Properties

Label 6762.2.a.bq.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.70156 q^{10} -1.00000 q^{12} -5.70156 q^{13} -3.70156 q^{15} +1.00000 q^{16} -1.00000 q^{18} +5.40312 q^{19} +3.70156 q^{20} -1.00000 q^{23} +1.00000 q^{24} +8.70156 q^{25} +5.70156 q^{26} -1.00000 q^{27} +0.298438 q^{29} +3.70156 q^{30} -6.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} +3.70156 q^{37} -5.40312 q^{38} +5.70156 q^{39} -3.70156 q^{40} -7.70156 q^{41} +5.70156 q^{43} +3.70156 q^{45} +1.00000 q^{46} -3.70156 q^{47} -1.00000 q^{48} -8.70156 q^{50} -5.70156 q^{52} +9.40312 q^{53} +1.00000 q^{54} -5.40312 q^{57} -0.298438 q^{58} +0.596876 q^{59} -3.70156 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -21.1047 q^{65} +4.00000 q^{67} +1.00000 q^{69} +7.40312 q^{71} -1.00000 q^{72} +5.40312 q^{73} -3.70156 q^{74} -8.70156 q^{75} +5.40312 q^{76} -5.70156 q^{78} +3.70156 q^{80} +1.00000 q^{81} +7.70156 q^{82} +2.59688 q^{83} -5.70156 q^{86} -0.298438 q^{87} +15.4031 q^{89} -3.70156 q^{90} -1.00000 q^{92} +6.00000 q^{93} +3.70156 q^{94} +20.0000 q^{95} +1.00000 q^{96} +1.10469 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} - 5 q^{13} - q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{19} + q^{20} - 2 q^{23} + 2 q^{24} + 11 q^{25} + 5 q^{26} - 2 q^{27} + 7 q^{29} + q^{30} - 12 q^{31} - 2 q^{32} + 2 q^{36} + q^{37} + 2 q^{38} + 5 q^{39} - q^{40} - 9 q^{41} + 5 q^{43} + q^{45} + 2 q^{46} - q^{47} - 2 q^{48} - 11 q^{50} - 5 q^{52} + 6 q^{53} + 2 q^{54} + 2 q^{57} - 7 q^{58} + 14 q^{59} - q^{60} - 20 q^{61} + 12 q^{62} + 2 q^{64} - 23 q^{65} + 8 q^{67} + 2 q^{69} + 2 q^{71} - 2 q^{72} - 2 q^{73} - q^{74} - 11 q^{75} - 2 q^{76} - 5 q^{78} + q^{80} + 2 q^{81} + 9 q^{82} + 18 q^{83} - 5 q^{86} - 7 q^{87} + 18 q^{89} - q^{90} - 2 q^{92} + 12 q^{93} + q^{94} + 40 q^{95} + 2 q^{96} - 17 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.70156 1.65539 0.827694 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.70156 −1.17054
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.70156 −1.58133 −0.790664 0.612250i \(-0.790265\pi\)
−0.790664 + 0.612250i \(0.790265\pi\)
\(14\) 0 0
\(15\) −3.70156 −0.955739
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.40312 1.23956 0.619781 0.784775i \(-0.287221\pi\)
0.619781 + 0.784775i \(0.287221\pi\)
\(20\) 3.70156 0.827694
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 8.70156 1.74031
\(26\) 5.70156 1.11817
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.298438 0.0554185 0.0277093 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(30\) 3.70156 0.675810
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.70156 0.608533 0.304267 0.952587i \(-0.401589\pi\)
0.304267 + 0.952587i \(0.401589\pi\)
\(38\) −5.40312 −0.876502
\(39\) 5.70156 0.912981
\(40\) −3.70156 −0.585268
\(41\) −7.70156 −1.20278 −0.601391 0.798955i \(-0.705387\pi\)
−0.601391 + 0.798955i \(0.705387\pi\)
\(42\) 0 0
\(43\) 5.70156 0.869480 0.434740 0.900556i \(-0.356840\pi\)
0.434740 + 0.900556i \(0.356840\pi\)
