Properties

Label 476.2.a.b.1.2
Level $476$
Weight $2$
Character 476.1
Self dual yes
Analytic conductor $3.801$
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] = [476,2,Mod(1,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, 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("476.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.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: \(3.80087913621\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 476.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{3} -1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} +O(q^{10})\) \(q+0.618034 q^{3} -1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} -5.23607 q^{11} -3.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} -0.618034 q^{21} +5.70820 q^{23} -2.38197 q^{25} -3.47214 q^{27} +7.70820 q^{29} -9.32624 q^{31} -3.23607 q^{33} +1.61803 q^{35} -8.47214 q^{37} -2.00000 q^{39} +11.0902 q^{41} -0.909830 q^{43} +4.23607 q^{45} +0.472136 q^{47} +1.00000 q^{49} +0.618034 q^{51} -13.7984 q^{53} +8.47214 q^{55} +0.291796 q^{57} +8.32624 q^{61} +2.61803 q^{63} +5.23607 q^{65} -1.90983 q^{67} +3.52786 q^{69} -6.94427 q^{71} +13.8541 q^{73} -1.47214 q^{75} +5.23607 q^{77} -14.9443 q^{79} +5.70820 q^{81} -11.4164 q^{83} -1.61803 q^{85} +4.76393 q^{87} +2.00000 q^{89} +3.23607 q^{91} -5.76393 q^{93} -0.763932 q^{95} +3.90983 q^{97} +13.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39} + 11 q^{41} - 13 q^{43} + 4 q^{45} - 8 q^{47} + 2 q^{49} - q^{51} - 3 q^{53} + 8 q^{55} + 14 q^{57} + q^{61} + 3 q^{63} + 6 q^{65} - 15 q^{67} + 16 q^{69} + 4 q^{71} + 21 q^{73} + 6 q^{75} + 6 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - q^{85} + 14 q^{87} + 4 q^{89} + 2 q^{91} - 16 q^{93} - 6 q^{95} + 19 q^{97} + 14 q^{99}+O(q^{100}) \) 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 0
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 0 0
\(5\) −1.61803 −0.723607 −0.361803 0.932254i \(-0.617839\pi\)
−0.361803 + 0.932254i \(0.617839\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 0.472136 0.108315 0.0541577 0.998532i \(-0.482753\pi\)
0.0541577 + 0.998532i \(0.482753\pi\)
\(20\) 0 0
\(21\) −0.618034 −0.134866
\(22\) 0 0
\(23\) 5.70820 1.19024 0.595121 0.803636i \(-0.297104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) 0 0
\(25\) −2.38197 −0.476393
\(26\) 0 0
\(27\) −3.47214 −0.668213
\(28\) 0 0
\(29\) 7.70820 1.43138 0.715689 0.698419i \(-0.246113\pi\)
0.715689 + 0.698419i \(0.246113\pi\)
\(30\) 0 0
\(31\) −9.32624 −1.67504 −0.837521 0.546405i \(-0.815996\pi\)
−0.837521 + 0.546405i \(0.815996\pi\)
\(32\) 0 0
\(33\) −3.23607 −0.563327
\(34\) 0 0
\(35\) 1.61803 0.273498
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 11.0902 1.73199 0.865997 0.500050i \(-0.166685\pi\)
0.865997 + 0.500050i \(0.166685\pi\)
\(42\) 0 0
\(43\) −0.909830 −0.138748 −0.0693739 0.997591i \(-0.522100\pi\)
−0.0693739 + 0.997591i \(0.522100\pi\)
\(44\) 0 0
\(45\) 4.23607 0.631476
\(46\) 0 0
\(47\) 0.472136 0.0688681 0.0344341 0.999407i \(-0.489037\pi\)
0.0344341 + 0.999407i \(0.489037\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.618034 0.0865421
\(52\) 0 0
\(53\) −13.7984 −1.89535 −0.947676 0.319233i \(-0.896575\pi\)
−0.947676 + 0.319233i \(0.896575\pi\)
\(54\) 0 0
\(55\) 8.47214 1.14238
\(56\) 0 0
\(57\) 0.291796 0.0386493
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.32624 1.06607 0.533033 0.846095i \(-0.321052\pi\)
0.533033 + 0.846095i \(0.321052\pi\)
\(62\) 0 0
\(63\) 2.61803 0.329841
\(64\) 0 0
\(65\) 5.23607 0.649454
\(66\) 0 0
\(67\) −1.90983 −0.233323 −0.116661 0.993172i \(-0.537219\pi\)
−0.116661 + 0.993172i \(0.537219\pi\)
\(68\) 0 0
\(69\) 3.52786 0.424705
\(70\) 0 0
\(71\) −6.94427 −0.824133 −0.412067 0.911154i \(-0.635193\pi\)
−0.412067 + 0.911154i \(0.635193\pi\)
\(72\) 0 0
\(73\) 13.8541 1.62150 0.810750 0.585393i \(-0.199060\pi\)
0.810750 + 0.585393i \(0.199060\pi\)
\(74\) 0 0
\(75\) −1.47214 −0.169988
\(76\) 0 0
\(77\) 5.23607 0.596705
\(78\) 0 0
\(79\) −14.9443 −1.68136 −0.840681 0.541531i \(-0.817845\pi\)
