943806i \(-0.607217\pi\)
−0.330499 + 0.943806i \(0.607217\pi\)
\(752\) −7.45340 −0.271798
\(753\) 7.75063 0.282449
\(754\) −1.92394 −0.0700656
\(755\) 0 0
\(756\) −14.0941 −0.512598
\(757\) −0.222816 −0.00809840 −0.00404920 0.999992i \(-0.501289\pi\)
−0.00404920 + 0.999992i \(0.501289\pi\)
\(758\) −3.55091 −0.128975
\(759\) 3.10560 0.112726
\(760\) 0 0
\(761\) −6.07798 −0.220327 −0.110163 0.993913i \(-0.535137\pi\)
−0.110163 + 0.993913i \(0.535137\pi\)
\(762\) −3.53942 −0.128220
\(763\) 31.8552 1.15323
\(764\) −11.4865 −0.415568
\(765\) 0 0
\(766\) 7.57507 0.273698
\(767\) 15.3599 0.554613
\(768\) −9.09411 −0.328156
\(769\) −14.6267 −0.527453 −0.263726 0.964598i \(-0.584952\pi\)
−0.263726 + 0.964598i \(0.584952\pi\)
\(770\) 0 0
\(771\) −11.3763 −0.409706
\(772\) 8.60015 0.309526
\(773\) −4.05728 −0.145930 −0.0729651 0.997334i \(-0.523246\pi\)
−0.0729651 + 0.997334i \(0.523246\pi\)
\(774\) 4.25965 0.153110
\(775\) 0 0
\(776\) 1.51275 0.0543044
\(777\) −8.54288 −0.306474
\(778\) 3.19136 0.114416
\(779\) −4.18060 −0.149786
\(780\) 0 0
\(781\) 11.7429 0.420192
\(782\) −5.55496 −0.198645
\(783\) 21.5211 0.769102
\(784\) −15.0495 −0.537482
\(785\) 0 0
\(786\) 1.47411 0.0525797
\(787\) 19.5657 0.697442 0.348721 0.937227i \(-0.386616\pi\)
0.348721 + 0.937227i \(0.386616\pi\)
\(788\) −31.9396 −1.13780
\(789\) 17.4776 0.622218
\(790\) 0 0
\(791\) 33.9028 1.20544
\(792\) 2.09080 0.0742934
\(793\) 0.844150 0.0299767
\(794\) 7.37163 0.261609
\(795\) 0 0
\(796\) 47.3370 1.67782
\(797\) 38.9051 1.37809 0.689046 0.724718i \(-0.258030\pi\)
0.689046 + 0.724718i \(0.258030\pi\)
\(798\) −0.335126 −0.0118633
\(799\) −10.8509 −0.383876
\(800\) 0 0
\(801\) −6.87561 −0.242938
\(802\) 7.09080 0.250385
\(803\) 2.55986 0.0903355
\(804\) −21.7657 −0.767617
\(805\) 0 0
\(806\) −0.702447 −0.0247426
\(807\) 19.8291 0.698017
\(808\) −16.1508 −0.568183
\(809\) −22.5730 −0.793625 −0.396812 0.917900i \(-0.629884\pi\)
−0.396812 + 0.917900i \(0.629884\pi\)
\(810\) 0 0
\(811\) −46.3605 −1.62794 −0.813968 0.580909i \(-0.802697\pi\)
−0.813968 + 0.580909i \(0.802697\pi\)
\(812\) −16.4359 −0.576789
\(813\) 10.6093 0.372083
\(814\) 1.41789 0.0496972
\(815\) 0 0
\(816\) −15.4494 −0.540836
\(817\) −7.31767 −0.256013
\(818\) 3.36393 0.117617
\(819\) −6.20105 −0.216682
\(820\) 0 0
\(821\) −17.8194 −0.621901 −0.310951 0.950426i \(-0.600647\pi\)
−0.310951 + 0.950426i \(0.600647\pi\)
\(822\) 1.12631 0.0392846
\(823\) −32.5394 −1.13425 −0.567126 0.823631i \(-0.691945\pi\)
−0.567126 + 0.823631i \(0.691945\pi\)
\(824\) −4.71394 −0.164218
\(825\) 0 0
\(826\) −4.12797 −0.143630
\(827\) 39.8256 1.38487 0.692436 0.721479i \(-0.256537\pi\)
0.692436 + 0.721479i \(0.256537\pi\)
\(828\) 19.4088 0.674502
\(829\) 38.7525 1.34593 0.672966 0.739674i \(-0.265020\pi\)
0.672966 + 0.739674i \(0.265020\pi\)
\(830\) 0 0
\(831\) 0.449354 0.0155879
\(832\) −10.2208 −0.354341
\(833\) −21.9095 −0.759118
\(834\) 0.716185 0.0247994
\(835\) 0 0
\(836\) −1.76809 −0.0611505
\(837\) 7.85756 0.271597
\(838\) 0.528402 0.0182533
\(839\) 2.76749 0.0955445 0.0477723 0.998858i \(-0.484788\pi\)