\(44\) 0 0
\(45\) 3.70156 0.551796
\(46\) 1.00000 0.147442
\(47\) −3.70156 −0.539928 −0.269964 0.962870i \(-0.587012\pi\)
−0.269964 + 0.962870i \(0.587012\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −8.70156 −1.23059
\(51\) 0 0
\(52\) −5.70156 −0.790664
\(53\) 9.40312 1.29162 0.645809 0.763499i \(-0.276520\pi\)
0.645809 + 0.763499i \(0.276520\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −5.40312 −0.715661
\(58\) −0.298438 −0.0391868
\(59\) 0.596876 0.0777066 0.0388533 0.999245i \(-0.487629\pi\)
0.0388533 + 0.999245i \(0.487629\pi\)
\(60\) −3.70156 −0.477870
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −21.1047 −2.61771
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.40312 0.878589 0.439295 0.898343i \(-0.355228\pi\)
0.439295 + 0.898343i \(0.355228\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.40312 0.632388 0.316194 0.948695i \(-0.397595\pi\)
0.316194 + 0.948695i \(0.397595\pi\)
\(74\) −3.70156 −0.430298
\(75\) −8.70156 −1.00477
\(76\) 5.40312 0.619781
\(77\) 0 0
\(78\) −5.70156 −0.645575
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 3.70156 0.413847
\(81\) 1.00000 0.111111
\(82\) 7.70156 0.850495
\(83\) 2.59688 0.285044 0.142522 0.989792i \(-0.454479\pi\)
0.142522 + 0.989792i \(0.454479\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.70156 −0.614815
\(87\) −0.298438 −0.0319959
\(88\) 0 0
\(89\) 15.4031 1.63273 0.816364 0.577538i \(-0.195986\pi\)
0.816364 + 0.577538i \(0.195986\pi\)
\(90\) −3.70156 −0.390179
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 3.70156 0.381787
\(95\) 20.0000 2.05196
\(96\) 1.00000 0.102062
\(97\) 1.10469 0.112164 0.0560820 0.998426i \(-0.482139\pi\)
0.0560820 + 0.998426i \(0.482139\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.70156 0.870156
\(101\) −4.59688 −0.457406 −0.228703 0.973496i \(-0.573449\pi\)
−0.228703 + 0.973496i \(0.573449\pi\)
\(102\) 0 0
\(103\) 17.1047 1.68537 0.842687 0.538403i \(-0.180972\pi\)
0.842687 + 0.538403i \(0.180972\pi\)
\(104\) 5.70156 0.559084
\(105\) 0 0
\(106\) −9.40312 −0.913312
\(107\) 14.8062 1.43137 0.715687 0.698421i \(-0.246114\pi\)
0.715687 + 0.698421i \(0.246114\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 15.7016 1.50394 0.751968 0.659199i \(-0.229105\pi\)
0.751968 + 0.659199i \(0.229105\pi\)
\(110\) 0 0
\(111\) −3.70156 −0.351337
\(112\) 0 0
\(113\) −4.29844 −0.404363 −0.202182 0.979348i \(-0.564803\pi\)
−0.202182 + 0.979348i \(0.564803\pi\)
\(114\) 5.40312 0.506049
\(115\) −3.70156 −0.345172
\(116\) 0.298438 0.0277093
\(117\) −5.70156 −0.527110
\(118\) −0.596876 −0.0549469
\(119\) 0 0
\(120\) 3.70156 0.337905
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) 7.70156 0.694426
\(124\) −6.00000 −0.538816
\(125\) 13.7016 1.22550
\(126\) 0 0
\(127\) 17.1047 1.51780 0.758898 0.651210i \(-0.225738\pi\)
0.758898 + 0.651210i \(0.225738\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.70156 −0.501995
\(130\) 21.1047 1.85100
\(131\) −15.4031 −1.34578 −0.672889 0.739744i \(-0.734947\pi\)