−0.840681 + 0.541531i \(0.817845\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) −11.4164 −1.25311 −0.626557 0.779376i \(-0.715536\pi\)
−0.626557 + 0.779376i \(0.715536\pi\)
\(84\) 0 0
\(85\) −1.61803 −0.175500
\(86\) 0 0
\(87\) 4.76393 0.510747
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 0 0
\(93\) −5.76393 −0.597692
\(94\) 0 0
\(95\) −0.763932 −0.0783778
\(96\) 0 0
\(97\) 3.90983 0.396983 0.198492 0.980103i \(-0.436396\pi\)
0.198492 + 0.980103i \(0.436396\pi\)
\(98\) 0 0
\(99\) 13.7082 1.37773
\(100\) 0 0
\(101\) 8.47214 0.843009 0.421505 0.906826i \(-0.361502\pi\)
0.421505 + 0.906826i \(0.361502\pi\)
\(102\) 0 0
\(103\) −11.4164 −1.12489 −0.562446 0.826834i \(-0.690140\pi\)
−0.562446 + 0.826834i \(0.690140\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 15.2361 1.47293 0.736463 0.676478i \(-0.236494\pi\)
0.736463 + 0.676478i \(0.236494\pi\)
\(108\) 0 0
\(109\) −12.4721 −1.19461 −0.597307 0.802013i \(-0.703763\pi\)
−0.597307 + 0.802013i \(0.703763\pi\)
\(110\) 0 0
\(111\) −5.23607 −0.496986
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −9.23607 −0.861268
\(116\) 0 0
\(117\) 8.47214 0.783249
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) 6.85410 0.618014
\(124\) 0 0
\(125\) 11.9443 1.06833
\(126\) 0 0
\(127\) −1.14590 −0.101682 −0.0508410 0.998707i \(-0.516190\pi\)
−0.0508410 + 0.998707i \(0.516190\pi\)
\(128\) 0 0
\(129\) −0.562306 −0.0495083
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −0.472136 −0.0409394
\(134\) 0 0
\(135\) 5.61803 0.483523
\(136\) 0 0
\(137\) 4.61803 0.394545 0.197273 0.980349i \(-0.436792\pi\)
0.197273 + 0.980349i \(0.436792\pi\)
\(138\) 0 0
\(139\) 15.0344 1.27520 0.637602 0.770366i \(-0.279926\pi\)
0.637602 + 0.770366i \(0.279926\pi\)
\(140\) 0 0
\(141\) 0.291796 0.0245737
\(142\) 0 0
\(143\) 16.9443 1.41695
\(144\) 0 0
\(145\) −12.4721 −1.03575
\(146\) 0 0
\(147\) 0.618034 0.0509746
\(148\) 0 0
\(149\) −11.3820 −0.932447 −0.466223 0.884667i \(-0.654386\pi\)
−0.466223 + 0.884667i \(0.654386\pi\)
\(150\) 0 0
\(151\) −21.8541 −1.77846 −0.889231 0.457459i \(-0.848760\pi\)
−0.889231 + 0.457459i \(0.848760\pi\)
\(152\) 0 0
\(153\) −2.61803 −0.211656
\(154\) 0 0
\(155\) 15.0902 1.21207
\(156\) 0 0
\(157\) 20.1803 1.61057 0.805283 0.592890i \(-0.202013\pi\)
0.805283 + 0.592890i \(0.202013\pi\)
\(158\) 0 0
\(159\) −8.52786 −0.676304
\(160\) 0 0
\(161\) −5.70820 −0.449869
\(162\) 0 0
\(163\) −7.70820 −0.603753 −0.301877 0.953347i \(-0.597613\pi\)
−0.301877 + 0.953347i \(0.597613\pi\)
\(164\) 0 0
\(165\) 5.23607 0.407627
\(166\) 0 0
\(167\) 16.6180 1.28594 0.642971 0.765890i \(-0.277702\pi\)
0.642971 + 0.765890i \(0.277702\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) −1.23607 −0.0945245
\(172\) 0 0
\(173\) −23.0902 −1.75551 −0.877757 0.479107i \(-0.840961\pi\)
−0.877757 + 0.479107i \(0.840961\pi\)
\(174\) 0 0
\(175\) 2.38197 0.180060
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.6180 −1.54106 −0.770532 0.637401i \(-0.780009\pi\)
−0.770532 + 0.637401i \(0.780009\pi\)
\(180\) 0 0
\(181\) −3.52786 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(182\) 0 0
\(183\) 5.14590 0.380396
\(184\) 0 0
\(185\) 13.7082 1.00785
\(186\) 0 0
\(187\) −5.23607 −0.382899
\(188\) 0 0
\(189\) 3.47214 0.252561
\(190\) 0 0
\(191\) −3.79837 −0.274841 −0.137420 0.990513i \(-0.543881\pi\)
−0.137420 + 0.990513i \(0.543881\pi\)
\(192\) 0 0
\(193\) 8.18034 0.588834 0.294417 0.955677i \(-0.404875\pi\)
0.294417 + 0.955677i \(0.404875\pi\)
\(194\) 0 0
\(195\) 3.23607 0.231740
\(196\) 0 0
\(197\) −3.70820 −0.264199 −0.132099 0.991236i \(-0.542172\pi\)
−0.132099 + 0.991236i \(0.542172\pi\)
\(198\) 0 0
\(199\) −0.909830 −0.0644961 −0.0322481 0.999480i \(-0.510267\pi\)
−0.0322481 + 0.999480i \(0.510267\pi\)
\(200\) 0 0
\(201\) −1.18034 −0.0832548
\(202\) 0 0
\(203\) −7.70820 −0.541010
\(204\) 0 0
\(205\) −17.9443 −1.25328
\(206\) 0 0
\(207\) −14.9443 −1.03870
\(208\) 0 0
\(209\) −2.47214 −0.171001
\(210\) 0 0
\(211\) −15.7082 −1.08140 −0.540699 0.841216i \(-0.681840\pi\)
