Properties

Label 1904.2.a.q.1.1
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.88301\) 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)
 
\(f(q)\) \(=\) \(q-2.88301 q^{3} -1.78180 q^{5} +1.00000 q^{7} +5.31175 q^{9} +O(q^{10})\) \(q-2.88301 q^{3} -1.78180 q^{5} +1.00000 q^{7} +5.31175 q^{9} -3.23607 q^{11} +2.47214 q^{13} +5.13695 q^{15} +1.00000 q^{17} -7.76602 q^{19} -2.88301 q^{21} +5.32962 q^{23} -1.82519 q^{25} -6.66481 q^{27} +10.5657 q^{29} +9.64137 q^{31} +9.32962 q^{33} -1.78180 q^{35} -1.62142 q^{37} -7.12720 q^{39} +2.82519 q^{41} -0.587035 q^{43} -9.46448 q^{45} -12.1871 q^{47} +1.00000 q^{49} -2.88301 q^{51} +3.40321 q^{53} +5.76602 q^{55} +22.3895 q^{57} +7.32962 q^{59} -3.82310 q^{61} +5.31175 q^{63} -4.40485 q^{65} -0.394361 q^{67} -15.3654 q^{69} +2.15137 q^{71} -14.3572 q^{73} +5.26205 q^{75} -3.23607 q^{77} -4.18710 q^{79} +3.27946 q^{81} -8.42108 q^{83} -1.78180 q^{85} -30.4610 q^{87} -17.0446 q^{89} +2.47214 q^{91} -27.7962 q^{93} +13.8375 q^{95} -7.72398 q^{97} -17.1892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9} - 4 q^{11} - 8 q^{13} - 12 q^{15} + 4 q^{17} - 14 q^{19} - 3 q^{21} - 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} + 8 q^{33} - q^{35} - 4 q^{37} - 4 q^{39} - 7 q^{41} - 19 q^{43} - 2 q^{45} - 8 q^{47} + 4 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{55} + 44 q^{57} - 23 q^{61} + 7 q^{63} - 8 q^{65} - 15 q^{67} - 2 q^{69} - 2 q^{71} - 5 q^{73} - 10 q^{75} - 4 q^{77} + 24 q^{79} - 8 q^{81} - 10 q^{83} - q^{85} - 16 q^{87} - 16 q^{89} - 8 q^{91} - 20 q^{93} - 22 q^{95} - 15 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.88301 −1.66451 −0.832254 0.554395i \(-0.812950\pi\)
−0.832254 + 0.554395i \(0.812950\pi\)
\(4\) 0 0
\(5\) −1.78180 −0.796845 −0.398422 0.917202i \(-0.630442\pi\)
−0.398422 + 0.917202i \(0.630442\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 5.31175 1.77058
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 2.47214 0.685647 0.342824 0.939400i \(-0.388617\pi\)
0.342824 + 0.939400i \(0.388617\pi\)
\(14\) 0 0
\(15\) 5.13695 1.32635
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) −7.76602 −1.78165 −0.890824 0.454349i \(-0.849872\pi\)
−0.890824 + 0.454349i \(0.849872\pi\)
\(20\) 0 0
\(21\) −2.88301 −0.629125
\(22\) 0 0
\(23\) 5.32962 1.11130 0.555651 0.831415i \(-0.312469\pi\)
0.555651 + 0.831415i \(0.312469\pi\)
\(24\) 0 0
\(25\) −1.82519 −0.365039
\(26\) 0 0
\(27\) −6.66481 −1.28264
\(28\) 0 0
\(29\) 10.5657 1.96200 0.980999 0.194010i \(-0.0621495\pi\)
0.980999 + 0.194010i \(0.0621495\pi\)
\(30\) 0 0
\(31\) 9.64137 1.73164 0.865821 0.500354i \(-0.166797\pi\)
0.865821 + 0.500354i \(0.166797\pi\)
\(32\) 0 0
\(33\) 9.32962 1.62408
\(34\) 0 0
\(35\) −1.78180 −0.301179
\(36\) 0 0
\(37\) −1.62142 −0.266559 −0.133280 0.991078i \(-0.542551\pi\)
−0.133280 + 0.991078i \(0.542551\pi\)
\(38\) 0 0
\(39\) −7.12720 −1.14126
\(40\) 0 0
\(41\) 2.82519 0.441221 0.220610 0.975362i \(-0.429195\pi\)
0.220610 + 0.975362i \(0.429195\pi\)
\(42\) 0 0
\(43\) −0.587035 −0.0895219 −0.0447610 0.998998i \(-0.514253\pi\)
−0.0447610 + 0.998998i \(0.514253\pi\)
\(44\) 0 0
\(45\) −9.46448 −1.41088
\(46\) 0 0
\(47\) −12.1871 −1.77767 −0.888836 0.458226i \(-0.848485\pi\)
−0.888836 + 0.458226i \(0.848485\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.88301 −0.403702
\(52\) 0 0
\(53\) 3.40321 0.467468 0.233734 0.972301i \(-0.424906\pi\)
0.233734 + 0.972301i \(0.424906\pi\)
\(54\) 0 0
\(55\) 5.76602 0.777490
\(56\) 0 0
\(57\) 22.3895 2.96557
\(58\) 0 0
\(59\) 7.32962 0.954235 0.477118 0.878839i \(-0.341682\pi\)
0.477118 + 0.878839i \(0.341682\pi\)
\(60\) 0 0
\(61\) −3.82310 −0.489498 −0.244749 0.969586i \(-0.578706\pi\)
−0.244749 + 0.969586i \(0.578706\pi\)
\(62\) 0 0
\(63\) 5.31175 0.669218
\(64\) 0 0
\(65\) −4.40485 −0.546354
\(66\) 0 0
\(67\) −0.394361 −0.0481788 −0.0240894 0.999710i \(-0.507669\pi\)
−0.0240894 + 0.999710i \(0.507669\pi\)
\(68\) 0 0
\(69\) −15.3654 −1.84977
\(70\) 0 0
\(71\) 2.15137 0.255321 0.127660 0.991818i \(-0.459253\pi\)
0.127660 + 0.991818i \(0.459253\pi\)
\(72\) 0 0
\(73\) −14.3572 −1.68039 −0.840194 0.542286i \(-0.817559\pi\)
