Properties

Label 1904.2.a.m.1.1
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
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] = [1904,2,Mod(1,1904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1904, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.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: \(15.2035165449\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 1904.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.93543 q^{3} +1.25410 q^{5} -1.00000 q^{7} +5.61676 q^{9} +O(q^{10})\) \(q-2.93543 q^{3} +1.25410 q^{5} -1.00000 q^{7} +5.61676 q^{9} -2.50820 q^{11} -3.68133 q^{15} -1.00000 q^{17} +3.87086 q^{19} +2.93543 q^{21} +4.37907 q^{23} -3.42723 q^{25} -7.68133 q^{27} -7.87086 q^{29} -1.25410 q^{31} +7.36266 q^{33} -1.25410 q^{35} +3.87086 q^{37} +9.28169 q^{41} +1.41082 q^{43} +7.04399 q^{45} +7.74173 q^{47} +1.00000 q^{49} +2.93543 q^{51} +1.38324 q^{53} -3.14554 q^{55} -11.3627 q^{57} -8.34625 q^{59} -11.6608 q^{61} -5.61676 q^{63} -10.9630 q^{67} -12.8545 q^{69} -0.475390 q^{71} +0.427229 q^{73} +10.0604 q^{75} +2.50820 q^{77} -8.00000 q^{79} +5.69774 q^{81} +1.83805 q^{83} -1.25410 q^{85} +23.1044 q^{87} -2.37907 q^{89} +3.68133 q^{93} +4.85446 q^{95} -1.12497 q^{97} -14.0880 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100}) \) 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
\(3\) −2.93543 −1.69477 −0.847386 0.530977i \(-0.821825\pi\)
−0.847386 + 0.530977i \(0.821825\pi\)
\(4\) 0 0
\(5\) 1.25410 0.560851 0.280426 0.959876i \(-0.409524\pi\)
0.280426 + 0.959876i \(0.409524\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 5.61676 1.87225
\(10\) 0 0
\(11\) −2.50820 −0.756252 −0.378126 0.925754i \(-0.623431\pi\)
−0.378126 + 0.925754i \(0.623431\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.68133 −0.950515
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 3.87086 0.888037 0.444019 0.896018i \(-0.353552\pi\)
0.444019 + 0.896018i \(0.353552\pi\)
\(20\) 0 0
\(21\) 2.93543 0.640564
\(22\) 0 0
\(23\) 4.37907 0.913099 0.456549 0.889698i \(-0.349085\pi\)
0.456549 + 0.889698i \(0.349085\pi\)
\(24\) 0 0
\(25\) −3.42723 −0.685446
\(26\) 0 0
\(27\) −7.68133 −1.47827
\(28\) 0 0
\(29\) −7.87086 −1.46158 −0.730791 0.682601i \(-0.760849\pi\)
−0.730791 + 0.682601i \(0.760849\pi\)
\(30\) 0 0
\(31\) −1.25410 −0.225243 −0.112622 0.993638i \(-0.535925\pi\)
−0.112622 + 0.993638i \(0.535925\pi\)
\(32\) 0 0
\(33\) 7.36266 1.28167
\(34\) 0 0
\(35\) −1.25410 −0.211982
\(36\) 0 0
\(37\) 3.87086 0.636366 0.318183 0.948029i \(-0.396927\pi\)
0.318183 + 0.948029i \(0.396927\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.28169 1.44956 0.724778 0.688982i \(-0.241942\pi\)
0.724778 + 0.688982i \(0.241942\pi\)
\(42\) 0 0
\(43\) 1.41082 0.215148 0.107574 0.994197i \(-0.465692\pi\)
0.107574 + 0.994197i \(0.465692\pi\)
\(44\) 0 0
\(45\) 7.04399 1.05006
\(46\) 0 0
\(47\) 7.74173 1.12925 0.564624 0.825348i \(-0.309021\pi\)
0.564624 + 0.825348i \(0.309021\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.93543 0.411043
\(52\) 0 0
\(53\) 1.38324 0.190002 0.0950011 0.995477i \(-0.469715\pi\)
0.0950011 + 0.995477i \(0.469715\pi\)
\(54\) 0 0
\(55\) −3.14554 −0.424145
\(56\) 0 0
\(57\) −11.3627 −1.50502
\(58\) 0 0
\(59\) −8.34625 −1.08659 −0.543295 0.839542i \(-0.682823\pi\)
−0.543295 + 0.839542i \(0.682823\pi\)
\(60\) 0 0
\(61\) −11.6608 −1.49301 −0.746503 0.665382i \(-0.768269\pi\)
−0.746503 + 0.665382i \(0.768269\pi\)
\(62\) 0 0
\(63\) −5.61676 −0.707646
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.9630 −1.33935 −0.669673 0.742656i \(-0.733566\pi\)
−0.669673 + 0.742656i \(0.733566\pi\)
\(68\) 0 0
\(69\) −12.8545 −1.54749
\(70\) 0 0
\(71\) −0.475390 −0.0564184 −0.0282092 0.999602i \(-0.508980\pi\)
−0.0282092 + 0.999602i \(0.508980\pi\)
\(72\) 0 0
\(73\) 0.427229 0.0500034 0.0250017 0.999687i \(-0.492041\pi\)
0.0250017 + 0.999687i \(0.492041\pi\)
\(74\) 0 0
\(75\) 10.0604 1.16167
\(76\) 0 0
\(77\) 2.50820 0.285836
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 5.69774 0.633082
\(82\) 0 0