−0.540699 + 0.841216i \(0.681840\pi\)
\(212\) 0 0
\(213\) −4.29180 −0.294069
\(214\) 0 0
\(215\) 1.47214 0.100399
\(216\) 0 0
\(217\) 9.32624 0.633106
\(218\) 0 0
\(219\) 8.56231 0.578587
\(220\) 0 0
\(221\) −3.23607 −0.217681
\(222\) 0 0
\(223\) −14.9443 −1.00074 −0.500371 0.865811i \(-0.666803\pi\)
−0.500371 + 0.865811i \(0.666803\pi\)
\(224\) 0 0
\(225\) 6.23607 0.415738
\(226\) 0 0
\(227\) −16.7426 −1.11125 −0.555624 0.831434i \(-0.687521\pi\)
−0.555624 + 0.831434i \(0.687521\pi\)
\(228\) 0 0
\(229\) −14.7639 −0.975628 −0.487814 0.872948i \(-0.662206\pi\)
−0.487814 + 0.872948i \(0.662206\pi\)
\(230\) 0 0
\(231\) 3.23607 0.212918
\(232\) 0 0
\(233\) −2.47214 −0.161955 −0.0809775 0.996716i \(-0.525804\pi\)
−0.0809775 + 0.996716i \(0.525804\pi\)
\(234\) 0 0
\(235\) −0.763932 −0.0498334
\(236\) 0 0
\(237\) −9.23607 −0.599947
\(238\) 0 0
\(239\) 23.0902 1.49358 0.746789 0.665061i \(-0.231594\pi\)
0.746789 + 0.665061i \(0.231594\pi\)
\(240\) 0 0
\(241\) −25.0344 −1.61261 −0.806305 0.591500i \(-0.798536\pi\)
−0.806305 + 0.591500i \(0.798536\pi\)
\(242\) 0 0
\(243\) 13.9443 0.894525
\(244\) 0 0
\(245\) −1.61803 −0.103372
\(246\) 0 0
\(247\) −1.52786 −0.0972157
\(248\) 0 0
\(249\) −7.05573 −0.447139
\(250\) 0 0
\(251\) 9.41641 0.594358 0.297179 0.954822i \(-0.403954\pi\)
0.297179 + 0.954822i \(0.403954\pi\)
\(252\) 0 0
\(253\) −29.8885 −1.87908
\(254\) 0 0
\(255\) −1.00000 −0.0626224
\(256\) 0 0
\(257\) 12.2918 0.766741 0.383371 0.923595i \(-0.374763\pi\)
0.383371 + 0.923595i \(0.374763\pi\)
\(258\) 0 0
\(259\) 8.47214 0.526433
\(260\) 0 0
\(261\) −20.1803 −1.24913
\(262\) 0 0
\(263\) 0.944272 0.0582263 0.0291132 0.999576i \(-0.490732\pi\)
0.0291132 + 0.999576i \(0.490732\pi\)
\(264\) 0 0
\(265\) 22.3262 1.37149
\(266\) 0 0
\(267\) 1.23607 0.0756461
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −9.70820 −0.589731 −0.294866 0.955539i \(-0.595275\pi\)
−0.294866 + 0.955539i \(0.595275\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 12.4721 0.752098
\(276\) 0 0
\(277\) 7.23607 0.434773 0.217387 0.976086i \(-0.430247\pi\)
0.217387 + 0.976086i \(0.430247\pi\)
\(278\) 0 0
\(279\) 24.4164 1.46177
\(280\) 0 0
\(281\) −20.0902 −1.19848 −0.599240 0.800570i \(-0.704530\pi\)
−0.599240 + 0.800570i \(0.704530\pi\)
\(282\) 0 0
\(283\) −24.5623 −1.46008 −0.730039 0.683406i \(-0.760498\pi\)
−0.730039 + 0.683406i \(0.760498\pi\)
\(284\) 0 0
\(285\) −0.472136 −0.0279669
\(286\) 0 0
\(287\) −11.0902 −0.654632
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 2.41641 0.141652
\(292\) 0 0
\(293\) −19.7082 −1.15137 −0.575683 0.817673i \(-0.695264\pi\)
−0.575683 + 0.817673i \(0.695264\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.1803 1.05493
\(298\) 0 0
\(299\) −18.4721 −1.06827
\(300\) 0 0
\(301\) 0.909830 0.0524417
\(302\) 0 0
\(303\) 5.23607 0.300804
\(304\) 0 0
\(305\) −13.4721 −0.771412
\(306\) 0 0
\(307\) 18.9443 1.08121 0.540603 0.841278i \(-0.318196\pi\)
0.540603 + 0.841278i \(0.318196\pi\)
\(308\) 0 0
\(309\) −7.05573 −0.401386
\(310\) 0 0
\(311\) 24.3820 1.38257 0.691287 0.722580i \(-0.257044\pi\)
0.691287 + 0.722580i \(0.257044\pi\)
\(312\) 0 0
\(313\) 32.2705 1.82404 0.912019 0.410149i \(-0.134523\pi\)
0.912019 + 0.410149i \(0.134523\pi\)
\(314\) 0 0
\(315\) −4.23607 −0.238675
\(316\) 0 0
\(317\) −5.81966 −0.326865 −0.163432 0.986555i \(-0.552257\pi\)
−0.163432 + 0.986555i \(0.552257\pi\)
\(318\) 0 0
\(319\) −40.3607 −2.25976
\(320\) 0 0
\(321\) 9.41641 0.525573
\(322\) 0 0
\(323\) 0.472136 0.0262703
\(324\) 0 0
\(325\) 7.70820 0.427574
\(326\) 0 0
\(327\) −7.70820 −0.426265
\(328\) 0 0
\(329\) −0.472136 −0.0260297
\(330\) 0 0
\(331\) −11.0344 −0.606508 −0.303254 0.952910i \(-0.598073\pi\)
−0.303254 + 0.952910i \(0.598073\pi\)
\(332\) 0 0
\(333\) 22.1803 1.21548
\(334\) 0 0
\(335\) 3.09017 0.168834
\(336\) 0 0
\(337\) 18.4721 1.00624 0.503121 0.864216i \(-0.332185\pi\)
0.503121 + 0.864216i \(0.332185\pi\)
\(338\) 0 0
\(339\) 3.70820 0.201402
\(340\) 0 0
\(341\) 48.8328 2.64445