−0.672889 + 0.739744i \(0.734947\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −3.70156 −0.318580
\(136\) 0 0
\(137\) 0.895314 0.0764918 0.0382459 0.999268i \(-0.487823\pi\)
0.0382459 + 0.999268i \(0.487823\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 21.1047 1.79008 0.895038 0.445990i \(-0.147148\pi\)
0.895038 + 0.445990i \(0.147148\pi\)
\(140\) 0 0
\(141\) 3.70156 0.311728
\(142\) −7.40312 −0.621256
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 1.10469 0.0917392
\(146\) −5.40312 −0.447166
\(147\) 0 0
\(148\) 3.70156 0.304267
\(149\) −17.4031 −1.42572 −0.712860 0.701307i \(-0.752600\pi\)
−0.712860 + 0.701307i \(0.752600\pi\)
\(150\) 8.70156 0.710480
\(151\) 9.10469 0.740929 0.370464 0.928847i \(-0.379199\pi\)
0.370464 + 0.928847i \(0.379199\pi\)
\(152\) −5.40312 −0.438251
\(153\) 0 0
\(154\) 0 0
\(155\) −22.2094 −1.78390
\(156\) 5.70156 0.456490
\(157\) 13.4031 1.06969 0.534843 0.844952i \(-0.320371\pi\)
0.534843 + 0.844952i \(0.320371\pi\)
\(158\) 0 0
\(159\) −9.40312 −0.745716
\(160\) −3.70156 −0.292634
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −7.70156 −0.601391
\(165\) 0 0
\(166\) −2.59688 −0.201557
\(167\) 21.4031 1.65622 0.828112 0.560563i \(-0.189415\pi\)
0.828112 + 0.560563i \(0.189415\pi\)
\(168\) 0 0
\(169\) 19.5078 1.50060
\(170\) 0 0
\(171\) 5.40312 0.413187
\(172\) 5.70156 0.434740
\(173\) 3.40312 0.258735 0.129367 0.991597i \(-0.458705\pi\)
0.129367 + 0.991597i \(0.458705\pi\)
\(174\) 0.298438 0.0226245
\(175\) 0 0
\(176\) 0 0
\(177\) −0.596876 −0.0448639
\(178\) −15.4031 −1.15451
\(179\) 5.70156 0.426155 0.213077 0.977035i \(-0.431651\pi\)
0.213077 + 0.977035i \(0.431651\pi\)
\(180\) 3.70156 0.275898
\(181\) −16.8062 −1.24920 −0.624599 0.780945i \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 1.00000 0.0737210
\(185\) 13.7016 1.00736
\(186\) −6.00000 −0.439941
\(187\) 0 0
\(188\) −3.70156 −0.269964
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) 22.8062 1.65020 0.825101 0.564985i \(-0.191118\pi\)
0.825101 + 0.564985i \(0.191118\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.10469 −0.511407 −0.255703 0.966755i \(-0.582307\pi\)
−0.255703 + 0.966755i \(0.582307\pi\)
\(194\) −1.10469 −0.0793119
\(195\) 21.1047 1.51134
\(196\) 0 0
\(197\) 6.50781 0.463662 0.231831 0.972756i \(-0.425528\pi\)
0.231831 + 0.972756i \(0.425528\pi\)
\(198\) 0 0
\(199\) −1.10469 −0.0783091 −0.0391546 0.999233i \(-0.512466\pi\)
−0.0391546 + 0.999233i \(0.512466\pi\)
\(200\) −8.70156 −0.615293
\(201\) −4.00000 −0.282138
\(202\) 4.59688 0.323435
\(203\) 0 0
\(204\) 0 0
\(205\) −28.5078 −1.99107
\(206\) −17.1047 −1.19174
\(207\) −1.00000 −0.0695048
\(208\) −5.70156 −0.395332
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 9.40312 0.645809
\(213\) −7.40312 −0.507254
\(214\) −14.8062 −1.01213
\(215\) 21.1047 1.43933
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −15.7016 −1.06344
\(219\) −5.40312 −0.365109
\(220\) 0 0
\(221\) 0 0
\(222\) 3.70156 0.248433