\(83\) 1.83805 0.201752 0.100876 0.994899i \(-0.467835\pi\)
0.100876 + 0.994899i \(0.467835\pi\)
\(84\) 0 0
\(85\) −1.25410 −0.136026
\(86\) 0 0
\(87\) 23.1044 2.47705
\(88\) 0 0
\(89\) −2.37907 −0.252181 −0.126090 0.992019i \(-0.540243\pi\)
−0.126090 + 0.992019i \(0.540243\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.68133 0.381736
\(94\) 0 0
\(95\) 4.85446 0.498057
\(96\) 0 0
\(97\) −1.12497 −0.114223 −0.0571115 0.998368i \(-0.518189\pi\)
−0.0571115 + 0.998368i \(0.518189\pi\)
\(98\) 0 0
\(99\) −14.0880 −1.41590
\(100\) 0 0
\(101\) −4.47539 −0.445318 −0.222659 0.974896i \(-0.571474\pi\)
−0.222659 + 0.974896i \(0.571474\pi\)
\(102\) 0 0
\(103\) 10.2499 1.00996 0.504978 0.863132i \(-0.331501\pi\)
0.504978 + 0.863132i \(0.331501\pi\)
\(104\) 0 0
\(105\) 3.68133 0.359261
\(106\) 0 0
\(107\) −15.7417 −1.52181 −0.760905 0.648863i \(-0.775245\pi\)
−0.760905 + 0.648863i \(0.775245\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −11.3627 −1.07850
\(112\) 0 0
\(113\) −16.0880 −1.51343 −0.756715 0.653745i \(-0.773197\pi\)
−0.756715 + 0.653745i \(0.773197\pi\)
\(114\) 0 0
\(115\) 5.49180 0.512113
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −4.70892 −0.428083
\(122\) 0 0
\(123\) −27.2458 −2.45667
\(124\) 0 0
\(125\) −10.5686 −0.945285
\(126\) 0 0
\(127\) −3.50403 −0.310933 −0.155466 0.987841i \(-0.549688\pi\)
−0.155466 + 0.987841i \(0.549688\pi\)
\(128\) 0 0
\(129\) −4.14137 −0.364628
\(130\) 0 0
\(131\) −17.1044 −1.49442 −0.747209 0.664589i \(-0.768606\pi\)
−0.747209 + 0.664589i \(0.768606\pi\)
\(132\) 0 0
\(133\) −3.87086 −0.335647
\(134\) 0 0
\(135\) −9.63317 −0.829091
\(136\) 0 0
\(137\) 17.0234 1.45441 0.727204 0.686421i \(-0.240819\pi\)
0.727204 + 0.686421i \(0.240819\pi\)
\(138\) 0 0
\(139\) 5.12497 0.434694 0.217347 0.976094i \(-0.430260\pi\)
0.217347 + 0.976094i \(0.430260\pi\)
\(140\) 0 0
\(141\) −22.7253 −1.91382
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.87086 −0.819731
\(146\) 0 0
\(147\) −2.93543 −0.242110
\(148\) 0 0
\(149\) −1.70191 −0.139426 −0.0697128 0.997567i \(-0.522208\pi\)
−0.0697128 + 0.997567i \(0.522208\pi\)
\(150\) 0 0
\(151\) −0.298094 −0.0242585 −0.0121293 0.999926i \(-0.503861\pi\)
−0.0121293 + 0.999926i \(0.503861\pi\)
\(152\) 0 0
\(153\) −5.61676 −0.454088
\(154\) 0 0
\(155\) −1.57277 −0.126328
\(156\) 0 0
\(157\) 20.3379 1.62314 0.811571 0.584253i \(-0.198613\pi\)
0.811571 + 0.584253i \(0.198613\pi\)
\(158\) 0 0
\(159\) −4.06040 −0.322011
\(160\) 0 0
\(161\) −4.37907 −0.345119
\(162\) 0 0
\(163\) −10.3463 −0.810381 −0.405191 0.914232i \(-0.632795\pi\)
−0.405191 + 0.914232i \(0.632795\pi\)
\(164\) 0 0
\(165\) 9.23353 0.718829
\(166\) 0 0
\(167\) −17.3145 −1.33984 −0.669918 0.742435i \(-0.733671\pi\)
−0.669918 + 0.742435i \(0.733671\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 21.7417 1.66263
\(172\) 0 0
\(173\) 2.04816 0.155719 0.0778594 0.996964i \(-0.475191\pi\)
0.0778594 + 0.996964i \(0.475191\pi\)
\(174\) 0 0
\(175\) 3.42723 0.259074
\(176\) 0 0
\(177\) 24.4999 1.84152
\(178\) 0 0
\(179\) −17.9107 −1.33871 −0.669354 0.742944i \(-0.733429\pi\)
−0.669354 + 0.742944i \(0.733429\pi\)
\(180\) 0 0
\(181\) −17.3955 −1.29300 −0.646498 0.762916i \(-0.723767\pi\)
−0.646498 + 0.762916i \(0.723767\pi\)
\(182\) 0 0
\(183\) 34.2294 2.53031
\(184\) 0 0
\(185\) 4.85446 0.356907
\(186\) 0 0
\(187\) 2.50820 0.183418
\(188\) 0 0
\(189\) 7.68133 0.558735
\(190\) 0 0
\(191\) −5.25410 −0.380173 −0.190087 0.981767i \(-0.560877\pi\)
−0.190087 + 0.981767i \(0.560877\pi\)
\(192\) 0 0
\(193\) −23.1372 −1.66545 −0.832726 0.553685i \(-0.813221\pi\)
−0.832726 + 0.553685i \(0.813221\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.03281 −0.287326 −0.143663 0.989627i \(-0.545888\pi\)
−0.143663 + 0.989627i \(0.545888\pi\)
\(198\) 0 0
\(199\) −1.31450 −0.0931825 −0.0465912 0.998914i \(-0.514836\pi\)
−0.0465912 + 0.998914i \(0.514836\pi\)
\(200\) 0 0
\(201\) 32.1812 2.26989
\(202\) 0 0
\(203\) 7.87086 0.552426
\(204\) 0 0
\(205\) 11.6402 0.812985
\(206\) 0 0
\(207\) 24.5962 1.70955
\(208\) 0 0
\(209\) −9.70892 −0.671580
\(210\) 0 0