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −5.70820 −0.307319
\(346\) 0 0
\(347\) 13.0557 0.700868 0.350434 0.936587i \(-0.386034\pi\)
0.350434 + 0.936587i \(0.386034\pi\)
\(348\) 0 0
\(349\) −24.4721 −1.30996 −0.654982 0.755645i \(-0.727324\pi\)
−0.654982 + 0.755645i \(0.727324\pi\)
\(350\) 0 0
\(351\) 11.2361 0.599737
\(352\) 0 0
\(353\) −17.2361 −0.917383 −0.458692 0.888595i \(-0.651682\pi\)
−0.458692 + 0.888595i \(0.651682\pi\)
\(354\) 0 0
\(355\) 11.2361 0.596349
\(356\) 0 0
\(357\) −0.618034 −0.0327098
\(358\) 0 0
\(359\) −0.562306 −0.0296774 −0.0148387 0.999890i \(-0.504723\pi\)
−0.0148387 + 0.999890i \(0.504723\pi\)
\(360\) 0 0
\(361\) −18.7771 −0.988268
\(362\) 0 0
\(363\) 10.1459 0.532522
\(364\) 0 0
\(365\) −22.4164 −1.17333
\(366\) 0 0
\(367\) 18.5066 0.966035 0.483018 0.875611i \(-0.339541\pi\)
0.483018 + 0.875611i \(0.339541\pi\)
\(368\) 0 0
\(369\) −29.0344 −1.51147
\(370\) 0 0
\(371\) 13.7984 0.716376
\(372\) 0 0
\(373\) −10.1459 −0.525335 −0.262667 0.964886i \(-0.584602\pi\)
−0.262667 + 0.964886i \(0.584602\pi\)
\(374\) 0 0
\(375\) 7.38197 0.381203
\(376\) 0 0
\(377\) −24.9443 −1.28470
\(378\) 0 0
\(379\) −20.6525 −1.06085 −0.530423 0.847733i \(-0.677967\pi\)
−0.530423 + 0.847733i \(0.677967\pi\)
\(380\) 0 0
\(381\) −0.708204 −0.0362824
\(382\) 0 0
\(383\) 0.652476 0.0333400 0.0166700 0.999861i \(-0.494694\pi\)
0.0166700 + 0.999861i \(0.494694\pi\)
\(384\) 0 0
\(385\) −8.47214 −0.431780
\(386\) 0 0
\(387\) 2.38197 0.121082
\(388\) 0 0
\(389\) −1.43769 −0.0728940 −0.0364470 0.999336i \(-0.511604\pi\)
−0.0364470 + 0.999336i \(0.511604\pi\)
\(390\) 0 0
\(391\) 5.70820 0.288676
\(392\) 0 0
\(393\) −4.94427 −0.249406
\(394\) 0 0
\(395\) 24.1803 1.21664
\(396\) 0 0
\(397\) 17.7984 0.893275 0.446637 0.894715i \(-0.352621\pi\)
0.446637 + 0.894715i \(0.352621\pi\)
\(398\) 0 0
\(399\) −0.291796 −0.0146081
\(400\) 0 0
\(401\) −1.52786 −0.0762979 −0.0381489 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(402\) 0 0
\(403\) 30.1803 1.50339
\(404\) 0 0
\(405\) −9.23607 −0.458944
\(406\) 0 0
\(407\) 44.3607 2.19888
\(408\) 0 0
\(409\) 14.1803 0.701173 0.350586 0.936530i \(-0.385982\pi\)
0.350586 + 0.936530i \(0.385982\pi\)
\(410\) 0 0
\(411\) 2.85410 0.140782
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 18.4721 0.906761
\(416\) 0 0
\(417\) 9.29180 0.455021
\(418\) 0 0
\(419\) −18.4508 −0.901383 −0.450691 0.892680i \(-0.648823\pi\)
−0.450691 + 0.892680i \(0.648823\pi\)
\(420\) 0 0
\(421\) 22.2148 1.08268 0.541341 0.840803i \(-0.317917\pi\)
0.541341 + 0.840803i \(0.317917\pi\)
\(422\) 0 0
\(423\) −1.23607 −0.0600997
\(424\) 0 0
\(425\) −2.38197 −0.115542
\(426\) 0 0
\(427\) −8.32624 −0.402935
\(428\) 0 0
\(429\) 10.4721 0.505599
\(430\) 0 0
\(431\) 22.9443 1.10519 0.552593 0.833451i \(-0.313638\pi\)
0.552593 + 0.833451i \(0.313638\pi\)
\(432\) 0 0
\(433\) −0.944272 −0.0453788 −0.0226894 0.999743i \(-0.507223\pi\)
−0.0226894 + 0.999743i \(0.507223\pi\)
\(434\) 0 0
\(435\) −7.70820 −0.369580
\(436\) 0 0
\(437\) 2.69505 0.128922
\(438\) 0 0
\(439\) 12.3262 0.588299 0.294150 0.955759i \(-0.404964\pi\)
0.294150 + 0.955759i \(0.404964\pi\)
\(440\) 0 0
\(441\) −2.61803 −0.124668
\(442\) 0 0
\(443\) −9.88854 −0.469819 −0.234909 0.972017i \(-0.575479\pi\)
−0.234909 + 0.972017i \(0.575479\pi\)
\(444\) 0 0
\(445\) −3.23607 −0.153404
\(446\) 0 0
\(447\) −7.03444 −0.332718
\(448\) 0 0
\(449\) 4.29180 0.202542 0.101271 0.994859i \(-0.467709\pi\)
0.101271 + 0.994859i \(0.467709\pi\)
\(450\) 0 0
\(451\) −58.0689 −2.73436
\(452\) 0 0
\(453\) −13.5066 −0.634594
\(454\) 0 0
\(455\) −5.23607 −0.245471
\(456\) 0 0
\(457\) −1.61803 −0.0756884 −0.0378442 0.999284i \(-0.512049\pi\)
−0.0378442 + 0.999284i \(0.512049\pi\)
\(458\) 0 0
\(459\) −3.47214 −0.162065
\(460\) 0 0
\(461\) 4.94427 0.230278 0.115139 0.993349i \(-0.463269\pi\)
0.115139 + 0.993349i \(0.463269\pi\)
\(462\) 0 0
\(463\) 16.0344 0.745184 0.372592 0.927995i \(-0.378469\pi\)
0.372592 + 0.927995i \(0.378469\pi\)
\(464\) 0 0
\(465\) 9.32624 0.432494
\(466\) 0 0
\(467\) 1.52786 0.0707011 0.0353506 0.999375i \(-0.488745\pi\)