0.0477723 + 0.998858i \(0.484788\pi\)
\(840\) 0 0
\(841\) −3.90302 −0.134587
\(842\) 3.71858 0.128151
\(843\) −22.1366 −0.762425
\(844\) 8.41624 0.289699
\(845\) 0 0
\(846\) −1.19269 −0.0410054
\(847\) −17.2054 −0.591183
\(848\) −9.82808 −0.337498
\(849\) 12.7995 0.439279
\(850\) 0 0
\(851\) 26.7385 0.916586
\(852\) −20.0248 −0.686037
\(853\) 24.1390 0.826503 0.413252 0.910617i \(-0.364393\pi\)
0.413252 + 0.910617i \(0.364393\pi\)
\(854\) −0.226865 −0.00776317
\(855\) 0 0
\(856\) −4.34555 −0.148528
\(857\) −3.16229 −0.108022 −0.0540109 0.998540i \(-0.517201\pi\)
−0.0540109 + 0.998540i \(0.517201\pi\)
\(858\) −0.280831 −0.00958743
\(859\) 4.86426 0.165967 0.0829833 0.996551i \(-0.473555\pi\)
0.0829833 + 0.996551i \(0.473555\pi\)
\(860\) 0 0
\(861\) −5.67264 −0.193323
\(862\) −1.08038 −0.0367978
\(863\) 4.54958 0.154870 0.0774348 0.996997i \(-0.475327\pi\)
0.0774348 + 0.996997i \(0.475327\pi\)
\(864\) −12.2182 −0.415671
\(865\) 0 0
\(866\) 8.18896 0.278272
\(867\) −8.85862 −0.300855
\(868\) −6.00092 −0.203684
\(869\) −1.45367 −0.0493122
\(870\) 0 0
\(871\) −21.7657 −0.737502
\(872\) 18.3156 0.620245
\(873\) −3.66487 −0.124037
\(874\) 1.04892 0.0354802
\(875\) 0 0
\(876\) −4.36526 −0.147488
\(877\) −18.6595 −0.630086 −0.315043 0.949077i \(-0.602019\pi\)
−0.315043 + 0.949077i \(0.602019\pi\)
\(878\) −7.96184 −0.268699
\(879\) −20.3153 −0.685217
\(880\) 0 0
\(881\) 19.4306 0.654632 0.327316 0.944915i \(-0.393856\pi\)
0.327316 + 0.944915i \(0.393856\pi\)
\(882\) −2.40821 −0.0810885
\(883\) 25.6437 0.862979 0.431490 0.902118i \(-0.357988\pi\)
0.431490 + 0.902118i \(0.357988\pi\)
\(884\) −15.9675 −0.537044
\(885\) 0 0
\(886\) −5.36839 −0.180354
\(887\) −31.0062 −1.04109 −0.520544 0.853835i \(-0.674271\pi\)
−0.520544 + 0.853835i \(0.674271\pi\)
\(888\) −4.91185 −0.164831
\(889\) −30.2368 −1.01411
\(890\) 0 0
\(891\) −3.30606 −0.110757
\(892\) −50.8753 −1.70343
\(893\) 2.04892 0.0685644
\(894\) 0.651728 0.0217970
\(895\) 0 0
\(896\) 12.3716 0.413305
\(897\) −5.29590 −0.176825
\(898\) 6.16421 0.205702
\(899\) 9.16315 0.305608
\(900\) 0 0
\(901\) −14.3080 −0.476668
\(902\) 0.941511 0.0313489
\(903\) −9.92931 −0.330427
\(904\) 19.4929 0.648324
\(905\) 0 0
\(906\) −2.02284 −0.0672042
\(907\) −17.7676 −0.589964 −0.294982 0.955503i \(-0.595314\pi\)
−0.294982 + 0.955503i \(0.595314\pi\)
\(908\) −28.4748 −0.944971
\(909\) 39.1280 1.29779
\(910\) 0 0
\(911\) 41.7313 1.38262 0.691309 0.722559i \(-0.257034\pi\)
0.691309 + 0.722559i \(0.257034\pi\)
\(912\) 2.91723 0.0965992
\(913\) −11.1564 −0.369224
\(914\) −3.24400 −0.107302
\(915\) 0 0
\(916\) 41.9614 1.38644
\(917\) 12.5931 0.415862
\(918\) −5.61894 −0.185453
\(919\) 30.4088 1.00309 0.501547 0.865130i \(-0.332764\pi\)
0.501547 + 0.865130i \(0.332764\pi\)
\(920\) 0 0
\(921\) 9.26828 0.305400
\(922\) 8.73630 0.287715
\(923\) −20.0248 −0.659123
\(924\) −2.39911 −0.0789249
\(925\) 0 0
\(926\) −3.62969 −0.119279
\(927\) 11.4203 0.375091
\(928\) −14.2483 −0.467724
\(929\) −28.8219 −0.945616 −0.472808 0.881165i \(-0.656760\pi\)
−0.472808 + 0.881165i \(0.656760\pi\)