\(211\) 10.8873 0.749511 0.374755 0.927124i \(-0.377727\pi\)
0.374755 + 0.927124i \(0.377727\pi\)
\(212\) 0 0
\(213\) 1.39547 0.0956164
\(214\) 0 0
\(215\) 1.76931 0.120666
\(216\) 0 0
\(217\) 1.25410 0.0851340
\(218\) 0 0
\(219\) −1.25410 −0.0847443
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.45898 −0.231631 −0.115815 0.993271i \(-0.536948\pi\)
−0.115815 + 0.993271i \(0.536948\pi\)
\(224\) 0 0
\(225\) −19.2499 −1.28333
\(226\) 0 0
\(227\) 23.8503 1.58300 0.791500 0.611170i \(-0.209301\pi\)
0.791500 + 0.611170i \(0.209301\pi\)
\(228\) 0 0
\(229\) 5.49180 0.362908 0.181454 0.983399i \(-0.441920\pi\)
0.181454 + 0.983399i \(0.441920\pi\)
\(230\) 0 0
\(231\) −7.36266 −0.484428
\(232\) 0 0
\(233\) −18.9753 −1.24311 −0.621555 0.783370i \(-0.713499\pi\)
−0.621555 + 0.783370i \(0.713499\pi\)
\(234\) 0 0
\(235\) 9.70892 0.633340
\(236\) 0 0
\(237\) 23.4835 1.52541
\(238\) 0 0
\(239\) 30.4272 1.96817 0.984087 0.177688i \(-0.0568617\pi\)
0.984087 + 0.177688i \(0.0568617\pi\)
\(240\) 0 0
\(241\) −10.8339 −0.697872 −0.348936 0.937147i \(-0.613457\pi\)
−0.348936 + 0.937147i \(0.613457\pi\)
\(242\) 0 0
\(243\) 6.31867 0.405343
\(244\) 0 0
\(245\) 1.25410 0.0801216
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.39547 −0.341924
\(250\) 0 0
\(251\) −10.1619 −0.641417 −0.320708 0.947178i \(-0.603921\pi\)
−0.320708 + 0.947178i \(0.603921\pi\)
\(252\) 0 0
\(253\) −10.9836 −0.690533
\(254\) 0 0
\(255\) 3.68133 0.230534
\(256\) 0 0
\(257\) 10.6373 0.663539 0.331769 0.943360i \(-0.392354\pi\)
0.331769 + 0.943360i \(0.392354\pi\)
\(258\) 0 0
\(259\) −3.87086 −0.240524
\(260\) 0 0
\(261\) −44.2088 −2.73645
\(262\) 0 0
\(263\) −5.96719 −0.367952 −0.183976 0.982931i \(-0.558897\pi\)
−0.183976 + 0.982931i \(0.558897\pi\)
\(264\) 0 0
\(265\) 1.73472 0.106563
\(266\) 0 0
\(267\) 6.98359 0.427389
\(268\) 0 0
\(269\) 26.1536 1.59461 0.797307 0.603574i \(-0.206257\pi\)
0.797307 + 0.603574i \(0.206257\pi\)
\(270\) 0 0
\(271\) −16.1208 −0.979269 −0.489634 0.871928i \(-0.662870\pi\)
−0.489634 + 0.871928i \(0.662870\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.59619 0.518370
\(276\) 0 0
\(277\) −26.7581 −1.60774 −0.803870 0.594805i \(-0.797229\pi\)
−0.803870 + 0.594805i \(0.797229\pi\)
\(278\) 0 0
\(279\) −7.04399 −0.421713
\(280\) 0 0
\(281\) 28.0674 1.67436 0.837181 0.546927i \(-0.184202\pi\)
0.837181 + 0.546927i \(0.184202\pi\)
\(282\) 0 0
\(283\) 30.1002 1.78927 0.894636 0.446795i \(-0.147435\pi\)
0.894636 + 0.446795i \(0.147435\pi\)
\(284\) 0 0
\(285\) −14.2499 −0.844093
\(286\) 0 0
\(287\) −9.28169 −0.547881
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 3.30226 0.193582
\(292\) 0 0
\(293\) −24.4999 −1.43130 −0.715649 0.698460i \(-0.753869\pi\)
−0.715649 + 0.698460i \(0.753869\pi\)
\(294\) 0 0
\(295\) −10.4671 −0.609415
\(296\) 0 0
\(297\) 19.2663 1.11795
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.41082 −0.0813184
\(302\) 0 0
\(303\) 13.1372 0.754713
\(304\) 0 0
\(305\) −14.6238 −0.837355
\(306\) 0 0
\(307\) −2.21712 −0.126538 −0.0632688 0.997997i \(-0.520153\pi\)
−0.0632688 + 0.997997i \(0.520153\pi\)
\(308\) 0 0
\(309\) −30.0880 −1.71165
\(310\) 0 0
\(311\) −13.2130 −0.749238 −0.374619 0.927179i \(-0.622226\pi\)
−0.374619 + 0.927179i \(0.622226\pi\)
\(312\) 0 0
\(313\) 33.3421 1.88461 0.942303 0.334761i \(-0.108656\pi\)
0.942303 + 0.334761i \(0.108656\pi\)
\(314\) 0 0
\(315\) −7.04399 −0.396884
\(316\) 0 0
\(317\) 28.8461 1.62016 0.810080 0.586320i \(-0.199424\pi\)
0.810080 + 0.586320i \(0.199424\pi\)
\(318\) 0 0
\(319\) 19.7417 1.10532
\(320\) 0 0
\(321\) 46.2088 2.57912
\(322\) 0 0
\(323\) −3.87086 −0.215381
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 41.0961 2.27262
\(328\) 0 0
\(329\) −7.74173 −0.426815
\(330\) 0 0
\(331\) −7.00417 −0.384984 −0.192492 0.981299i \(-0.561657\pi\)
−0.192492 + 0.981299i \(0.561657\pi\)
\(332\) 0 0
\(333\) 21.7417 1.19144
\(334\) 0 0
\(335\) −13.7487 −0.751174
\(336\) 0 0
\(337\) 28.8461 1.57135 0.785674 0.618641i \(-0.212316\pi\)
0.785674 + 0.618641i \(0.212316\pi\)
\(338\) 0 0
\(339\) 47.2252 2.56492