0.0353506 + 0.999375i \(0.488745\pi\)
\(468\) 0 0
\(469\) 1.90983 0.0881878
\(470\) 0 0
\(471\) 12.4721 0.574686
\(472\) 0 0
\(473\) 4.76393 0.219046
\(474\) 0 0
\(475\) −1.12461 −0.0516007
\(476\) 0 0
\(477\) 36.1246 1.65403
\(478\) 0 0
\(479\) 9.27051 0.423580 0.211790 0.977315i \(-0.432071\pi\)
0.211790 + 0.977315i \(0.432071\pi\)
\(480\) 0 0
\(481\) 27.4164 1.25008
\(482\) 0 0
\(483\) −3.52786 −0.160523
\(484\) 0 0
\(485\) −6.32624 −0.287260
\(486\) 0 0
\(487\) 22.3607 1.01326 0.506630 0.862164i \(-0.330891\pi\)
0.506630 + 0.862164i \(0.330891\pi\)
\(488\) 0 0
\(489\) −4.76393 −0.215432
\(490\) 0 0
\(491\) 10.0902 0.455363 0.227681 0.973736i \(-0.426885\pi\)
0.227681 + 0.973736i \(0.426885\pi\)
\(492\) 0 0
\(493\) 7.70820 0.347160
\(494\) 0 0
\(495\) −22.1803 −0.996932
\(496\) 0 0
\(497\) 6.94427 0.311493
\(498\) 0 0
\(499\) 34.9443 1.56432 0.782160 0.623077i \(-0.214118\pi\)
0.782160 + 0.623077i \(0.214118\pi\)
\(500\) 0 0
\(501\) 10.2705 0.458853
\(502\) 0 0
\(503\) −8.38197 −0.373733 −0.186867 0.982385i \(-0.559833\pi\)
−0.186867 + 0.982385i \(0.559833\pi\)
\(504\) 0 0
\(505\) −13.7082 −0.610007
\(506\) 0 0
\(507\) −1.56231 −0.0693844
\(508\) 0 0
\(509\) −5.23607 −0.232085 −0.116042 0.993244i \(-0.537021\pi\)
−0.116042 + 0.993244i \(0.537021\pi\)
\(510\) 0 0
\(511\) −13.8541 −0.612869
\(512\) 0 0
\(513\) −1.63932 −0.0723778
\(514\) 0 0
\(515\) 18.4721 0.813980
\(516\) 0 0
\(517\) −2.47214 −0.108724
\(518\) 0 0
\(519\) −14.2705 −0.626406
\(520\) 0 0
\(521\) 15.7426 0.689698 0.344849 0.938658i \(-0.387930\pi\)
0.344849 + 0.938658i \(0.387930\pi\)
\(522\) 0 0
\(523\) 11.8197 0.516838 0.258419 0.966033i \(-0.416799\pi\)
0.258419 + 0.966033i \(0.416799\pi\)
\(524\) 0 0
\(525\) 1.47214 0.0642493
\(526\) 0 0
\(527\) −9.32624 −0.406257
\(528\) 0 0
\(529\) 9.58359 0.416678
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −35.8885 −1.55451
\(534\) 0 0
\(535\) −24.6525 −1.06582
\(536\) 0 0
\(537\) −12.7426 −0.549886
\(538\) 0 0
\(539\) −5.23607 −0.225533
\(540\) 0 0
\(541\) −0.763932 −0.0328440 −0.0164220 0.999865i \(-0.505228\pi\)
−0.0164220 + 0.999865i \(0.505228\pi\)
\(542\) 0 0
\(543\) −2.18034 −0.0935673
\(544\) 0 0
\(545\) 20.1803 0.864431
\(546\) 0 0
\(547\) −9.34752 −0.399671 −0.199836 0.979829i \(-0.564041\pi\)
−0.199836 + 0.979829i \(0.564041\pi\)
\(548\) 0 0
\(549\) −21.7984 −0.930332
\(550\) 0 0
\(551\) 3.63932 0.155040
\(552\) 0 0
\(553\) 14.9443 0.635495
\(554\) 0 0
\(555\) 8.47214 0.359622
\(556\) 0 0
\(557\) −10.3607 −0.438996 −0.219498 0.975613i \(-0.570442\pi\)
−0.219498 + 0.975613i \(0.570442\pi\)
\(558\) 0 0
\(559\) 2.94427 0.124529
\(560\) 0 0
\(561\) −3.23607 −0.136627
\(562\) 0 0
\(563\) 11.1246 0.468846 0.234423 0.972135i \(-0.424680\pi\)
0.234423 + 0.972135i \(0.424680\pi\)
\(564\) 0 0
\(565\) −9.70820 −0.408427
\(566\) 0 0
\(567\) −5.70820 −0.239722
\(568\) 0 0
\(569\) −5.49342 −0.230296 −0.115148 0.993348i \(-0.536734\pi\)
−0.115148 + 0.993348i \(0.536734\pi\)
\(570\) 0 0
\(571\) −0.472136 −0.0197583 −0.00987914 0.999951i \(-0.503145\pi\)
−0.00987914 + 0.999951i \(0.503145\pi\)
\(572\) 0 0
\(573\) −2.34752 −0.0980692
\(574\) 0 0
\(575\) −13.5967 −0.567024
\(576\) 0 0
\(577\) −23.1246 −0.962690 −0.481345 0.876531i \(-0.659852\pi\)
−0.481345 + 0.876531i \(0.659852\pi\)
\(578\) 0 0
\(579\) 5.05573 0.210109
\(580\) 0 0
\(581\) 11.4164 0.473632
\(582\) 0 0
\(583\) 72.2492 2.99226
\(584\) 0 0
\(585\) −13.7082 −0.566764
\(586\) 0 0
\(587\) 10.8328 0.447118 0.223559 0.974690i \(-0.428232\pi\)
0.223559 + 0.974690i \(0.428232\pi\)
\(588\) 0 0
\(589\) −4.40325 −0.181433
\(590\) 0 0
\(591\) −2.29180 −0.0942719
\(592\) 0 0
\(593\) 22.3607 0.918243 0.459122 0.888373i \(-0.348164\pi\)
0.459122 + 0.888373i \(0.348164\pi\)
\(594\) 0 0
\(595\) 1.61803 0.0663329
\(596\) 0 0
\(597\) −0.562306 −0.0230136
\(598\) 0 0
\(599\) 24.9098 1.01779 0.508894 0.860829i \(-0.330055\pi\)
0.508894 + 0.860829i \(0.330055\pi\)
\(600\) 0 0
\(601\) −32.2492 −1.31547 −0.657737 0.753248i \(-0.728486\pi\)