\(223\) 20.8062 1.39329 0.696645 0.717416i \(-0.254675\pi\)
0.696645 + 0.717416i \(0.254675\pi\)
\(224\) 0 0
\(225\) 8.70156 0.580104
\(226\) 4.29844 0.285928
\(227\) 26.5078 1.75939 0.879693 0.475543i \(-0.157748\pi\)
0.879693 + 0.475543i \(0.157748\pi\)
\(228\) −5.40312 −0.357831
\(229\) −28.2094 −1.86413 −0.932064 0.362294i \(-0.881994\pi\)
−0.932064 + 0.362294i \(0.881994\pi\)
\(230\) 3.70156 0.244074
\(231\) 0 0
\(232\) −0.298438 −0.0195934
\(233\) 28.8062 1.88716 0.943580 0.331145i \(-0.107435\pi\)
0.943580 + 0.331145i \(0.107435\pi\)
\(234\) 5.70156 0.372723
\(235\) −13.7016 −0.893791
\(236\) 0.596876 0.0388533
\(237\) 0 0
\(238\) 0 0
\(239\) −18.8062 −1.21648 −0.608238 0.793755i \(-0.708123\pi\)
−0.608238 + 0.793755i \(0.708123\pi\)
\(240\) −3.70156 −0.238935
\(241\) −5.10469 −0.328822 −0.164411 0.986392i \(-0.552572\pi\)
−0.164411 + 0.986392i \(0.552572\pi\)
\(242\) 11.0000 0.707107
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −7.70156 −0.491034
\(247\) −30.8062 −1.96015
\(248\) 6.00000 0.381000
\(249\) −2.59688 −0.164570
\(250\) −13.7016 −0.866563
\(251\) −13.9109 −0.878050 −0.439025 0.898475i \(-0.644676\pi\)
−0.439025 + 0.898475i \(0.644676\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −17.1047 −1.07324
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −23.6125 −1.47291 −0.736454 0.676488i \(-0.763501\pi\)
−0.736454 + 0.676488i \(0.763501\pi\)
\(258\) 5.70156 0.354964
\(259\) 0 0
\(260\) −21.1047 −1.30886
\(261\) 0.298438 0.0184728
\(262\) 15.4031 0.951608
\(263\) −16.5078 −1.01792 −0.508958 0.860792i \(-0.669969\pi\)
−0.508958 + 0.860792i \(0.669969\pi\)
\(264\) 0 0
\(265\) 34.8062 2.13813
\(266\) 0 0
\(267\) −15.4031 −0.942656
\(268\) 4.00000 0.244339
\(269\) 3.40312 0.207492 0.103746 0.994604i \(-0.466917\pi\)
0.103746 + 0.994604i \(0.466917\pi\)
\(270\) 3.70156 0.225270
\(271\) −17.4031 −1.05716 −0.528582 0.848882i \(-0.677276\pi\)
−0.528582 + 0.848882i \(0.677276\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.895314 −0.0540879
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −13.4031 −0.805316 −0.402658 0.915351i \(-0.631914\pi\)
−0.402658 + 0.915351i \(0.631914\pi\)
\(278\) −21.1047 −1.26577
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 4.29844 0.256423 0.128212 0.991747i \(-0.459076\pi\)
0.128212 + 0.991747i \(0.459076\pi\)
\(282\) −3.70156 −0.220425
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 7.40312 0.439295
\(285\) −20.0000 −1.18470
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) −1.10469 −0.0648694
\(291\) −1.10469 −0.0647579
\(292\) 5.40312 0.316194
\(293\) 3.19375 0.186581 0.0932905 0.995639i \(-0.470261\pi\)
0.0932905 + 0.995639i \(0.470261\pi\)
\(294\) 0 0
\(295\) 2.20937 0.128635
\(296\) −3.70156 −0.215149
\(297\) 0 0
\(298\) 17.4031 1.00814
\(299\) 5.70156 0.329730
\(300\) −8.70156 −0.502385
\(301\) 0 0
\(302\) −9.10469 −0.523916
\(303\) 4.59688 0.264084
\(304\) 5.40312 0.309890
\(305\) −37.0156 −2.11951
\(306\) 0 0