\(340\) 0 0
\(341\) 3.14554 0.170341
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −16.1208 −0.867915
\(346\) 0 0
\(347\) 16.1208 0.865410 0.432705 0.901536i \(-0.357559\pi\)
0.432705 + 0.901536i \(0.357559\pi\)
\(348\) 0 0
\(349\) −19.9037 −1.06542 −0.532710 0.846298i \(-0.678826\pi\)
−0.532710 + 0.846298i \(0.678826\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.1127 0.591471 0.295735 0.955270i \(-0.404435\pi\)
0.295735 + 0.955270i \(0.404435\pi\)
\(354\) 0 0
\(355\) −0.596187 −0.0316423
\(356\) 0 0
\(357\) −2.93543 −0.155360
\(358\) 0 0
\(359\) 6.96302 0.367494 0.183747 0.982974i \(-0.441177\pi\)
0.183747 + 0.982974i \(0.441177\pi\)
\(360\) 0 0
\(361\) −4.01641 −0.211390
\(362\) 0 0
\(363\) 13.8227 0.725504
\(364\) 0 0
\(365\) 0.535789 0.0280445
\(366\) 0 0
\(367\) −3.60558 −0.188210 −0.0941050 0.995562i \(-0.529999\pi\)
−0.0941050 + 0.995562i \(0.529999\pi\)
\(368\) 0 0
\(369\) 52.1330 2.71394
\(370\) 0 0
\(371\) −1.38324 −0.0718141
\(372\) 0 0
\(373\) −18.2981 −0.947439 −0.473720 0.880676i \(-0.657089\pi\)
−0.473720 + 0.880676i \(0.657089\pi\)
\(374\) 0 0
\(375\) 31.0234 1.60204
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.23353 0.0633620 0.0316810 0.999498i \(-0.489914\pi\)
0.0316810 + 0.999498i \(0.489914\pi\)
\(380\) 0 0
\(381\) 10.2859 0.526960
\(382\) 0 0
\(383\) −5.55742 −0.283971 −0.141986 0.989869i \(-0.545349\pi\)
−0.141986 + 0.989869i \(0.545349\pi\)
\(384\) 0 0
\(385\) 3.14554 0.160312
\(386\) 0 0
\(387\) 7.92425 0.402812
\(388\) 0 0
\(389\) 5.38324 0.272941 0.136470 0.990644i \(-0.456424\pi\)
0.136470 + 0.990644i \(0.456424\pi\)
\(390\) 0 0
\(391\) −4.37907 −0.221459
\(392\) 0 0
\(393\) 50.2088 2.53270
\(394\) 0 0
\(395\) −10.0328 −0.504806
\(396\) 0 0
\(397\) −27.1801 −1.36413 −0.682066 0.731291i \(-0.738918\pi\)
−0.682066 + 0.731291i \(0.738918\pi\)
\(398\) 0 0
\(399\) 11.3627 0.568845
\(400\) 0 0
\(401\) 4.44258 0.221852 0.110926 0.993829i \(-0.464618\pi\)
0.110926 + 0.993829i \(0.464618\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 7.14554 0.355065
\(406\) 0 0
\(407\) −9.70892 −0.481253
\(408\) 0 0
\(409\) 9.26634 0.458191 0.229095 0.973404i \(-0.426423\pi\)
0.229095 + 0.973404i \(0.426423\pi\)
\(410\) 0 0
\(411\) −49.9711 −2.46489
\(412\) 0 0
\(413\) 8.34625 0.410692
\(414\) 0 0
\(415\) 2.30510 0.113153
\(416\) 0 0
\(417\) −15.0440 −0.736707
\(418\) 0 0
\(419\) −8.36683 −0.408746 −0.204373 0.978893i \(-0.565516\pi\)
−0.204373 + 0.978893i \(0.565516\pi\)
\(420\) 0 0
\(421\) −25.3473 −1.23535 −0.617676 0.786432i \(-0.711926\pi\)
−0.617676 + 0.786432i \(0.711926\pi\)
\(422\) 0 0
\(423\) 43.4835 2.11424
\(424\) 0 0
\(425\) 3.42723 0.166245
\(426\) 0 0
\(427\) 11.6608 0.564303
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.1291 0.680577 0.340288 0.940321i \(-0.389475\pi\)
0.340288 + 0.940321i \(0.389475\pi\)
\(432\) 0 0
\(433\) 2.25827 0.108526 0.0542628 0.998527i \(-0.482719\pi\)
0.0542628 + 0.998527i \(0.482719\pi\)
\(434\) 0 0
\(435\) 28.9753 1.38926
\(436\) 0 0
\(437\) 16.9508 0.810866
\(438\) 0 0
\(439\) −35.8779 −1.71236 −0.856179 0.516680i \(-0.827168\pi\)
−0.856179 + 0.516680i \(0.827168\pi\)
\(440\) 0 0
\(441\) 5.61676 0.267465
\(442\) 0 0
\(443\) −19.4835 −0.925687 −0.462844 0.886440i \(-0.653171\pi\)
−0.462844 + 0.886440i \(0.653171\pi\)
\(444\) 0 0
\(445\) −2.98359 −0.141436
\(446\) 0 0
\(447\) 4.99583 0.236295
\(448\) 0 0
\(449\) 4.79095 0.226099 0.113049 0.993589i \(-0.463938\pi\)
0.113049 + 0.993589i \(0.463938\pi\)
\(450\) 0 0
\(451\) −23.2804 −1.09623
\(452\) 0 0
\(453\) 0.875034 0.0411127
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.1494 1.55067 0.775333 0.631553i \(-0.217582\pi\)
0.775333 + 0.631553i \(0.217582\pi\)
\(458\) 0 0
\(459\) 7.68133 0.358534
\(460\) 0 0
\(461\) 33.3955 1.55538 0.777691 0.628647i \(-0.216391\pi\)
0.777691 + 0.628647i \(0.216391\pi\)
\(462\) 0 0
\(463\) −11.2817 −0.524304 −0.262152 0.965027i \(-0.584432\pi\)
−0.262152 + 0.965027i \(0.584432\pi\)
\(464\) 0 0
\(465\) 4.61676 0.214097
\(466\) 0 0
\(467\) 7.45065 0.344775 0.172387 0.985029i \(-0.444852\pi\)