−0.840194 + 0.542286i \(0.817559\pi\)
\(74\) 0 0
\(75\) 5.26205 0.607609
\(76\) 0 0
\(77\) −3.23607 −0.368784
\(78\) 0 0
\(79\) −4.18710 −0.471086 −0.235543 0.971864i \(-0.575687\pi\)
−0.235543 + 0.971864i \(0.575687\pi\)
\(80\) 0 0
\(81\) 3.27946 0.364385
\(82\) 0 0
\(83\) −8.42108 −0.924334 −0.462167 0.886793i \(-0.652928\pi\)
−0.462167 + 0.886793i \(0.652928\pi\)
\(84\) 0 0
\(85\) −1.78180 −0.193263
\(86\) 0 0
\(87\) −30.4610 −3.26576
\(88\) 0 0
\(89\) −17.0446 −1.80672 −0.903361 0.428880i \(-0.858908\pi\)
−0.903361 + 0.428880i \(0.858908\pi\)
\(90\) 0 0
\(91\) 2.47214 0.259150
\(92\) 0 0
\(93\) −27.7962 −2.88233
\(94\) 0 0
\(95\) 13.8375 1.41970
\(96\) 0 0
\(97\) −7.72398 −0.784251 −0.392126 0.919912i \(-0.628260\pi\)
−0.392126 + 0.919912i \(0.628260\pi\)
\(98\) 0 0
\(99\) −17.1892 −1.72758
\(100\) 0 0
\(101\) −13.6834 −1.36155 −0.680775 0.732492i \(-0.738357\pi\)
−0.680775 + 0.732492i \(0.738357\pi\)
\(102\) 0 0
\(103\) −6.90854 −0.680718 −0.340359 0.940295i \(-0.610549\pi\)
−0.340359 + 0.940295i \(0.610549\pi\)
\(104\) 0 0
\(105\) 5.13695 0.501315
\(106\) 0 0
\(107\) 14.9868 1.44883 0.724413 0.689366i \(-0.242111\pi\)
0.724413 + 0.689366i \(0.242111\pi\)
\(108\) 0 0
\(109\) −19.3874 −1.85698 −0.928490 0.371358i \(-0.878892\pi\)
−0.928490 + 0.371358i \(0.878892\pi\)
\(110\) 0 0
\(111\) 4.67456 0.443690
\(112\) 0 0
\(113\) 6.26971 0.589805 0.294902 0.955527i \(-0.404713\pi\)
0.294902 + 0.955527i \(0.404713\pi\)
\(114\) 0 0
\(115\) −9.49631 −0.885536
\(116\) 0 0
\(117\) 13.1314 1.21400
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −8.14506 −0.734416
\(124\) 0 0
\(125\) 12.1611 1.08772
\(126\) 0 0
\(127\) −0.362807 −0.0321939 −0.0160970 0.999870i \(-0.505124\pi\)
−0.0160970 + 0.999870i \(0.505124\pi\)
\(128\) 0 0
\(129\) 1.69243 0.149010
\(130\) 0 0
\(131\) 1.94218 0.169689 0.0848446 0.996394i \(-0.472961\pi\)
0.0848446 + 0.996394i \(0.472961\pi\)
\(132\) 0 0
\(133\) −7.76602 −0.673400
\(134\) 0 0
\(135\) 11.8754 1.02207
\(136\) 0 0
\(137\) 5.17481 0.442114 0.221057 0.975261i \(-0.429049\pi\)
0.221057 + 0.975261i \(0.429049\pi\)
\(138\) 0 0
\(139\) −4.13559 −0.350776 −0.175388 0.984499i \(-0.556118\pi\)
−0.175388 + 0.984499i \(0.556118\pi\)
\(140\) 0 0
\(141\) 35.1356 2.95895
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −18.8259 −1.56341
\(146\) 0 0
\(147\) −2.88301 −0.237787
\(148\) 0 0
\(149\) −2.17013 −0.177784 −0.0888921 0.996041i \(-0.528333\pi\)
−0.0888921 + 0.996041i \(0.528333\pi\)
\(150\) 0 0
\(151\) −3.05917 −0.248952 −0.124476 0.992223i \(-0.539725\pi\)
−0.124476 + 0.992223i \(0.539725\pi\)
\(152\) 0 0
\(153\) 5.31175 0.429430
\(154\) 0 0
\(155\) −17.1790 −1.37985
\(156\) 0 0
\(157\) 9.95313 0.794346 0.397173 0.917744i \(-0.369991\pi\)
0.397173 + 0.917744i \(0.369991\pi\)
\(158\) 0 0
\(159\) −9.81151 −0.778103
\(160\) 0 0
\(161\) 5.32962 0.420033
\(162\) 0 0
\(163\) −15.6613 −1.22669 −0.613345 0.789815i \(-0.710176\pi\)
−0.613345 + 0.789815i \(0.710176\pi\)
\(164\) 0 0
\(165\) −16.6235 −1.29414
\(166\) 0 0
\(167\) 8.11908 0.628273 0.314137 0.949378i \(-0.398285\pi\)
0.314137 + 0.949378i \(0.398285\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) −41.2512 −3.15456
\(172\) 0 0
\(173\) 12.3640 0.940018 0.470009 0.882662i \(-0.344251\pi\)
0.470009 + 0.882662i \(0.344251\pi\)
\(174\) 0 0
\(175\) −1.82519 −0.137972
\(176\) 0 0
\(177\) −21.1314 −1.58833
\(178\) 0 0
\(179\) 18.1233 1.35460 0.677298 0.735709i \(-0.263151\pi\)
0.677298 + 0.735709i \(0.263151\pi\)
\(180\) 0 0
\(181\) −18.4143 −1.36873 −0.684363 0.729142i \(-0.739919\pi\)
−0.684363 + 0.729142i \(0.739919\pi\)
\(182\) 0 0
\(183\) 11.0220 0.814773
\(184\) 0 0
\(185\) 2.88904 0.212406
\(186\) 0 0
\(187\) −3.23607 −0.236645
\(188\) 0 0
\(189\) −6.66481 −0.484794
\(190\) 0 0
\(191\) −10.7013 −0.774318 −0.387159 0.922013i \(-0.626543\pi\)
−0.387159 + 0.922013i \(0.626543\pi\)
\(192\) 0 0
\(193\) −1.51254 −0.108875 −0.0544376 0.998517i \(-0.517337\pi\)
−0.0544376 + 0.998517i \(0.517337\pi\)
\(194\) 0 0
\(195\) 12.6992 0.909411
\(196\) 0 0
\(197\) −7.27180 −0.518094 −0.259047 0.965865i \(-0.583409\pi\)
−0.259047 + 0.965865i \(0.583409\pi\)
\(198\) 0 0