\(307\) −5.10469 −0.291340 −0.145670 0.989333i \(-0.546534\pi\)
−0.145670 + 0.989333i \(0.546534\pi\)
\(308\) 0 0
\(309\) −17.1047 −0.973052
\(310\) 22.2094 1.26141
\(311\) 16.2094 0.919149 0.459575 0.888139i \(-0.348002\pi\)
0.459575 + 0.888139i \(0.348002\pi\)
\(312\) −5.70156 −0.322787
\(313\) 4.59688 0.259831 0.129915 0.991525i \(-0.458529\pi\)
0.129915 + 0.991525i \(0.458529\pi\)
\(314\) −13.4031 −0.756382
\(315\) 0 0
\(316\) 0 0
\(317\) 7.70156 0.432563 0.216281 0.976331i \(-0.430607\pi\)
0.216281 + 0.976331i \(0.430607\pi\)
\(318\) 9.40312 0.527301
\(319\) 0 0
\(320\) 3.70156 0.206924
\(321\) −14.8062 −0.826404
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −49.6125 −2.75201
\(326\) 12.0000 0.664619
\(327\) −15.7016 −0.868298
\(328\) 7.70156 0.425248
\(329\) 0 0
\(330\) 0 0
\(331\) −17.6125 −0.968070 −0.484035 0.875049i \(-0.660829\pi\)
−0.484035 + 0.875049i \(0.660829\pi\)
\(332\) 2.59688 0.142522
\(333\) 3.70156 0.202844
\(334\) −21.4031 −1.17113
\(335\) 14.8062 0.808952
\(336\) 0 0
\(337\) 17.4031 0.948009 0.474004 0.880523i \(-0.342808\pi\)
0.474004 + 0.880523i \(0.342808\pi\)
\(338\) −19.5078 −1.06109
\(339\) 4.29844 0.233459
\(340\) 0 0
\(341\) 0 0
\(342\) −5.40312 −0.292167
\(343\) 0 0
\(344\) −5.70156 −0.307408
\(345\) 3.70156 0.199285
\(346\) −3.40312 −0.182953
\(347\) 20.5078 1.10092 0.550458 0.834863i \(-0.314453\pi\)
0.550458 + 0.834863i \(0.314453\pi\)
\(348\) −0.298438 −0.0159979
\(349\) −34.2094 −1.83119 −0.915593 0.402107i \(-0.868278\pi\)
−0.915593 + 0.402107i \(0.868278\pi\)
\(350\) 0 0
\(351\) 5.70156 0.304327
\(352\) 0 0
\(353\) 33.3141 1.77313 0.886564 0.462606i \(-0.153085\pi\)
0.886564 + 0.462606i \(0.153085\pi\)
\(354\) 0.596876 0.0317236
\(355\) 27.4031 1.45441
\(356\) 15.4031 0.816364
\(357\) 0 0
\(358\) −5.70156 −0.301337
\(359\) 9.70156 0.512029 0.256014 0.966673i \(-0.417591\pi\)
0.256014 + 0.966673i \(0.417591\pi\)
\(360\) −3.70156 −0.195089
\(361\) 10.1938 0.536513
\(362\) 16.8062 0.883317
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) −10.0000 −0.522708
\(367\) −15.9109 −0.830544 −0.415272 0.909697i \(-0.636314\pi\)
−0.415272 + 0.909697i \(0.636314\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −7.70156 −0.400927
\(370\) −13.7016 −0.712310
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −4.80625 −0.248858 −0.124429 0.992229i \(-0.539710\pi\)
−0.124429 + 0.992229i \(0.539710\pi\)
\(374\) 0 0
\(375\) −13.7016 −0.707546
\(376\) 3.70156 0.190893
\(377\) −1.70156 −0.0876349
\(378\) 0 0
\(379\) −17.7016 −0.909268 −0.454634 0.890678i \(-0.650230\pi\)
−0.454634 + 0.890678i \(0.650230\pi\)
\(380\) 20.0000 1.02598
\(381\) −17.1047 −0.876300
\(382\) −22.8062 −1.16687
\(383\) −6.80625 −0.347783 −0.173892 0.984765i \(-0.555634\pi\)
−0.173892 + 0.984765i \(0.555634\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 7.10469 0.361619
\(387\) 5.70156 0.289827
\(388\) 1.10469 0.0560820
\(389\) 36.8062 1.86615 0.933075 0.359681i \(-0.117114\pi\)
0.933075 + 0.359681i \(0.117114\pi\)