−0.657737 + 0.753248i \(0.728486\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) 0 0
\(605\) −26.5623 −1.07991
\(606\) 0 0
\(607\) −33.1459 −1.34535 −0.672675 0.739938i \(-0.734855\pi\)
−0.672675 + 0.739938i \(0.734855\pi\)
\(608\) 0 0
\(609\) −4.76393 −0.193044
\(610\) 0 0
\(611\) −1.52786 −0.0618108
\(612\) 0 0
\(613\) 2.56231 0.103491 0.0517453 0.998660i \(-0.483522\pi\)
0.0517453 + 0.998660i \(0.483522\pi\)
\(614\) 0 0
\(615\) −11.0902 −0.447199
\(616\) 0 0
\(617\) 41.0132 1.65113 0.825564 0.564309i \(-0.190857\pi\)
0.825564 + 0.564309i \(0.190857\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −19.8197 −0.795336
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) −7.41641 −0.296656
\(626\) 0 0
\(627\) −1.52786 −0.0610170
\(628\) 0 0
\(629\) −8.47214 −0.337806
\(630\) 0 0
\(631\) −25.9098 −1.03145 −0.515727 0.856753i \(-0.672478\pi\)
−0.515727 + 0.856753i \(0.672478\pi\)
\(632\) 0 0
\(633\) −9.70820 −0.385866
\(634\) 0 0
\(635\) 1.85410 0.0735778
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 18.1803 0.719203
\(640\) 0 0
\(641\) 38.9443 1.53821 0.769103 0.639125i \(-0.220703\pi\)
0.769103 + 0.639125i \(0.220703\pi\)
\(642\) 0 0
\(643\) 9.85410 0.388608 0.194304 0.980941i \(-0.437755\pi\)
0.194304 + 0.980941i \(0.437755\pi\)
\(644\) 0 0
\(645\) 0.909830 0.0358245
\(646\) 0 0
\(647\) −25.2361 −0.992132 −0.496066 0.868285i \(-0.665223\pi\)
−0.496066 + 0.868285i \(0.665223\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5.76393 0.225906
\(652\) 0 0
\(653\) −23.5279 −0.920716 −0.460358 0.887733i \(-0.652279\pi\)
−0.460358 + 0.887733i \(0.652279\pi\)
\(654\) 0 0
\(655\) 12.9443 0.505775
\(656\) 0 0
\(657\) −36.2705 −1.41505
\(658\) 0 0
\(659\) −40.7426 −1.58711 −0.793554 0.608500i \(-0.791772\pi\)
−0.793554 + 0.608500i \(0.791772\pi\)
\(660\) 0 0
\(661\) −22.8328 −0.888094 −0.444047 0.896004i \(-0.646458\pi\)
−0.444047 + 0.896004i \(0.646458\pi\)
\(662\) 0 0
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 0.763932 0.0296240
\(666\) 0 0
\(667\) 44.0000 1.70369
\(668\) 0 0
\(669\) −9.23607 −0.357087
\(670\) 0 0
\(671\) −43.5967 −1.68303
\(672\) 0 0
\(673\) −5.52786 −0.213083 −0.106542 0.994308i \(-0.533978\pi\)
−0.106542 + 0.994308i \(0.533978\pi\)
\(674\) 0 0
\(675\) 8.27051 0.318332
\(676\) 0 0
\(677\) 37.7771 1.45189 0.725946 0.687752i \(-0.241402\pi\)
0.725946 + 0.687752i \(0.241402\pi\)
\(678\) 0 0
\(679\) −3.90983 −0.150046
\(680\) 0 0
\(681\) −10.3475 −0.396518
\(682\) 0 0
\(683\) 5.81966 0.222683 0.111342 0.993782i \(-0.464485\pi\)
0.111342 + 0.993782i \(0.464485\pi\)
\(684\) 0 0
\(685\) −7.47214 −0.285496
\(686\) 0 0
\(687\) −9.12461 −0.348126
\(688\) 0 0
\(689\) 44.6525 1.70112
\(690\) 0 0
\(691\) 46.5066 1.76919 0.884597 0.466357i \(-0.154434\pi\)
0.884597 + 0.466357i \(0.154434\pi\)
\(692\) 0 0
\(693\) −13.7082 −0.520732
\(694\) 0 0
\(695\) −24.3262 −0.922747
\(696\) 0 0
\(697\) 11.0902 0.420070
\(698\) 0 0
\(699\) −1.52786 −0.0577891
\(700\) 0 0
\(701\) −18.5836 −0.701893 −0.350946 0.936396i \(-0.614140\pi\)
−0.350946 + 0.936396i \(0.614140\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) −0.472136 −0.0177817
\(706\) 0 0
\(707\) −8.47214 −0.318627
\(708\) 0 0
\(709\) 9.34752 0.351054 0.175527 0.984475i \(-0.443837\pi\)
0.175527 + 0.984475i \(0.443837\pi\)
\(710\) 0 0
\(711\) 39.1246 1.46729
\(712\) 0 0
\(713\) −53.2361 −1.99371
\(714\) 0 0
\(715\) −27.4164 −1.02532
\(716\) 0 0
\(717\) 14.2705 0.532942
\(718\) 0 0
\(719\) −45.8541 −1.71007 −0.855035 0.518571i \(-0.826464\pi\)
−0.855035 + 0.518571i \(0.826464\pi\)
\(720\) 0 0
\(721\) 11.4164 0.425169
\(722\) 0 0
\(723\) −15.4721 −0.575415
\(724\) 0 0
\(725\) −18.3607 −0.681899
\(726\) 0 0
\(727\) 7.52786 0.279193 0.139597 0.990208i \(-0.455419\pi\)
0.139597 + 0.990208i \(0.455419\pi\)
\(728\) 0 0
\(729\) −8.50658 −0.315058
\(730\) 0 0
\(731\) −0.909830 −0.0336513
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −21.5623 −0.793182 −0.396591 0.917995i \(-0.629807\pi\)
−0.396591 + 0.917995i \(0.629807\pi\)
\(740\) 0 0
\(741\) −0.944272 −0.0346887