0.172387 + 0.985029i \(0.444852\pi\)
\(468\) 0 0
\(469\) 10.9630 0.506225
\(470\) 0 0
\(471\) −59.7006 −2.75086
\(472\) 0 0
\(473\) −3.53863 −0.162706
\(474\) 0 0
\(475\) −13.2663 −0.608701
\(476\) 0 0
\(477\) 7.76931 0.355732
\(478\) 0 0
\(479\) −20.8943 −0.954684 −0.477342 0.878718i \(-0.658400\pi\)
−0.477342 + 0.878718i \(0.658400\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.8545 0.584898
\(484\) 0 0
\(485\) −1.41082 −0.0640621
\(486\) 0 0
\(487\) 3.90368 0.176893 0.0884463 0.996081i \(-0.471810\pi\)
0.0884463 + 0.996081i \(0.471810\pi\)
\(488\) 0 0
\(489\) 30.3707 1.37341
\(490\) 0 0
\(491\) −12.9958 −0.586494 −0.293247 0.956037i \(-0.594736\pi\)
−0.293247 + 0.956037i \(0.594736\pi\)
\(492\) 0 0
\(493\) 7.87086 0.354486
\(494\) 0 0
\(495\) −17.6678 −0.794107
\(496\) 0 0
\(497\) 0.475390 0.0213241
\(498\) 0 0
\(499\) 24.3379 1.08951 0.544757 0.838594i \(-0.316622\pi\)
0.544757 + 0.838594i \(0.316622\pi\)
\(500\) 0 0
\(501\) 50.8255 2.27072
\(502\) 0 0
\(503\) −38.0122 −1.69488 −0.847441 0.530890i \(-0.821858\pi\)
−0.847441 + 0.530890i \(0.821858\pi\)
\(504\) 0 0
\(505\) −5.61259 −0.249757
\(506\) 0 0
\(507\) 38.1606 1.69477
\(508\) 0 0
\(509\) 41.3132 1.83117 0.915587 0.402120i \(-0.131726\pi\)
0.915587 + 0.402120i \(0.131726\pi\)
\(510\) 0 0
\(511\) −0.427229 −0.0188995
\(512\) 0 0
\(513\) −29.7334 −1.31276
\(514\) 0 0
\(515\) 12.8545 0.566435
\(516\) 0 0
\(517\) −19.4178 −0.853995
\(518\) 0 0
\(519\) −6.01224 −0.263908
\(520\) 0 0
\(521\) 40.9271 1.79305 0.896524 0.442995i \(-0.146084\pi\)
0.896524 + 0.442995i \(0.146084\pi\)
\(522\) 0 0
\(523\) −20.2088 −0.883668 −0.441834 0.897097i \(-0.645672\pi\)
−0.441834 + 0.897097i \(0.645672\pi\)
\(524\) 0 0
\(525\) −10.0604 −0.439072
\(526\) 0 0
\(527\) 1.25410 0.0546295
\(528\) 0 0
\(529\) −3.82376 −0.166251
\(530\) 0 0
\(531\) −46.8789 −2.03437
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −19.7417 −0.853509
\(536\) 0 0
\(537\) 52.5756 2.26881
\(538\) 0 0
\(539\) −2.50820 −0.108036
\(540\) 0 0
\(541\) −28.0328 −1.20522 −0.602612 0.798034i \(-0.705873\pi\)
−0.602612 + 0.798034i \(0.705873\pi\)
\(542\) 0 0
\(543\) 51.0632 2.19133
\(544\) 0 0
\(545\) −17.5574 −0.752077
\(546\) 0 0
\(547\) 10.3051 0.440614 0.220307 0.975431i \(-0.429294\pi\)
0.220307 + 0.975431i \(0.429294\pi\)
\(548\) 0 0
\(549\) −65.4957 −2.79529
\(550\) 0 0
\(551\) −30.4671 −1.29794
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −14.2499 −0.604876
\(556\) 0 0
\(557\) 6.32390 0.267952 0.133976 0.990985i \(-0.457225\pi\)
0.133976 + 0.990985i \(0.457225\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −7.36266 −0.310852
\(562\) 0 0
\(563\) 11.1784 0.471112 0.235556 0.971861i \(-0.424309\pi\)
0.235556 + 0.971861i \(0.424309\pi\)
\(564\) 0 0
\(565\) −20.1760 −0.848809
\(566\) 0 0
\(567\) −5.69774 −0.239282
\(568\) 0 0
\(569\) −21.8779 −0.917168 −0.458584 0.888651i \(-0.651643\pi\)
−0.458584 + 0.888651i \(0.651643\pi\)
\(570\) 0 0
\(571\) −32.9096 −1.37723 −0.688613 0.725129i \(-0.741780\pi\)
−0.688613 + 0.725129i \(0.741780\pi\)
\(572\) 0 0
\(573\) 15.4231 0.644308
\(574\) 0 0
\(575\) −15.0081 −0.625880
\(576\) 0 0
\(577\) −36.5327 −1.52088 −0.760438 0.649411i \(-0.775016\pi\)
−0.760438 + 0.649411i \(0.775016\pi\)
\(578\) 0 0
\(579\) 67.9177 2.82256
\(580\) 0 0
\(581\) −1.83805 −0.0762552
\(582\) 0 0
\(583\) −3.46944 −0.143690
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 34.4999 1.42396 0.711981 0.702199i \(-0.247798\pi\)
0.711981 + 0.702199i \(0.247798\pi\)
\(588\) 0 0
\(589\) −4.85446 −0.200024
\(590\) 0 0
\(591\) 11.8381 0.486952
\(592\) 0 0
\(593\) 17.8381 0.732521 0.366260 0.930512i \(-0.380638\pi\)
0.366260 + 0.930512i \(0.380638\pi\)
\(594\) 0 0
\(595\) 1.25410 0.0514132
\(596\) 0 0
\(597\) 3.85863 0.157923
\(598\) 0 0
\(599\) −39.6196 −1.61881 −0.809407 0.587249i \(-0.800211\pi\)
−0.809407 + 0.587249i \(0.800211\pi\)
\(600\) 0 0
\(601\) −2.25827 −0.0921168 −0.0460584 0.998939i \(-0.514666\pi\)
−0.0460584 + 0.998939i \(0.514666\pi\)
\(602\) 0 0
\(603\) −61.5767 −2.50760
\(604\) 0 0