\(390\) −21.1047 −1.06868
\(391\) 0 0
\(392\) 0 0
\(393\) 15.4031 0.776985
\(394\) −6.50781 −0.327859
\(395\) 0 0
\(396\) 0 0
\(397\) −6.20937 −0.311639 −0.155820 0.987786i \(-0.549802\pi\)
−0.155820 + 0.987786i \(0.549802\pi\)
\(398\) 1.10469 0.0553729
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) 28.8062 1.43852 0.719258 0.694743i \(-0.244482\pi\)
0.719258 + 0.694743i \(0.244482\pi\)
\(402\) 4.00000 0.199502
\(403\) 34.2094 1.70409
\(404\) −4.59688 −0.228703
\(405\) 3.70156 0.183932
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.2094 1.59265 0.796325 0.604868i \(-0.206774\pi\)
0.796325 + 0.604868i \(0.206774\pi\)
\(410\) 28.5078 1.40790
\(411\) −0.895314 −0.0441626
\(412\) 17.1047 0.842687
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 9.61250 0.471859
\(416\) 5.70156 0.279542
\(417\) −21.1047 −1.03350
\(418\) 0 0
\(419\) 2.59688 0.126866 0.0634328 0.997986i \(-0.479795\pi\)
0.0634328 + 0.997986i \(0.479795\pi\)
\(420\) 0 0
\(421\) 15.7016 0.765247 0.382624 0.923904i \(-0.375021\pi\)
0.382624 + 0.923904i \(0.375021\pi\)
\(422\) 12.0000 0.584151
\(423\) −3.70156 −0.179976
\(424\) −9.40312 −0.456656
\(425\) 0 0
\(426\) 7.40312 0.358683
\(427\) 0 0
\(428\) 14.8062 0.715687
\(429\) 0 0
\(430\) −21.1047 −1.01776
\(431\) 5.10469 0.245884 0.122942 0.992414i \(-0.460767\pi\)
0.122942 + 0.992414i \(0.460767\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −17.1047 −0.821999 −0.410999 0.911636i \(-0.634820\pi\)
−0.410999 + 0.911636i \(0.634820\pi\)
\(434\) 0 0
\(435\) −1.10469 −0.0529657
\(436\) 15.7016 0.751968
\(437\) −5.40312 −0.258466
\(438\) 5.40312 0.258171
\(439\) −3.19375 −0.152429 −0.0762147 0.997091i \(-0.524283\pi\)
−0.0762147 + 0.997091i \(0.524283\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.1047 0.812668 0.406334 0.913725i \(-0.366807\pi\)
0.406334 + 0.913725i \(0.366807\pi\)
\(444\) −3.70156 −0.175668
\(445\) 57.0156 2.70280
\(446\) −20.8062 −0.985204
\(447\) 17.4031 0.823140
\(448\) 0 0
\(449\) 27.6125 1.30311 0.651557 0.758600i \(-0.274116\pi\)
0.651557 + 0.758600i \(0.274116\pi\)
\(450\) −8.70156 −0.410196
\(451\) 0 0
\(452\) −4.29844 −0.202182
\(453\) −9.10469 −0.427775
\(454\) −26.5078 −1.24407
\(455\) 0 0
\(456\) 5.40312 0.253024
\(457\) −16.2094 −0.758242 −0.379121 0.925347i \(-0.623774\pi\)
−0.379121 + 0.925347i \(0.623774\pi\)
\(458\) 28.2094 1.31814
\(459\) 0 0
\(460\) −3.70156 −0.172586
\(461\) −27.4031 −1.27629 −0.638145 0.769916i \(-0.720298\pi\)
−0.638145 + 0.769916i \(0.720298\pi\)
\(462\) 0 0
\(463\) −13.1047 −0.609026 −0.304513 0.952508i \(-0.598494\pi\)
−0.304513 + 0.952508i \(0.598494\pi\)
\(464\) 0.298438 0.0138546
\(465\) 22.2094 1.02993
\(466\) −28.8062 −1.33442
\(467\) 8.29844 0.384006 0.192003 0.981394i \(-0.438502\pi\)
0.192003 + 0.981394i \(0.438502\pi\)
\(468\) −5.70156 −0.263555
\(469\) 0 0
\(470\) 13.7016 0.632006
\(471\) −13.4031 −0.617583
\(472\) −0.596876 −0.0274734
\(473\) 0 0
\(474\) 0 0
\(475\) 47.0156 2.15722
\(476\) 0 0
\(477\) 9.40312 0.430539