\(199\) 23.1832 1.64341 0.821706 0.569912i \(-0.193023\pi\)
0.821706 + 0.569912i \(0.193023\pi\)
\(200\) 0 0
\(201\) 1.13695 0.0801940
\(202\) 0 0
\(203\) 10.5657 0.741566
\(204\) 0 0
\(205\) −5.03393 −0.351585
\(206\) 0 0
\(207\) 28.3096 1.96765
\(208\) 0 0
\(209\) 25.1314 1.73837
\(210\) 0 0
\(211\) −11.1535 −0.767836 −0.383918 0.923367i \(-0.625425\pi\)
−0.383918 + 0.923367i \(0.625425\pi\)
\(212\) 0 0
\(213\) −6.20242 −0.424983
\(214\) 0 0
\(215\) 1.04598 0.0713351
\(216\) 0 0
\(217\) 9.64137 0.654499
\(218\) 0 0
\(219\) 41.3921 2.79702
\(220\) 0 0
\(221\) 2.47214 0.166294
\(222\) 0 0
\(223\) 15.5636 1.04222 0.521108 0.853491i \(-0.325519\pi\)
0.521108 + 0.853491i \(0.325519\pi\)
\(224\) 0 0
\(225\) −9.69497 −0.646332
\(226\) 0 0
\(227\) −15.3649 −1.01980 −0.509902 0.860232i \(-0.670318\pi\)
−0.509902 + 0.860232i \(0.670318\pi\)
\(228\) 0 0
\(229\) −14.9085 −0.985184 −0.492592 0.870260i \(-0.663950\pi\)
−0.492592 + 0.870260i \(0.663950\pi\)
\(230\) 0 0
\(231\) 9.32962 0.613844
\(232\) 0 0
\(233\) 12.9085 0.845666 0.422833 0.906208i \(-0.361036\pi\)
0.422833 + 0.906208i \(0.361036\pi\)
\(234\) 0 0
\(235\) 21.7150 1.41653
\(236\) 0 0
\(237\) 12.0715 0.784126
\(238\) 0 0
\(239\) −7.05917 −0.456620 −0.228310 0.973589i \(-0.573320\pi\)
−0.228310 + 0.973589i \(0.573320\pi\)
\(240\) 0 0
\(241\) −22.0309 −1.41914 −0.709568 0.704637i \(-0.751110\pi\)
−0.709568 + 0.704637i \(0.751110\pi\)
\(242\) 0 0
\(243\) 10.5397 0.676122
\(244\) 0 0
\(245\) −1.78180 −0.113835
\(246\) 0 0
\(247\) −19.1987 −1.22158
\(248\) 0 0
\(249\) 24.2781 1.53856
\(250\) 0 0
\(251\) −28.1267 −1.77534 −0.887671 0.460479i \(-0.847678\pi\)
−0.887671 + 0.460479i \(0.847678\pi\)
\(252\) 0 0
\(253\) −17.2470 −1.08431
\(254\) 0 0
\(255\) 5.13695 0.321688
\(256\) 0 0
\(257\) 9.04459 0.564186 0.282093 0.959387i \(-0.408971\pi\)
0.282093 + 0.959387i \(0.408971\pi\)
\(258\) 0 0
\(259\) −1.62142 −0.100750
\(260\) 0 0
\(261\) 56.1223 3.47388
\(262\) 0 0
\(263\) −14.6592 −0.903927 −0.451964 0.892036i \(-0.649276\pi\)
−0.451964 + 0.892036i \(0.649276\pi\)
\(264\) 0 0
\(265\) −6.06384 −0.372499
\(266\) 0 0
\(267\) 49.1397 3.00730
\(268\) 0 0
\(269\) −9.75926 −0.595032 −0.297516 0.954717i \(-0.596158\pi\)
−0.297516 + 0.954717i \(0.596158\pi\)
\(270\) 0 0
\(271\) −15.1602 −0.920918 −0.460459 0.887681i \(-0.652315\pi\)
−0.460459 + 0.887681i \(0.652315\pi\)
\(272\) 0 0
\(273\) −7.12720 −0.431357
\(274\) 0 0
\(275\) 5.90645 0.356172
\(276\) 0 0
\(277\) 10.7324 0.644846 0.322423 0.946596i \(-0.395503\pi\)
0.322423 + 0.946596i \(0.395503\pi\)
\(278\) 0 0
\(279\) 51.2126 3.06602
\(280\) 0 0
\(281\) −2.06245 −0.123036 −0.0615179 0.998106i \(-0.519594\pi\)
−0.0615179 + 0.998106i \(0.519594\pi\)
\(282\) 0 0
\(283\) 15.4364 0.917597 0.458798 0.888540i \(-0.348280\pi\)
0.458798 + 0.888540i \(0.348280\pi\)
\(284\) 0 0
\(285\) −39.8936 −2.36310
\(286\) 0 0
\(287\) 2.82519 0.166766
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 22.2683 1.30539
\(292\) 0 0
\(293\) 2.28503 0.133493 0.0667465 0.997770i \(-0.478738\pi\)
0.0667465 + 0.997770i \(0.478738\pi\)
\(294\) 0 0
\(295\) −13.0599 −0.760377
\(296\) 0 0
\(297\) 21.5678 1.25149
\(298\) 0 0
\(299\) 13.1755 0.761961
\(300\) 0 0
\(301\) −0.587035 −0.0338361
\(302\) 0 0
\(303\) 39.4494 2.26631
\(304\) 0 0
\(305\) 6.81200 0.390054
\(306\) 0 0
\(307\) 20.6277 1.17728 0.588642 0.808394i \(-0.299663\pi\)
0.588642 + 0.808394i \(0.299663\pi\)
\(308\) 0 0
\(309\) 19.9174 1.13306
\(310\) 0 0
\(311\) −18.2250 −1.03344 −0.516721 0.856154i \(-0.672848\pi\)
−0.516721 + 0.856154i \(0.672848\pi\)
\(312\) 0 0
\(313\) −15.6609 −0.885205 −0.442602 0.896718i \(-0.645945\pi\)
−0.442602 + 0.896718i \(0.645945\pi\)
\(314\) 0 0
\(315\) −9.46448 −0.533263
\(316\) 0 0
\(317\) 1.75508 0.0985750 0.0492875 0.998785i \(-0.484305\pi\)
0.0492875 + 0.998785i \(0.484305\pi\)
\(318\) 0 0
\(319\) −34.1913 −1.91434
\(320\) 0 0
\(321\) −43.2070 −2.41158
\(322\) 0 0
\(323\) −7.76602 −0.432113
\(324\) 0 0
\(325\) −4.51212 −0.250288
\(326\) 0 0
\(327\) 55.8942 3.09096
\(328\) 0 0
\(329\) −12.1871 −0.671897
\(330\) 0 0
\(331\) 14.2829 0.785059 0.392530 0.919739i \(-0.371600\pi\)