\(605\) −5.90546 −0.240091
\(606\) 0 0
\(607\) −40.0039 −1.62371 −0.811854 0.583860i \(-0.801542\pi\)
−0.811854 + 0.583860i \(0.801542\pi\)
\(608\) 0 0
\(609\) −23.1044 −0.936237
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.88310 0.0760578 0.0380289 0.999277i \(-0.487892\pi\)
0.0380289 + 0.999277i \(0.487892\pi\)
\(614\) 0 0
\(615\) −34.1690 −1.37783
\(616\) 0 0
\(617\) −6.85446 −0.275950 −0.137975 0.990436i \(-0.544059\pi\)
−0.137975 + 0.990436i \(0.544059\pi\)
\(618\) 0 0
\(619\) −16.3463 −0.657011 −0.328506 0.944502i \(-0.606545\pi\)
−0.328506 + 0.944502i \(0.606545\pi\)
\(620\) 0 0
\(621\) −33.6371 −1.34981
\(622\) 0 0
\(623\) 2.37907 0.0953153
\(624\) 0 0
\(625\) 3.88204 0.155282
\(626\) 0 0
\(627\) 28.4999 1.13817
\(628\) 0 0
\(629\) −3.87086 −0.154341
\(630\) 0 0
\(631\) −2.27051 −0.0903875 −0.0451938 0.998978i \(-0.514391\pi\)
−0.0451938 + 0.998978i \(0.514391\pi\)
\(632\) 0 0
\(633\) −31.9588 −1.27025
\(634\) 0 0
\(635\) −4.39442 −0.174387
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.67015 −0.105630
\(640\) 0 0
\(641\) −33.4423 −1.32089 −0.660446 0.750874i \(-0.729633\pi\)
−0.660446 + 0.750874i \(0.729633\pi\)
\(642\) 0 0
\(643\) 12.7100 0.501232 0.250616 0.968087i \(-0.419367\pi\)
0.250616 + 0.968087i \(0.419367\pi\)
\(644\) 0 0
\(645\) −5.19370 −0.204502
\(646\) 0 0
\(647\) −29.9917 −1.17909 −0.589547 0.807734i \(-0.700694\pi\)
−0.589547 + 0.807734i \(0.700694\pi\)
\(648\) 0 0
\(649\) 20.9341 0.821735
\(650\) 0 0
\(651\) −3.68133 −0.144283
\(652\) 0 0
\(653\) 7.01641 0.274573 0.137287 0.990531i \(-0.456162\pi\)
0.137287 + 0.990531i \(0.456162\pi\)
\(654\) 0 0
\(655\) −21.4506 −0.838146
\(656\) 0 0
\(657\) 2.39964 0.0936190
\(658\) 0 0
\(659\) 26.9875 1.05128 0.525642 0.850706i \(-0.323825\pi\)
0.525642 + 0.850706i \(0.323825\pi\)
\(660\) 0 0
\(661\) −4.16195 −0.161881 −0.0809405 0.996719i \(-0.525792\pi\)
−0.0809405 + 0.996719i \(0.525792\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.85446 −0.188248
\(666\) 0 0
\(667\) −34.4671 −1.33457
\(668\) 0 0
\(669\) 10.1536 0.392561
\(670\) 0 0
\(671\) 29.2475 1.12909
\(672\) 0 0
\(673\) −18.1760 −0.700632 −0.350316 0.936632i \(-0.613926\pi\)
−0.350316 + 0.936632i \(0.613926\pi\)
\(674\) 0 0
\(675\) 26.3257 1.01328
\(676\) 0 0
\(677\) 4.57171 0.175705 0.0878526 0.996133i \(-0.472000\pi\)
0.0878526 + 0.996133i \(0.472000\pi\)
\(678\) 0 0
\(679\) 1.12497 0.0431722
\(680\) 0 0
\(681\) −70.0109 −2.68282
\(682\) 0 0
\(683\) 6.15361 0.235461 0.117731 0.993046i \(-0.462438\pi\)
0.117731 + 0.993046i \(0.462438\pi\)
\(684\) 0 0
\(685\) 21.3491 0.815707
\(686\) 0 0
\(687\) −16.1208 −0.615047
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 29.2405 1.11236 0.556181 0.831061i \(-0.312266\pi\)
0.556181 + 0.831061i \(0.312266\pi\)
\(692\) 0 0
\(693\) 14.0880 0.535158
\(694\) 0 0
\(695\) 6.42723 0.243799
\(696\) 0 0
\(697\) −9.28169 −0.351569
\(698\) 0 0
\(699\) 55.7006 2.10679
\(700\) 0 0
\(701\) −14.4342 −0.545174 −0.272587 0.962131i \(-0.587879\pi\)
−0.272587 + 0.962131i \(0.587879\pi\)
\(702\) 0 0
\(703\) 14.9836 0.565117
\(704\) 0 0
\(705\) −28.4999 −1.07337
\(706\) 0 0
\(707\) 4.47539 0.168314
\(708\) 0 0
\(709\) −15.3955 −0.578189 −0.289095 0.957301i \(-0.593354\pi\)
−0.289095 + 0.957301i \(0.593354\pi\)
\(710\) 0 0
\(711\) −44.9341 −1.68516
\(712\) 0 0
\(713\) −5.49180 −0.205669
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −89.3171 −3.33561
\(718\) 0 0
\(719\) −3.02342 −0.112754 −0.0563772 0.998410i \(-0.517955\pi\)
−0.0563772 + 0.998410i \(0.517955\pi\)
\(720\) 0 0
\(721\) −10.2499 −0.381727
\(722\) 0 0
\(723\) 31.8021 1.18273
\(724\) 0 0
\(725\) 26.9753 1.00184
\(726\) 0 0
\(727\) −2.12914 −0.0789653 −0.0394826 0.999220i \(-0.512571\pi\)
−0.0394826 + 0.999220i \(0.512571\pi\)
\(728\) 0 0
\(729\) −35.6412 −1.32005
\(730\) 0 0
\(731\) −1.41082 −0.0521811
\(732\) 0 0
\(733\) 11.3627 0.419689 0.209845 0.977735i \(-0.432704\pi\)
0.209845 + 0.977735i \(0.432704\pi\)
\(734\) 0 0
\(735\) −3.68133 −0.135788
\(736\) 0 0
\(737\) 27.4975 1.01288
\(738\) 0 0
\(739\) 46.6995 1.71787 0.858935 0.512085i \(-0.171127\pi\)