\(742\) 0 0
\(743\) −26.6525 −0.977785 −0.488892 0.872344i \(-0.662599\pi\)
−0.488892 + 0.872344i \(0.662599\pi\)
\(744\) 0 0
\(745\) 18.4164 0.674725
\(746\) 0 0
\(747\) 29.8885 1.09356
\(748\) 0 0
\(749\) −15.2361 −0.556714
\(750\) 0 0
\(751\) −6.94427 −0.253400 −0.126700 0.991941i \(-0.540439\pi\)
−0.126700 + 0.991941i \(0.540439\pi\)
\(752\) 0 0
\(753\) 5.81966 0.212080
\(754\) 0 0
\(755\) 35.3607 1.28691
\(756\) 0 0
\(757\) −29.5623 −1.07446 −0.537230 0.843436i \(-0.680529\pi\)
−0.537230 + 0.843436i \(0.680529\pi\)
\(758\) 0 0
\(759\) −18.4721 −0.670496
\(760\) 0 0
\(761\) 7.81966 0.283462 0.141731 0.989905i \(-0.454733\pi\)
0.141731 + 0.989905i \(0.454733\pi\)
\(762\) 0 0
\(763\) 12.4721 0.451522
\(764\) 0 0
\(765\) 4.23607 0.153155
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 33.1246 1.19450 0.597252 0.802054i \(-0.296259\pi\)
0.597252 + 0.802054i \(0.296259\pi\)
\(770\) 0 0
\(771\) 7.59675 0.273590
\(772\) 0 0
\(773\) −21.4164 −0.770295 −0.385147 0.922855i \(-0.625849\pi\)
−0.385147 + 0.922855i \(0.625849\pi\)
\(774\) 0 0
\(775\) 22.2148 0.797979
\(776\) 0 0
\(777\) 5.23607 0.187843
\(778\) 0 0
\(779\) 5.23607 0.187602
\(780\) 0 0
\(781\) 36.3607 1.30109
\(782\) 0 0
\(783\) −26.7639 −0.956465
\(784\) 0 0
\(785\) −32.6525 −1.16542
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 0 0
\(789\) 0.583592 0.0207764
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −26.9443 −0.956819
\(794\) 0 0
\(795\) 13.7984 0.489378
\(796\) 0 0
\(797\) −11.4164 −0.404390 −0.202195 0.979345i \(-0.564807\pi\)
−0.202195 + 0.979345i \(0.564807\pi\)
\(798\) 0 0
\(799\) 0.472136 0.0167030
\(800\) 0 0
\(801\) −5.23607 −0.185007
\(802\) 0 0
\(803\) −72.5410 −2.55992
\(804\) 0 0
\(805\) 9.23607 0.325529
\(806\) 0 0
\(807\) −6.18034 −0.217558
\(808\) 0 0
\(809\) 31.4853 1.10696 0.553482 0.832861i \(-0.313299\pi\)
0.553482 + 0.832861i \(0.313299\pi\)
\(810\) 0 0
\(811\) −32.0344 −1.12488 −0.562441 0.826838i \(-0.690138\pi\)
−0.562441 + 0.826838i \(0.690138\pi\)
\(812\) 0 0
\(813\) −6.00000 −0.210429
\(814\) 0 0
\(815\) 12.4721 0.436880
\(816\) 0 0
\(817\) −0.429563 −0.0150285
\(818\) 0 0
\(819\) −8.47214 −0.296040
\(820\) 0 0
\(821\) −30.3607 −1.05960 −0.529798 0.848124i \(-0.677732\pi\)
−0.529798 + 0.848124i \(0.677732\pi\)
\(822\) 0 0
\(823\) −37.4853 −1.30666 −0.653328 0.757075i \(-0.726628\pi\)
−0.653328 + 0.757075i \(0.726628\pi\)
\(824\) 0 0
\(825\) 7.70820 0.268365
\(826\) 0 0
\(827\) 36.4721 1.26826 0.634130 0.773226i \(-0.281358\pi\)
0.634130 + 0.773226i \(0.281358\pi\)
\(828\) 0 0
\(829\) 16.1803 0.561966 0.280983 0.959713i \(-0.409339\pi\)
0.280983 + 0.959713i \(0.409339\pi\)
\(830\) 0 0
\(831\) 4.47214 0.155137
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −26.8885 −0.930516
\(836\) 0 0
\(837\) 32.3820 1.11928
\(838\) 0 0
\(839\) −22.4721 −0.775824 −0.387912 0.921696i \(-0.626804\pi\)
−0.387912 + 0.921696i \(0.626804\pi\)
\(840\) 0 0
\(841\) 30.4164 1.04884
\(842\) 0 0
\(843\) −12.4164 −0.427644
\(844\) 0 0
\(845\) 4.09017 0.140706
\(846\) 0 0
\(847\) −16.4164 −0.564074
\(848\) 0 0
\(849\) −15.1803 −0.520988
\(850\) 0 0
\(851\) −48.3607 −1.65778
\(852\) 0 0
\(853\) 6.36068 0.217786 0.108893 0.994054i \(-0.465269\pi\)
0.108893 + 0.994054i \(0.465269\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −3.90983 −0.133557 −0.0667786 0.997768i \(-0.521272\pi\)
−0.0667786 + 0.997768i \(0.521272\pi\)
\(858\) 0 0
\(859\) −2.94427 −0.100457 −0.0502286 0.998738i \(-0.515995\pi\)
−0.0502286 + 0.998738i \(0.515995\pi\)
\(860\) 0 0
\(861\) −6.85410 −0.233587
\(862\) 0 0
\(863\) −30.6738 −1.04415 −0.522074 0.852900i \(-0.674841\pi\)
−0.522074 + 0.852900i \(0.674841\pi\)
\(864\) 0 0
\(865\) 37.3607 1.27030
\(866\) 0 0
\(867\) 0.618034 0.0209895
\(868\) 0 0
\(869\) 78.2492 2.65442
\(870\) 0 0
\(871\) 6.18034 0.209413
\(872\) 0 0
\(873\) −10.2361 −0.346438
\(874\) 0 0
\(875\) −11.9443 −0.403790
\(876\) 0 0
\(877\) 31.7082 1.07071 0.535355 0.844627i \(-0.320178\pi\)
0.535355 + 0.844627i \(0.320178\pi\)
\(878\) 0 0
\(879\) −12.1803 −0.410833