\(930\) 0 0
\(931\) 4.13706 0.135587
\(932\) −52.5824 −1.72239
\(933\) 14.7095 0.481567
\(934\) 8.20270 0.268401
\(935\) 0 0
\(936\) −3.56538 −0.116538
\(937\) −50.4601 −1.64846 −0.824230 0.566255i \(-0.808392\pi\)
−0.824230 + 0.566255i \(0.808392\pi\)
\(938\) 5.84953 0.190994
\(939\) 15.1661 0.494928
\(940\) 0 0
\(941\) 1.10082 0.0358857 0.0179428 0.999839i \(-0.494288\pi\)
0.0179428 + 0.999839i \(0.494288\pi\)
\(942\) 2.84117 0.0925702
\(943\) 17.7549 0.578180
\(944\) 35.9335 1.16954
\(945\) 0 0
\(946\) 1.64801 0.0535813
\(947\) −13.9353 −0.452836 −0.226418 0.974030i \(-0.572701\pi\)
−0.226418 + 0.974030i \(0.572701\pi\)
\(948\) 2.47889 0.0805107
\(949\) −4.36526 −0.141702
\(950\) 0 0
\(951\) −16.2620 −0.527333
\(952\) 8.71751 0.282536
\(953\) −41.3400 −1.33913 −0.669567 0.742751i \(-0.733520\pi\)
−0.669567 + 0.742751i \(0.733520\pi\)
\(954\) −1.57268 −0.0509174
\(955\) 0 0
\(956\) 22.3582 0.723117
\(957\) 3.66334 0.118419
\(958\) −6.08111 −0.196472
\(959\) 9.62192 0.310708
\(960\) 0 0
\(961\) −27.6544 −0.892079
\(962\) −2.41789 −0.0779561
\(963\) 10.5278 0.339254
\(964\) −22.9178 −0.738133
\(965\) 0 0
\(966\) 1.42327 0.0457930
\(967\) −5.26875 −0.169432 −0.0847158 0.996405i \(-0.526998\pi\)
−0.0847158 + 0.996405i \(0.526998\pi\)
\(968\) −9.89248 −0.317956
\(969\) 4.24698 0.136433
\(970\) 0 0
\(971\) −5.15346 −0.165382 −0.0826911 0.996575i \(-0.526351\pi\)
−0.0826911 + 0.996575i \(0.526351\pi\)
\(972\) 30.6270 0.982361
\(973\) 6.11828 0.196143
\(974\) 7.30186 0.233967
\(975\) 0 0
\(976\) 1.97484 0.0632130
\(977\) 4.77612 0.152802 0.0764008 0.997077i \(-0.475657\pi\)
0.0764008 + 0.997077i \(0.475657\pi\)
\(978\) −3.86831 −0.123695
\(979\) −2.66009 −0.0850168
\(980\) 0 0
\(981\) −44.3726 −1.41671
\(982\) −8.96482 −0.286079
\(983\) −28.9758 −0.924186 −0.462093 0.886832i \(-0.652901\pi\)
−0.462093 + 0.886832i \(0.652901\pi\)
\(984\) −3.26157 −0.103975
\(985\) 0 0
\(986\) −6.55257 −0.208676
\(987\) 2.78017 0.0884937
\(988\) 3.01507 0.0959220
\(989\) 31.0780 0.988222
\(990\) 0 0
\(991\) −38.5042 −1.22313 −0.611564 0.791195i \(-0.709459\pi\)
−0.611564 + 0.791195i \(0.709459\pi\)
\(992\) −5.20219 −0.165170
\(993\) 3.82908 0.121512
\(994\) 5.38165 0.170696
\(995\) 0 0
\(996\) 19.0248 0.602822
\(997\) 10.1491 0.321427 0.160713 0.987001i \(-0.448621\pi\)
0.160713 + 0.987001i \(0.448621\pi\)
\(998\) −1.93735 −0.0613256
\(999\) 27.0465 0.855714
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 475.2.a.h.1.1 yes 3
3.2 odd 2 4275.2.a.z.1.3 3
4.3 odd 2 7600.2.a.bn.1.3 3
5.2 odd 4 475.2.b.c.324.3 6
5.3 odd 4 475.2.b.c.324.4 6
5.4 even 2 475.2.a.d.1.3 3
15.14 odd 2 4275.2.a.bn.1.1 3
19.18 odd 2 9025.2.a.w.1.3 3
20.19 odd 2 7600.2.a.bw.1.1 3
95.94 odd 2 9025.2.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 5.4 even 2
475.2.a.h.1.1 yes 3 1.1 even 1 trivial
475.2.b.c.324.3 6 5.2 odd 4
475.2.b.c.324.4 6 5.3 odd 4
4275.2.a.z.1.3 3 3.2 odd 2
4275.2.a.bn.1.1 3 15.14 odd 2
7600.2.a.bn.1.3 3 4.3 odd 2
7600.2.a.bw.1.1 3 20.19 odd 2
9025.2.a.w.1.3 3 19.18 odd 2
9025.2.a.be.1.1 3 95.94 odd 2