0.392530 + 0.919739i \(0.371600\pi\)
\(332\) 0 0
\(333\) −8.61256 −0.471965
\(334\) 0 0
\(335\) 0.702671 0.0383910
\(336\) 0 0
\(337\) 24.9290 1.35797 0.678983 0.734154i \(-0.262421\pi\)
0.678983 + 0.734154i \(0.262421\pi\)
\(338\) 0 0
\(339\) −18.0756 −0.981734
\(340\) 0 0
\(341\) −31.2001 −1.68958
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 27.3780 1.47398
\(346\) 0 0
\(347\) −35.1934 −1.88928 −0.944640 0.328110i \(-0.893588\pi\)
−0.944640 + 0.328110i \(0.893588\pi\)
\(348\) 0 0
\(349\) 2.71029 0.145079 0.0725394 0.997366i \(-0.476890\pi\)
0.0725394 + 0.997366i \(0.476890\pi\)
\(350\) 0 0
\(351\) −16.4763 −0.879441
\(352\) 0 0
\(353\) −12.3580 −0.657749 −0.328874 0.944374i \(-0.606669\pi\)
−0.328874 + 0.944374i \(0.606669\pi\)
\(354\) 0 0
\(355\) −3.83331 −0.203451
\(356\) 0 0
\(357\) −2.88301 −0.152585
\(358\) 0 0
\(359\) 5.04942 0.266498 0.133249 0.991083i \(-0.457459\pi\)
0.133249 + 0.991083i \(0.457459\pi\)
\(360\) 0 0
\(361\) 41.3111 2.17427
\(362\) 0 0
\(363\) 1.52184 0.0798758
\(364\) 0 0
\(365\) 25.5817 1.33901
\(366\) 0 0
\(367\) −25.9566 −1.35492 −0.677461 0.735559i \(-0.736920\pi\)
−0.677461 + 0.735559i \(0.736920\pi\)
\(368\) 0 0
\(369\) 15.0067 0.781219
\(370\) 0 0
\(371\) 3.40321 0.176686
\(372\) 0 0
\(373\) −8.23890 −0.426594 −0.213297 0.976987i \(-0.568420\pi\)
−0.213297 + 0.976987i \(0.568420\pi\)
\(374\) 0 0
\(375\) −35.0606 −1.81052
\(376\) 0 0
\(377\) 26.1198 1.34524
\(378\) 0 0
\(379\) −11.7755 −0.604866 −0.302433 0.953171i \(-0.597799\pi\)
−0.302433 + 0.953171i \(0.597799\pi\)
\(380\) 0 0
\(381\) 1.04598 0.0535871
\(382\) 0 0
\(383\) 23.2827 1.18969 0.594846 0.803839i \(-0.297213\pi\)
0.594846 + 0.803839i \(0.297213\pi\)
\(384\) 0 0
\(385\) 5.76602 0.293864
\(386\) 0 0
\(387\) −3.11818 −0.158506
\(388\) 0 0
\(389\) −11.3717 −0.576566 −0.288283 0.957545i \(-0.593084\pi\)
−0.288283 + 0.957545i \(0.593084\pi\)
\(390\) 0 0
\(391\) 5.32962 0.269530
\(392\) 0 0
\(393\) −5.59933 −0.282449
\(394\) 0 0
\(395\) 7.46058 0.375382
\(396\) 0 0
\(397\) 25.0645 1.25795 0.628977 0.777424i \(-0.283474\pi\)
0.628977 + 0.777424i \(0.283474\pi\)
\(398\) 0 0
\(399\) 22.3895 1.12088
\(400\) 0 0
\(401\) 7.49631 0.374348 0.187174 0.982327i \(-0.440067\pi\)
0.187174 + 0.982327i \(0.440067\pi\)
\(402\) 0 0
\(403\) 23.8348 1.18730
\(404\) 0 0
\(405\) −5.84334 −0.290358
\(406\) 0 0
\(407\) 5.24701 0.260085
\(408\) 0 0
\(409\) −11.4657 −0.566941 −0.283470 0.958981i \(-0.591486\pi\)
−0.283470 + 0.958981i \(0.591486\pi\)
\(410\) 0 0
\(411\) −14.9190 −0.735901
\(412\) 0 0
\(413\) 7.32962 0.360667
\(414\) 0 0
\(415\) 15.0047 0.736550
\(416\) 0 0
\(417\) 11.9230 0.583870
\(418\) 0 0
\(419\) −3.40998 −0.166588 −0.0832942 0.996525i \(-0.526544\pi\)
−0.0832942 + 0.996525i \(0.526544\pi\)
\(420\) 0 0
\(421\) 13.9524 0.679998 0.339999 0.940426i \(-0.389573\pi\)
0.339999 + 0.940426i \(0.389573\pi\)
\(422\) 0 0
\(423\) −64.7349 −3.14752
\(424\) 0 0
\(425\) −1.82519 −0.0885349
\(426\) 0 0
\(427\) −3.82310 −0.185013
\(428\) 0 0
\(429\) 23.0641 1.11354
\(430\) 0 0
\(431\) −27.7702 −1.33764 −0.668822 0.743423i \(-0.733201\pi\)
−0.668822 + 0.743423i \(0.733201\pi\)
\(432\) 0 0
\(433\) −16.1913 −0.778103 −0.389052 0.921216i \(-0.627197\pi\)
−0.389052 + 0.921216i \(0.627197\pi\)
\(434\) 0 0
\(435\) 54.2754 2.60230
\(436\) 0 0
\(437\) −41.3899 −1.97995
\(438\) 0 0
\(439\) 14.5266 0.693318 0.346659 0.937991i \(-0.387316\pi\)
0.346659 + 0.937991i \(0.387316\pi\)
\(440\) 0 0
\(441\) 5.31175 0.252941
\(442\) 0 0
\(443\) −33.5993 −1.59635 −0.798176 0.602424i \(-0.794202\pi\)
−0.798176 + 0.602424i \(0.794202\pi\)
\(444\) 0 0
\(445\) 30.3700 1.43968
\(446\) 0 0
\(447\) 6.25652 0.295923
\(448\) 0 0
\(449\) −13.7191 −0.647447 −0.323723 0.946152i \(-0.604935\pi\)
−0.323723 + 0.946152i \(0.604935\pi\)
\(450\) 0 0
\(451\) −9.14252 −0.430504
\(452\) 0 0
\(453\) 8.81962 0.414382
\(454\) 0 0
\(455\) −4.40485 −0.206503
\(456\) 0 0
\(457\) 9.18874 0.429831 0.214916 0.976633i \(-0.431052\pi\)
0.214916 + 0.976633i \(0.431052\pi\)
\(458\) 0 0
\(459\) −6.66481 −0.311087
\(460\) 0 0
\(461\) −27.3338 −1.27306 −0.636531 0.771251i \(-0.719631\pi\)
−0.636531 + 0.771251i \(0.719631\pi\)