0.858935 + 0.512085i \(0.171127\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.0245 1.61510 0.807551 0.589798i \(-0.200793\pi\)
0.807551 + 0.589798i \(0.200793\pi\)
\(744\) 0 0
\(745\) −2.13436 −0.0781970
\(746\) 0 0
\(747\) 10.3239 0.377732
\(748\) 0 0
\(749\) 15.7417 0.575190
\(750\) 0 0
\(751\) −49.6678 −1.81240 −0.906201 0.422847i \(-0.861031\pi\)
−0.906201 + 0.422847i \(0.861031\pi\)
\(752\) 0 0
\(753\) 29.8297 1.08706
\(754\) 0 0
\(755\) −0.373840 −0.0136054
\(756\) 0 0
\(757\) −40.7651 −1.48163 −0.740817 0.671707i \(-0.765562\pi\)
−0.740817 + 0.671707i \(0.765562\pi\)
\(758\) 0 0
\(759\) 32.2416 1.17030
\(760\) 0 0
\(761\) 26.6618 0.966490 0.483245 0.875485i \(-0.339458\pi\)
0.483245 + 0.875485i \(0.339458\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) −7.04399 −0.254676
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −23.9260 −0.862795 −0.431397 0.902162i \(-0.641979\pi\)
−0.431397 + 0.902162i \(0.641979\pi\)
\(770\) 0 0
\(771\) −31.2252 −1.12455
\(772\) 0 0
\(773\) 34.7253 1.24898 0.624492 0.781032i \(-0.285306\pi\)
0.624492 + 0.781032i \(0.285306\pi\)
\(774\) 0 0
\(775\) 4.29809 0.154392
\(776\) 0 0
\(777\) 11.3627 0.407633
\(778\) 0 0
\(779\) 35.9282 1.28726
\(780\) 0 0
\(781\) 1.19237 0.0426665
\(782\) 0 0
\(783\) 60.4587 2.16062
\(784\) 0 0
\(785\) 25.5058 0.910342
\(786\) 0 0
\(787\) −33.5386 −1.19552 −0.597762 0.801674i \(-0.703943\pi\)
−0.597762 + 0.801674i \(0.703943\pi\)
\(788\) 0 0
\(789\) 17.5163 0.623596
\(790\) 0 0
\(791\) 16.0880 0.572023
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −5.09215 −0.180600
\(796\) 0 0
\(797\) −3.13509 −0.111050 −0.0555252 0.998457i \(-0.517683\pi\)
−0.0555252 + 0.998457i \(0.517683\pi\)
\(798\) 0 0
\(799\) −7.74173 −0.273883
\(800\) 0 0
\(801\) −13.3627 −0.472146
\(802\) 0 0
\(803\) −1.07158 −0.0378151
\(804\) 0 0
\(805\) −5.49180 −0.193560
\(806\) 0 0
\(807\) −76.7722 −2.70251
\(808\) 0 0
\(809\) 34.4999 1.21295 0.606475 0.795102i \(-0.292583\pi\)
0.606475 + 0.795102i \(0.292583\pi\)
\(810\) 0 0
\(811\) −2.08097 −0.0730729 −0.0365364 0.999332i \(-0.511633\pi\)
−0.0365364 + 0.999332i \(0.511633\pi\)
\(812\) 0 0
\(813\) 47.3215 1.65964
\(814\) 0 0
\(815\) −12.9753 −0.454503
\(816\) 0 0
\(817\) 5.46110 0.191060
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.64568 −0.301736 −0.150868 0.988554i \(-0.548207\pi\)
−0.150868 + 0.988554i \(0.548207\pi\)
\(822\) 0 0
\(823\) −7.52461 −0.262291 −0.131146 0.991363i \(-0.541866\pi\)
−0.131146 + 0.991363i \(0.541866\pi\)
\(824\) 0 0
\(825\) −25.2335 −0.878519
\(826\) 0 0
\(827\) −40.8545 −1.42065 −0.710324 0.703875i \(-0.751452\pi\)
−0.710324 + 0.703875i \(0.751452\pi\)
\(828\) 0 0
\(829\) −38.9836 −1.35396 −0.676978 0.736003i \(-0.736711\pi\)
−0.676978 + 0.736003i \(0.736711\pi\)
\(830\) 0 0
\(831\) 78.5467 2.72475
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −21.7141 −0.751449
\(836\) 0 0
\(837\) 9.63317 0.332971
\(838\) 0 0
\(839\) 36.7581 1.26903 0.634516 0.772910i \(-0.281200\pi\)
0.634516 + 0.772910i \(0.281200\pi\)
\(840\) 0 0
\(841\) 32.9505 1.13622
\(842\) 0 0
\(843\) −82.3900 −2.83766
\(844\) 0 0
\(845\) −16.3033 −0.560851
\(846\) 0 0
\(847\) 4.70892 0.161800
\(848\) 0 0
\(849\) −88.3572 −3.03241
\(850\) 0 0
\(851\) 16.9508 0.581065
\(852\) 0 0
\(853\) 20.2968 0.694948 0.347474 0.937690i \(-0.387039\pi\)
0.347474 + 0.937690i \(0.387039\pi\)
\(854\) 0 0
\(855\) 27.2663 0.932489
\(856\) 0 0
\(857\) −20.8255 −0.711387 −0.355694 0.934603i \(-0.615755\pi\)
−0.355694 + 0.934603i \(0.615755\pi\)
\(858\) 0 0
\(859\) 35.7641 1.22025 0.610127 0.792303i \(-0.291118\pi\)
0.610127 + 0.792303i \(0.291118\pi\)
\(860\) 0 0
\(861\) 27.2458 0.928533
\(862\) 0 0
\(863\) 16.5616 0.563763 0.281882 0.959449i \(-0.409041\pi\)
0.281882 + 0.959449i \(0.409041\pi\)
\(864\) 0 0
\(865\) 2.56860 0.0873351
\(866\) 0 0
\(867\) −2.93543 −0.0996925
\(868\) 0 0
\(869\) 20.0656 0.680680
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −6.31867 −0.213855
\(874\) 0 0
\(875\) 10.5686 0.357284
\(876\) 0 0