\(880\) 0 0
\(881\) −35.3262 −1.19017 −0.595086 0.803662i \(-0.702882\pi\)
−0.595086 + 0.803662i \(0.702882\pi\)
\(882\) 0 0
\(883\) −31.9787 −1.07617 −0.538085 0.842891i \(-0.680852\pi\)
−0.538085 + 0.842891i \(0.680852\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.5066 0.722120 0.361060 0.932543i \(-0.382415\pi\)
0.361060 + 0.932543i \(0.382415\pi\)
\(888\) 0 0
\(889\) 1.14590 0.0384322
\(890\) 0 0
\(891\) −29.8885 −1.00130
\(892\) 0 0
\(893\) 0.222912 0.00745948
\(894\) 0 0
\(895\) 33.3607 1.11512
\(896\) 0 0
\(897\) −11.4164 −0.381183
\(898\) 0 0
\(899\) −71.8885 −2.39762
\(900\) 0 0
\(901\) −13.7984 −0.459690
\(902\) 0 0
\(903\) 0.562306 0.0187124
\(904\) 0 0
\(905\) 5.70820 0.189747
\(906\) 0 0
\(907\) 31.3050 1.03946 0.519732 0.854329i \(-0.326032\pi\)
0.519732 + 0.854329i \(0.326032\pi\)
\(908\) 0 0
\(909\) −22.1803 −0.735675
\(910\) 0 0
\(911\) −5.52786 −0.183146 −0.0915732 0.995798i \(-0.529190\pi\)
−0.0915732 + 0.995798i \(0.529190\pi\)
\(912\) 0 0
\(913\) 59.7771 1.97833
\(914\) 0 0
\(915\) −8.32624 −0.275257
\(916\) 0 0
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 39.1591 1.29174 0.645869 0.763448i \(-0.276495\pi\)
0.645869 + 0.763448i \(0.276495\pi\)
\(920\) 0 0
\(921\) 11.7082 0.385798
\(922\) 0 0
\(923\) 22.4721 0.739679
\(924\) 0 0
\(925\) 20.1803 0.663525
\(926\) 0 0
\(927\) 29.8885 0.981669
\(928\) 0 0
\(929\) −12.2705 −0.402582 −0.201291 0.979531i \(-0.564514\pi\)
−0.201291 + 0.979531i \(0.564514\pi\)
\(930\) 0 0
\(931\) 0.472136 0.0154736
\(932\) 0 0
\(933\) 15.0689 0.493333
\(934\) 0 0
\(935\) 8.47214 0.277068
\(936\) 0 0
\(937\) 57.3050 1.87207 0.936036 0.351905i \(-0.114466\pi\)
0.936036 + 0.351905i \(0.114466\pi\)
\(938\) 0 0
\(939\) 19.9443 0.650857
\(940\) 0 0
\(941\) −0.978714 −0.0319052 −0.0159526 0.999873i \(-0.505078\pi\)
−0.0159526 + 0.999873i \(0.505078\pi\)
\(942\) 0 0
\(943\) 63.3050 2.06149
\(944\) 0 0
\(945\) −5.61803 −0.182755
\(946\) 0 0
\(947\) 47.1246 1.53134 0.765672 0.643231i \(-0.222407\pi\)
0.765672 + 0.643231i \(0.222407\pi\)
\(948\) 0 0
\(949\) −44.8328 −1.45533
\(950\) 0 0
\(951\) −3.59675 −0.116633
\(952\) 0 0
\(953\) 16.6738 0.540116 0.270058 0.962844i \(-0.412957\pi\)
0.270058 + 0.962844i \(0.412957\pi\)
\(954\) 0 0
\(955\) 6.14590 0.198877
\(956\) 0 0
\(957\) −24.9443 −0.806334
\(958\) 0 0
\(959\) −4.61803 −0.149124
\(960\) 0 0
\(961\) 55.9787 1.80576
\(962\) 0 0
\(963\) −39.8885 −1.28539
\(964\) 0 0
\(965\) −13.2361 −0.426084
\(966\) 0 0
\(967\) 13.2016 0.424536 0.212268 0.977212i \(-0.431915\pi\)
0.212268 + 0.977212i \(0.431915\pi\)
\(968\) 0 0
\(969\) 0.291796 0.00937384
\(970\) 0 0
\(971\) −19.1246 −0.613738 −0.306869 0.951752i \(-0.599281\pi\)
−0.306869 + 0.951752i \(0.599281\pi\)
\(972\) 0 0
\(973\) −15.0344 −0.481982
\(974\) 0 0
\(975\) 4.76393 0.152568
\(976\) 0 0
\(977\) 49.2148 1.57452 0.787260 0.616621i \(-0.211499\pi\)
0.787260 + 0.616621i \(0.211499\pi\)
\(978\) 0 0
\(979\) −10.4721 −0.334691
\(980\) 0 0
\(981\) 32.6525 1.04251
\(982\) 0 0
\(983\) 18.0902 0.576987 0.288493 0.957482i \(-0.406846\pi\)
0.288493 + 0.957482i \(0.406846\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −0.291796 −0.00928797
\(988\) 0 0
\(989\) −5.19350 −0.165144
\(990\) 0 0
\(991\) 52.2492 1.65975 0.829876 0.557948i \(-0.188411\pi\)
0.829876 + 0.557948i \(0.188411\pi\)
\(992\) 0 0
\(993\) −6.81966 −0.216415
\(994\) 0 0
\(995\) 1.47214 0.0466698
\(996\) 0 0
\(997\) −0.0344419 −0.00109078 −0.000545392 1.00000i \(-0.500174\pi\)
−0.000545392 1.00000i \(0.500174\pi\)
\(998\) 0 0
\(999\) 29.4164 0.930694
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 476.2.a.b.1.2 2
3.2 odd 2 4284.2.a.m.1.2 2
4.3 odd 2 1904.2.a.j.1.1 2
7.6 odd 2 3332.2.a.l.1.1 2
8.3 odd 2 7616.2.a.p.1.2 2
8.5 even 2 7616.2.a.u.1.1 2
17.16 even 2 8092.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.b.1.2 2 1.1 even 1 trivial
1904.2.a.j.1.1 2 4.3 odd 2
3332.2.a.l.1.1 2 7.6 odd 2
4284.2.a.m.1.2 2 3.2 odd 2
7616.2.a.p.1.2 2 8.3 odd 2
7616.2.a.u.1.1 2 8.5 even 2
8092.2.a.m.1.1 2 17.16 even 2