\(462\) 0 0
\(463\) −7.36191 −0.342137 −0.171069 0.985259i \(-0.554722\pi\)
−0.171069 + 0.985259i \(0.554722\pi\)
\(464\) 0 0
\(465\) 49.5272 2.29677
\(466\) 0 0
\(467\) 28.4805 1.31792 0.658960 0.752178i \(-0.270997\pi\)
0.658960 + 0.752178i \(0.270997\pi\)
\(468\) 0 0
\(469\) −0.394361 −0.0182099
\(470\) 0 0
\(471\) −28.6950 −1.32219
\(472\) 0 0
\(473\) 1.89968 0.0873476
\(474\) 0 0
\(475\) 14.1745 0.650370
\(476\) 0 0
\(477\) 18.0770 0.827691
\(478\) 0 0
\(479\) −7.72259 −0.352854 −0.176427 0.984314i \(-0.556454\pi\)
−0.176427 + 0.984314i \(0.556454\pi\)
\(480\) 0 0
\(481\) −4.00836 −0.182766
\(482\) 0 0
\(483\) −15.3654 −0.699148
\(484\) 0 0
\(485\) 13.7626 0.624927
\(486\) 0 0
\(487\) −29.0172 −1.31490 −0.657448 0.753500i \(-0.728364\pi\)
−0.657448 + 0.753500i \(0.728364\pi\)
\(488\) 0 0
\(489\) 45.1518 2.04183
\(490\) 0 0
\(491\) −14.2250 −0.641964 −0.320982 0.947085i \(-0.604013\pi\)
−0.320982 + 0.947085i \(0.604013\pi\)
\(492\) 0 0
\(493\) 10.5657 0.475855
\(494\) 0 0
\(495\) 30.6277 1.37661
\(496\) 0 0
\(497\) 2.15137 0.0965021
\(498\) 0 0
\(499\) 21.6971 0.971294 0.485647 0.874155i \(-0.338584\pi\)
0.485647 + 0.874155i \(0.338584\pi\)
\(500\) 0 0
\(501\) −23.4074 −1.04577
\(502\) 0 0
\(503\) 26.4700 1.18024 0.590120 0.807316i \(-0.299081\pi\)
0.590120 + 0.807316i \(0.299081\pi\)
\(504\) 0 0
\(505\) 24.3811 1.08494
\(506\) 0 0
\(507\) 19.8597 0.882002
\(508\) 0 0
\(509\) 37.3380 1.65498 0.827488 0.561483i \(-0.189769\pi\)
0.827488 + 0.561483i \(0.189769\pi\)
\(510\) 0 0
\(511\) −14.3572 −0.635127
\(512\) 0 0
\(513\) 51.7591 2.28522
\(514\) 0 0
\(515\) 12.3096 0.542427
\(516\) 0 0
\(517\) 39.4383 1.73449
\(518\) 0 0
\(519\) −35.6456 −1.56467
\(520\) 0 0
\(521\) 40.7287 1.78436 0.892179 0.451681i \(-0.149176\pi\)
0.892179 + 0.451681i \(0.149176\pi\)
\(522\) 0 0
\(523\) −29.7056 −1.29894 −0.649468 0.760389i \(-0.725008\pi\)
−0.649468 + 0.760389i \(0.725008\pi\)
\(524\) 0 0
\(525\) 5.26205 0.229655
\(526\) 0 0
\(527\) 9.64137 0.419985
\(528\) 0 0
\(529\) 5.40485 0.234993
\(530\) 0 0
\(531\) 38.9331 1.68955
\(532\) 0 0
\(533\) 6.98426 0.302522
\(534\) 0 0
\(535\) −26.7034 −1.15449
\(536\) 0 0
\(537\) −52.2496 −2.25473
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −45.4631 −1.95461 −0.977305 0.211835i \(-0.932056\pi\)
−0.977305 + 0.211835i \(0.932056\pi\)
\(542\) 0 0
\(543\) 53.0887 2.27825
\(544\) 0 0
\(545\) 34.5445 1.47972
\(546\) 0 0
\(547\) 10.3113 0.440879 0.220440 0.975401i \(-0.429251\pi\)
0.220440 + 0.975401i \(0.429251\pi\)
\(548\) 0 0
\(549\) −20.3074 −0.866698
\(550\) 0 0
\(551\) −82.0534 −3.49559
\(552\) 0 0
\(553\) −4.18710 −0.178054
\(554\) 0 0
\(555\) −8.32913 −0.353552
\(556\) 0 0
\(557\) 16.8770 0.715101 0.357550 0.933894i \(-0.383612\pi\)
0.357550 + 0.933894i \(0.383612\pi\)
\(558\) 0 0
\(559\) −1.45123 −0.0613805
\(560\) 0 0
\(561\) 9.32962 0.393897
\(562\) 0 0
\(563\) 35.6197 1.50119 0.750597 0.660761i \(-0.229766\pi\)
0.750597 + 0.660761i \(0.229766\pi\)
\(564\) 0 0
\(565\) −11.1714 −0.469983
\(566\) 0 0
\(567\) 3.27946 0.137724
\(568\) 0 0
\(569\) 32.2578 1.35232 0.676159 0.736755i \(-0.263643\pi\)
0.676159 + 0.736755i \(0.263643\pi\)
\(570\) 0 0
\(571\) −35.4946 −1.48540 −0.742702 0.669622i \(-0.766456\pi\)
−0.742702 + 0.669622i \(0.766456\pi\)
\(572\) 0 0
\(573\) 30.8519 1.28886
\(574\) 0 0
\(575\) −9.72758 −0.405668
\(576\) 0 0
\(577\) 33.1356 1.37945 0.689726 0.724071i \(-0.257731\pi\)
0.689726 + 0.724071i \(0.257731\pi\)
\(578\) 0 0
\(579\) 4.36068 0.181224
\(580\) 0 0
\(581\) −8.42108 −0.349365
\(582\) 0 0
\(583\) −11.0130 −0.456113
\(584\) 0 0
\(585\) −23.3975 −0.967366
\(586\) 0 0
\(587\) 35.1397 1.45037 0.725186 0.688553i \(-0.241754\pi\)
0.725186 + 0.688553i \(0.241754\pi\)
\(588\) 0 0
\(589\) −74.8751 −3.08518
\(590\) 0 0
\(591\) 20.9647 0.862372
\(592\) 0 0
\(593\) −30.0729 −1.23495 −0.617474 0.786591i \(-0.711844\pi\)
−0.617474 + 0.786591i \(0.711844\pi\)
\(594\) 0 0
\(595\) −1.78180 −0.0730466
\(596\) 0 0
\(597\) −66.8373 −2.73547
\(598\) 0 0
\(599\) −20.6464 −0.843591 −0.421796 0.906691i \(-0.638600\pi\)
−0.421796 + 0.906691i \(0.638600\pi\)
\(600\) 0 0
\(601\) −20.5961 −0.840134 −0.420067 0.907493i \(-0.637993\pi\)