\(877\) 52.2744 1.76518 0.882591 0.470142i \(-0.155797\pi\)
0.882591 + 0.470142i \(0.155797\pi\)
\(878\) 0 0
\(879\) 71.9177 2.42572
\(880\) 0 0
\(881\) 7.85029 0.264483 0.132241 0.991218i \(-0.457783\pi\)
0.132241 + 0.991218i \(0.457783\pi\)
\(882\) 0 0
\(883\) 36.9135 1.24224 0.621120 0.783716i \(-0.286678\pi\)
0.621120 + 0.783716i \(0.286678\pi\)
\(884\) 0 0
\(885\) 30.7253 1.03282
\(886\) 0 0
\(887\) −49.8196 −1.67278 −0.836389 0.548136i \(-0.815338\pi\)
−0.836389 + 0.548136i \(0.815338\pi\)
\(888\) 0 0
\(889\) 3.50403 0.117522
\(890\) 0 0
\(891\) −14.2911 −0.478769
\(892\) 0 0
\(893\) 29.9672 1.00281
\(894\) 0 0
\(895\) −22.4618 −0.750816
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.87086 0.329212
\(900\) 0 0
\(901\) −1.38324 −0.0460823
\(902\) 0 0
\(903\) 4.14137 0.137816
\(904\) 0 0
\(905\) −21.8157 −0.725178
\(906\) 0 0
\(907\) −8.70297 −0.288977 −0.144489 0.989506i \(-0.546154\pi\)
−0.144489 + 0.989506i \(0.546154\pi\)
\(908\) 0 0
\(909\) −25.1372 −0.833748
\(910\) 0 0
\(911\) 56.8378 1.88312 0.941560 0.336846i \(-0.109360\pi\)
0.941560 + 0.336846i \(0.109360\pi\)
\(912\) 0 0
\(913\) −4.61021 −0.152576
\(914\) 0 0
\(915\) 42.9271 1.41913
\(916\) 0 0
\(917\) 17.1044 0.564837
\(918\) 0 0
\(919\) 23.3473 0.770157 0.385078 0.922884i \(-0.374174\pi\)
0.385078 + 0.922884i \(0.374174\pi\)
\(920\) 0 0
\(921\) 6.50820 0.214453
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −13.2663 −0.436195
\(926\) 0 0
\(927\) 57.5714 1.89089
\(928\) 0 0
\(929\) 16.8667 0.553378 0.276689 0.960959i \(-0.410763\pi\)
0.276689 + 0.960959i \(0.410763\pi\)
\(930\) 0 0
\(931\) 3.87086 0.126862
\(932\) 0 0
\(933\) 38.7857 1.26979
\(934\) 0 0
\(935\) 3.14554 0.102870
\(936\) 0 0
\(937\) 1.55530 0.0508096 0.0254048 0.999677i \(-0.491913\pi\)
0.0254048 + 0.999677i \(0.491913\pi\)
\(938\) 0 0
\(939\) −97.8734 −3.19398
\(940\) 0 0
\(941\) 40.3226 1.31448 0.657239 0.753682i \(-0.271724\pi\)
0.657239 + 0.753682i \(0.271724\pi\)
\(942\) 0 0
\(943\) 40.6451 1.32359
\(944\) 0 0
\(945\) 9.63317 0.313367
\(946\) 0 0
\(947\) 12.7337 0.413788 0.206894 0.978363i \(-0.433664\pi\)
0.206894 + 0.978363i \(0.433664\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −84.6758 −2.74580
\(952\) 0 0
\(953\) −2.80941 −0.0910056 −0.0455028 0.998964i \(-0.514489\pi\)
−0.0455028 + 0.998964i \(0.514489\pi\)
\(954\) 0 0
\(955\) −6.58918 −0.213221
\(956\) 0 0
\(957\) −57.9505 −1.87327
\(958\) 0 0
\(959\) −17.0234 −0.549715
\(960\) 0 0
\(961\) −29.4272 −0.949265
\(962\) 0 0
\(963\) −88.4176 −2.84922
\(964\) 0 0
\(965\) −29.0164 −0.934071
\(966\) 0 0
\(967\) 42.4106 1.36383 0.681916 0.731431i \(-0.261147\pi\)
0.681916 + 0.731431i \(0.261147\pi\)
\(968\) 0 0
\(969\) 11.3627 0.365021
\(970\) 0 0
\(971\) 11.8709 0.380954 0.190477 0.981692i \(-0.438997\pi\)
0.190477 + 0.981692i \(0.438997\pi\)
\(972\) 0 0
\(973\) −5.12497 −0.164299
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.5837 0.946466 0.473233 0.880937i \(-0.343087\pi\)
0.473233 + 0.880937i \(0.343087\pi\)
\(978\) 0 0
\(979\) 5.96719 0.190712
\(980\) 0 0
\(981\) −78.6347 −2.51061
\(982\) 0 0
\(983\) −33.8196 −1.07868 −0.539339 0.842089i \(-0.681326\pi\)
−0.539339 + 0.842089i \(0.681326\pi\)
\(984\) 0 0
\(985\) −5.05756 −0.161147
\(986\) 0 0
\(987\) 22.7253 0.723355
\(988\) 0 0
\(989\) 6.17809 0.196452
\(990\) 0 0
\(991\) −48.1208 −1.52861 −0.764304 0.644856i \(-0.776917\pi\)
−0.764304 + 0.644856i \(0.776917\pi\)
\(992\) 0 0
\(993\) 20.5603 0.652460
\(994\) 0 0
\(995\) −1.64852 −0.0522615
\(996\) 0 0
\(997\) 56.2403 1.78115 0.890574 0.454839i \(-0.150303\pi\)
0.890574 + 0.454839i \(0.150303\pi\)
\(998\) 0 0
\(999\) −29.7334 −0.940723
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 1904.2.a.m.1.1 3
4.3 odd 2 952.2.a.f.1.3 3
8.3 odd 2 7616.2.a.bc.1.1 3
8.5 even 2 7616.2.a.be.1.3 3
12.11 even 2 8568.2.a.x.1.2 3
28.27 even 2 6664.2.a.j.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.f.1.3 3 4.3 odd 2
1904.2.a.m.1.1 3 1.1 even 1 trivial
6664.2.a.j.1.1 3 28.27 even 2
7616.2.a.bc.1.1 3 8.3 odd 2
7616.2.a.be.1.3 3 8.5 even 2
8568.2.a.x.1.2 3 12.11 even 2