−0.420067 + 0.907493i \(0.637993\pi\)
\(602\) 0 0
\(603\) −2.09475 −0.0853047
\(604\) 0 0
\(605\) 0.940548 0.0382387
\(606\) 0 0
\(607\) 14.9199 0.605582 0.302791 0.953057i \(-0.402082\pi\)
0.302791 + 0.953057i \(0.402082\pi\)
\(608\) 0 0
\(609\) −30.4610 −1.23434
\(610\) 0 0
\(611\) −30.1282 −1.21886
\(612\) 0 0
\(613\) −24.9575 −1.00802 −0.504011 0.863697i \(-0.668143\pi\)
−0.504011 + 0.863697i \(0.668143\pi\)
\(614\) 0 0
\(615\) 14.5129 0.585215
\(616\) 0 0
\(617\) −7.19596 −0.289698 −0.144849 0.989454i \(-0.546270\pi\)
−0.144849 + 0.989454i \(0.546270\pi\)
\(618\) 0 0
\(619\) 17.7770 0.714517 0.357258 0.934006i \(-0.383712\pi\)
0.357258 + 0.934006i \(0.383712\pi\)
\(620\) 0 0
\(621\) −35.5209 −1.42540
\(622\) 0 0
\(623\) −17.0446 −0.682877
\(624\) 0 0
\(625\) −12.5427 −0.501708
\(626\) 0 0
\(627\) −72.4540 −2.89354
\(628\) 0 0
\(629\) −1.62142 −0.0646501
\(630\) 0 0
\(631\) −35.3428 −1.40698 −0.703488 0.710708i \(-0.748375\pi\)
−0.703488 + 0.710708i \(0.748375\pi\)
\(632\) 0 0
\(633\) 32.1556 1.27807
\(634\) 0 0
\(635\) 0.646450 0.0256536
\(636\) 0 0
\(637\) 2.47214 0.0979496
\(638\) 0 0
\(639\) 11.4275 0.452067
\(640\) 0 0
\(641\) −3.66153 −0.144622 −0.0723108 0.997382i \(-0.523037\pi\)
−0.0723108 + 0.997382i \(0.523037\pi\)
\(642\) 0 0
\(643\) −29.6783 −1.17040 −0.585199 0.810890i \(-0.698984\pi\)
−0.585199 + 0.810890i \(0.698984\pi\)
\(644\) 0 0
\(645\) −3.01557 −0.118738
\(646\) 0 0
\(647\) −28.9758 −1.13916 −0.569579 0.821937i \(-0.692894\pi\)
−0.569579 + 0.821937i \(0.692894\pi\)
\(648\) 0 0
\(649\) −23.7191 −0.931058
\(650\) 0 0
\(651\) −27.7962 −1.08942
\(652\) 0 0
\(653\) 19.2003 0.751367 0.375684 0.926748i \(-0.377408\pi\)
0.375684 + 0.926748i \(0.377408\pi\)
\(654\) 0 0
\(655\) −3.46058 −0.135216
\(656\) 0 0
\(657\) −76.2621 −2.97527
\(658\) 0 0
\(659\) 5.13350 0.199973 0.0999865 0.994989i \(-0.468120\pi\)
0.0999865 + 0.994989i \(0.468120\pi\)
\(660\) 0 0
\(661\) −34.8839 −1.35683 −0.678413 0.734681i \(-0.737332\pi\)
−0.678413 + 0.734681i \(0.737332\pi\)
\(662\) 0 0
\(663\) −7.12720 −0.276797
\(664\) 0 0
\(665\) 13.8375 0.536595
\(666\) 0 0
\(667\) 56.3111 2.18037
\(668\) 0 0
\(669\) −44.8700 −1.73478
\(670\) 0 0
\(671\) 12.3718 0.477609
\(672\) 0 0
\(673\) −46.2976 −1.78464 −0.892320 0.451403i \(-0.850924\pi\)
−0.892320 + 0.451403i \(0.850924\pi\)
\(674\) 0 0
\(675\) 12.1646 0.468214
\(676\) 0 0
\(677\) −8.29450 −0.318784 −0.159392 0.987215i \(-0.550953\pi\)
−0.159392 + 0.987215i \(0.550953\pi\)
\(678\) 0 0
\(679\) −7.72398 −0.296419
\(680\) 0 0
\(681\) 44.2972 1.69747
\(682\) 0 0
\(683\) 43.0198 1.64611 0.823053 0.567964i \(-0.192269\pi\)
0.823053 + 0.567964i \(0.192269\pi\)
\(684\) 0 0
\(685\) −9.22047 −0.352296
\(686\) 0 0
\(687\) 42.9815 1.63985
\(688\) 0 0
\(689\) 8.41321 0.320518
\(690\) 0 0
\(691\) 36.3817 1.38403 0.692013 0.721885i \(-0.256724\pi\)
0.692013 + 0.721885i \(0.256724\pi\)
\(692\) 0 0
\(693\) −17.1892 −0.652963
\(694\) 0 0
\(695\) 7.36880 0.279514
\(696\) 0 0
\(697\) 2.82519 0.107012
\(698\) 0 0
\(699\) −37.2155 −1.40762
\(700\) 0 0
\(701\) 18.5920 0.702208 0.351104 0.936336i \(-0.385806\pi\)
0.351104 + 0.936336i \(0.385806\pi\)
\(702\) 0 0
\(703\) 12.5920 0.474915
\(704\) 0 0
\(705\) −62.6045 −2.35782
\(706\) 0 0
\(707\) −13.6834 −0.514618
\(708\) 0 0
\(709\) 11.0871 0.416384 0.208192 0.978088i \(-0.433242\pi\)
0.208192 + 0.978088i \(0.433242\pi\)
\(710\) 0 0
\(711\) −22.2409 −0.834098
\(712\) 0 0
\(713\) 51.3849 1.92438
\(714\) 0 0
\(715\) 14.2544 0.533084
\(716\) 0 0
\(717\) 20.3517 0.760047
\(718\) 0 0
\(719\) −0.357237 −0.0133227 −0.00666135 0.999978i \(-0.502120\pi\)
−0.00666135 + 0.999978i \(0.502120\pi\)
\(720\) 0 0
\(721\) −6.90854 −0.257287
\(722\) 0 0
\(723\) 63.5153 2.36216
\(724\) 0 0
\(725\) −19.2844 −0.716205
\(726\) 0 0
\(727\) 3.07614 0.114088 0.0570439 0.998372i \(-0.481833\pi\)
0.0570439 + 0.998372i \(0.481833\pi\)
\(728\) 0 0
\(729\) −40.2245 −1.48980
\(730\) 0 0
\(731\) −0.587035 −0.0217123
\(732\) 0 0
\(733\) 17.9930 0.664588 0.332294 0.943176i \(-0.392177\pi\)
0.332294 + 0.943176i \(0.392177\pi\)
\(734\) 0 0
\(735\) 5.13695 0.189479
\(736\) 0 0
\(737\) 1.27618 0.0470086
\(738\) 0 0
\(739\) 8.47140 0.311625 0.155813 0.987787i \(-0.450200\pi\)
0.155813 + 0.987787i \(0.450200\pi\)
\(740\) 0 0
\(741\) 55.3500 2.03333
\(742\) 0 0
\(743\) −23.1587 −0.849612 −0.424806 0.905284i \(-0.639658\pi\)
−0.424806 + 0.905284i \(0.639658\pi\)
\(744\) 0 0
\(745\) 3.86674 0.141666
\(746\) 0 0
\(747\) −44.7307 −1.63661
\(748\) 0 0
\(749\) 14.9868 0.547605
\(750\) 0 0
\(751\) 19.3849 0.707363 0.353682 0.935366i \(-0.384930\pi\)
0.353682 + 0.935366i \(0.384930\pi\)
\(752\) 0 0
\(753\) 81.0896 2.95507
\(754\) 0 0
\(755\) 5.45083 0.198376
\(756\) 0 0
\(757\) 48.9669 1.77973 0.889866 0.456222i \(-0.150798\pi\)
0.889866 + 0.456222i \(0.150798\pi\)
\(758\) 0 0
\(759\) 49.7233 1.80484
\(760\) 0 0
\(761\) −25.9354 −0.940158 −0.470079 0.882624i \(-0.655775\pi\)
−0.470079 + 0.882624i \(0.655775\pi\)
\(762\) 0 0
\(763\) −19.3874 −0.701872
\(764\) 0 0
\(765\) −9.46448 −0.342189
\(766\) 0 0
\(767\) 18.1198 0.654269
\(768\) 0 0
\(769\) −5.37649 −0.193881 −0.0969407 0.995290i \(-0.530906\pi\)
−0.0969407 + 0.995290i \(0.530906\pi\)
\(770\) 0 0
\(771\) −26.0756 −0.939092
\(772\) 0 0
\(773\) −3.82642 −0.137627 −0.0688135 0.997630i \(-0.521921\pi\)
−0.0688135 + 0.997630i \(0.521921\pi\)
\(774\) 0 0
\(775\) −17.5974 −0.632116
\(776\) 0 0
\(777\) 4.67456 0.167699
\(778\) 0 0
\(779\) −21.9405 −0.786100
\(780\) 0 0
\(781\) −6.96198 −0.249119
\(782\) 0 0
\(783\) −70.4183 −2.51654
\(784\) 0 0
\(785\) −17.7345 −0.632970
\(786\) 0 0
\(787\) 1.50487 0.0536427 0.0268214 0.999640i \(-0.491461\pi\)
0.0268214 + 0.999640i \(0.491461\pi\)
\(788\) 0 0
\(789\) 42.2628 1.50459
\(790\) 0 0
\(791\) 6.26971 0.222925
\(792\) 0 0
\(793\) −9.45123 −0.335623
\(794\) 0 0
\(795\) 17.4821 0.620027
\(796\) 0 0
\(797\) 28.4448 1.00757 0.503783 0.863830i \(-0.331941\pi\)
0.503783 + 0.863830i \(0.331941\pi\)
\(798\) 0 0
\(799\) −12.1871 −0.431149
\(800\) 0 0
\(801\) −90.5366 −3.19896
\(802\) 0 0
\(803\) 46.4610 1.63957
\(804\) 0 0
\(805\) −9.49631 −0.334701
\(806\) 0 0
\(807\) 28.1360 0.990436
\(808\) 0 0
\(809\) −29.5952 −1.04051 −0.520255 0.854011i \(-0.674163\pi\)
−0.520255 + 0.854011i \(0.674163\pi\)
\(810\) 0 0
\(811\) −49.2818 −1.73052 −0.865259 0.501325i \(-0.832846\pi\)
−0.865259 + 0.501325i \(0.832846\pi\)
\(812\) 0 0
\(813\) 43.7071 1.53287
\(814\) 0 0
\(815\) 27.9053 0.977481
\(816\) 0 0
\(817\) 4.55892 0.159497
\(818\) 0 0
\(819\) 13.1314 0.458847
\(820\) 0 0
\(821\) −39.0114 −1.36151 −0.680753 0.732513i \(-0.738347\pi\)
−0.680753 + 0.732513i \(0.738347\pi\)
\(822\) 0 0
\(823\) 36.4772 1.27152 0.635758 0.771888i \(-0.280688\pi\)
0.635758 + 0.771888i \(0.280688\pi\)
\(824\) 0 0
\(825\) −17.0284 −0.592851
\(826\) 0 0
\(827\) 7.06846 0.245795 0.122897 0.992419i \(-0.460781\pi\)
0.122897 + 0.992419i \(0.460781\pi\)
\(828\) 0 0
\(829\) −19.5952 −0.680568 −0.340284 0.940323i \(-0.610523\pi\)
−0.340284 + 0.940323i \(0.610523\pi\)
\(830\) 0 0
\(831\) −30.9416 −1.07335
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −14.4666 −0.500636
\(836\) 0 0
\(837\) −64.2579 −2.22108
\(838\) 0 0
\(839\) −5.42576 −0.187318 −0.0936589 0.995604i \(-0.529856\pi\)
−0.0936589 + 0.995604i \(0.529856\pi\)
\(840\) 0 0
\(841\) 82.6338 2.84944
\(842\) 0 0
\(843\) 5.94608 0.204794
\(844\) 0 0
\(845\) 12.2740 0.422238
\(846\) 0 0
\(847\) −0.527864 −0.0181376
\(848\) 0 0
\(849\) −44.5032 −1.52735
\(850\) 0 0
\(851\) −8.64153 −0.296228
\(852\) 0 0
\(853\) 8.52577 0.291917 0.145958 0.989291i \(-0.453373\pi\)
0.145958 + 0.989291i \(0.453373\pi\)
\(854\) 0 0
\(855\) 73.5013 2.51369
\(856\) 0 0
\(857\) 28.9710 0.989630 0.494815 0.868998i \(-0.335236\pi\)
0.494815 + 0.868998i \(0.335236\pi\)
\(858\) 0 0
\(859\) −9.67776 −0.330201 −0.165100 0.986277i \(-0.552795\pi\)
−0.165100 + 0.986277i \(0.552795\pi\)
\(860\) 0 0
\(861\) −8.14506 −0.277583
\(862\) 0 0
\(863\) −9.67293 −0.329270 −0.164635 0.986355i \(-0.552645\pi\)
−0.164635 + 0.986355i \(0.552645\pi\)
\(864\) 0 0
\(865\) −22.0302 −0.749048
\(866\) 0 0
\(867\) −2.88301 −0.0979122
\(868\) 0 0
\(869\) 13.5498 0.459644
\(870\) 0 0
\(871\) −0.974913 −0.0330337
\(872\) 0 0
\(873\) −41.0279 −1.38858
\(874\) 0 0
\(875\) 12.1611 0.411121
\(876\) 0 0
\(877\) −40.8080 −1.37799 −0.688995 0.724766i \(-0.741948\pi\)
−0.688995 + 0.724766i \(0.741948\pi\)
\(878\) 0 0
\(879\) −6.58777 −0.222200
\(880\) 0 0
\(881\) 0.544255 0.0183364 0.00916821 0.999958i \(-0.497082\pi\)
0.00916821 + 0.999958i \(0.497082\pi\)
\(882\) 0 0
\(883\) −7.31126 −0.246043 −0.123022 0.992404i \(-0.539258\pi\)
−0.123022 + 0.992404i \(0.539258\pi\)
\(884\) 0 0
\(885\) 37.6519 1.26565
\(886\) 0 0
\(887\) −13.8600 −0.465374 −0.232687 0.972552i \(-0.574752\pi\)
−0.232687 + 0.972552i \(0.574752\pi\)
\(888\) 0 0
\(889\) −0.362807 −0.0121682
\(890\) 0 0
\(891\) −10.6126 −0.355534
\(892\) 0 0
\(893\) 94.6453 3.16718
\(894\) 0 0
\(895\) −32.2920 −1.07940
\(896\) 0 0
\(897\) −37.9852 −1.26829
\(898\) 0 0
\(899\) 101.868 3.39748
\(900\) 0 0
\(901\) 3.40321 0.113378
\(902\) 0 0
\(903\) 1.69243 0.0563205
\(904\) 0 0
\(905\) 32.8106 1.09066
\(906\) 0 0
\(907\) −50.8411 −1.68815 −0.844075 0.536225i \(-0.819850\pi\)
−0.844075 + 0.536225i \(0.819850\pi\)
\(908\) 0 0
\(909\) −72.6829 −2.41074
\(910\) 0 0
\(911\) −25.8459 −0.856311 −0.428156 0.903705i \(-0.640836\pi\)
−0.428156 + 0.903705i \(0.640836\pi\)
\(912\) 0 0
\(913\) 27.2512 0.901883
\(914\) 0 0
\(915\) −19.6391 −0.649248
\(916\) 0 0
\(917\) 1.94218 0.0641365
\(918\) 0 0
\(919\) −44.8471 −1.47937 −0.739684 0.672954i \(-0.765025\pi\)
−0.739684 + 0.672954i \(0.765025\pi\)
\(920\) 0 0
\(921\) −59.4699 −1.95960
\(922\) 0 0
\(923\) 5.31848 0.175060
\(924\) 0 0
\(925\) 2.95940 0.0973044
\(926\) 0 0
\(927\) −36.6964 −1.20527
\(928\) 0 0
\(929\) −44.2402 −1.45147 −0.725737 0.687972i \(-0.758501\pi\)
−0.725737 + 0.687972i \(0.758501\pi\)
\(930\) 0 0
\(931\) −7.76602 −0.254521
\(932\) 0 0
\(933\) 52.5428 1.72017
\(934\) 0 0
\(935\) 5.76602 0.188569
\(936\) 0 0
\(937\) −41.2813 −1.34860 −0.674300 0.738457i \(-0.735555\pi\)
−0.674300 + 0.738457i \(0.735555\pi\)
\(938\) 0 0
\(939\) 45.1505 1.47343
\(940\) 0 0
\(941\) −11.1816 −0.364509 −0.182254 0.983251i \(-0.558339\pi\)
−0.182254 + 0.983251i \(0.558339\pi\)
\(942\) 0 0
\(943\) 15.0572 0.490330
\(944\) 0 0
\(945\) 11.8754 0.386305
\(946\) 0 0
\(947\) −18.5810 −0.603802 −0.301901 0.953339i \(-0.597621\pi\)
−0.301901 + 0.953339i \(0.597621\pi\)
\(948\) 0 0
\(949\) −35.4930 −1.15215
\(950\) 0 0
\(951\) −5.05991 −0.164079
\(952\) 0 0
\(953\) 13.3359 0.431993 0.215997 0.976394i \(-0.430700\pi\)
0.215997 + 0.976394i \(0.430700\pi\)
\(954\) 0 0
\(955\) 19.0675 0.617011
\(956\) 0 0
\(957\) 98.5739 3.18644
\(958\) 0 0
\(959\) 5.17481 0.167103
\(960\) 0 0
\(961\) 61.9561 1.99858
\(962\) 0 0
\(963\) 79.6060 2.56527
\(964\) 0 0
\(965\) 2.69505 0.0867567
\(966\) 0 0
\(967\) −10.9877 −0.353341 −0.176670 0.984270i \(-0.556533\pi\)
−0.176670 + 0.984270i \(0.556533\pi\)
\(968\) 0 0
\(969\) 22.3895 0.719255
\(970\) 0 0
\(971\) −47.3831 −1.52059 −0.760297 0.649575i \(-0.774947\pi\)
−0.760297 + 0.649575i \(0.774947\pi\)
\(972\) 0 0
\(973\) −4.13559 −0.132581
\(974\) 0 0
\(975\) 13.0085 0.416606
\(976\) 0 0
\(977\) 1.49148 0.0477166 0.0238583 0.999715i \(-0.492405\pi\)
0.0238583 + 0.999715i \(0.492405\pi\)
\(978\) 0 0
\(979\) 55.1574 1.76284
\(980\) 0 0
\(981\) −102.981 −3.28794
\(982\) 0 0
\(983\) 53.9441 1.72055 0.860275 0.509830i \(-0.170292\pi\)
0.860275 + 0.509830i \(0.170292\pi\)
\(984\) 0 0
\(985\) 12.9569 0.412841
\(986\) 0 0
\(987\) 35.1356 1.11838
\(988\) 0 0
\(989\) −3.12867 −0.0994860
\(990\) 0 0
\(991\) 51.2401 1.62769 0.813847 0.581079i \(-0.197369\pi\)
0.813847 + 0.581079i \(0.197369\pi\)
\(992\) 0 0
\(993\) −41.1778 −1.30674
\(994\) 0 0
\(995\) −41.3077 −1.30954
\(996\) 0 0
\(997\) −12.0544 −0.381766 −0.190883 0.981613i \(-0.561135\pi\)
−0.190883 + 0.981613i \(0.561135\pi\)
\(998\) 0 0
\(999\) 10.8064 0.341900
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.q.1.1 4
4.3 odd 2 952.2.a.g.1.4 4
8.3 odd 2 7616.2.a.bj.1.1 4
8.5 even 2 7616.2.a.bp.1.4 4
12.11 even 2 8568.2.a.bj.1.3 4
28.27 even 2 6664.2.a.o.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.4 4 4.3 odd 2
1904.2.a.q.1.1 4 1.1 even 1 trivial
6664.2.a.o.1.1 4 28.27 even 2
7616.2.a.bj.1.1 4 8.3 odd 2
7616.2.a.bp.1.4 4 8.5 even 2
8568.2.a.bj.1.3 4 12.11 even 2