\(478\) 18.8062 0.860178
\(479\) −7.40312 −0.338257 −0.169129 0.985594i \(-0.554095\pi\)
−0.169129 + 0.985594i \(0.554095\pi\)
\(480\) 3.70156 0.168952
\(481\) −21.1047 −0.962291
\(482\) 5.10469 0.232512
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 4.08907 0.185675
\(486\) 1.00000 0.0453609
\(487\) −13.1047 −0.593830 −0.296915 0.954904i \(-0.595958\pi\)
−0.296915 + 0.954904i \(0.595958\pi\)
\(488\) 10.0000 0.452679
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −17.6125 −0.794841 −0.397420 0.917637i \(-0.630095\pi\)
−0.397420 + 0.917637i \(0.630095\pi\)
\(492\) 7.70156 0.347213
\(493\) 0 0
\(494\) 30.8062 1.38604
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 2.59688 0.116369
\(499\) 37.0156 1.65705 0.828523 0.559954i \(-0.189181\pi\)
0.828523 + 0.559954i \(0.189181\pi\)
\(500\) 13.7016 0.612752
\(501\) −21.4031 −0.956221
\(502\) 13.9109 0.620875
\(503\) 11.4031 0.508440 0.254220 0.967146i \(-0.418181\pi\)
0.254220 + 0.967146i \(0.418181\pi\)
\(504\) 0 0
\(505\) −17.0156 −0.757185
\(506\) 0 0
\(507\) −19.5078 −0.866372
\(508\) 17.1047 0.758898
\(509\) 14.8062 0.656275 0.328138 0.944630i \(-0.393579\pi\)
0.328138 + 0.944630i \(0.393579\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −5.40312 −0.238554
\(514\) 23.6125 1.04150
\(515\) 63.3141 2.78995
\(516\) −5.70156 −0.250997
\(517\) 0 0
\(518\) 0 0
\(519\) −3.40312 −0.149381
\(520\) 21.1047 0.925502
\(521\) −15.4031 −0.674823 −0.337412 0.941357i \(-0.609551\pi\)
−0.337412 + 0.941357i \(0.609551\pi\)
\(522\) −0.298438 −0.0130623
\(523\) 38.4187 1.67993 0.839967 0.542637i \(-0.182574\pi\)
0.839967 + 0.542637i \(0.182574\pi\)
\(524\) −15.4031 −0.672889
\(525\) 0 0
\(526\) 16.5078 0.719775
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −34.8062 −1.51189
\(531\) 0.596876 0.0259022
\(532\) 0 0
\(533\) 43.9109 1.90199
\(534\) 15.4031 0.666558
\(535\) 54.8062 2.36948
\(536\) −4.00000 −0.172774
\(537\) −5.70156 −0.246041
\(538\) −3.40312 −0.146719
\(539\) 0 0
\(540\) −3.70156 −0.159290
\(541\) −20.2094 −0.868869 −0.434434 0.900703i \(-0.643052\pi\)
−0.434434 + 0.900703i \(0.643052\pi\)
\(542\) 17.4031 0.747528
\(543\) 16.8062 0.721225
\(544\) 0 0
\(545\) 58.1203 2.48960
\(546\) 0 0
\(547\) −14.2094 −0.607549 −0.303774 0.952744i \(-0.598247\pi\)
−0.303774 + 0.952744i \(0.598247\pi\)
\(548\) 0.895314 0.0382459
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 1.61250 0.0686947
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 13.4031 0.569444
\(555\) −13.7016 −0.581599
\(556\) 21.1047 0.895038
\(557\) 21.4031 0.906879 0.453440 0.891287i \(-0.350197\pi\)
0.453440 + 0.891287i \(0.350197\pi\)
\(558\) 6.00000 0.254000
\(559\) −32.5078 −1.37493
\(560\) 0 0
\(561\) 0 0
\(562\) −4.29844 −0.181319
\(563\) −44.7172 −1.88460 −0.942302 0.334763i \(-0.891344\pi\)
−0.942302 + 0.334763i \(0.891344\pi\)
\(564\) 3.70156 0.155864
\(565\) −15.9109 −0.669378
\(566\) −10.0000 −0.420331
\(567\) 0 0
\(568\) −7.40312 −0.310628
\(569\) 11.1047 0.465533 0.232766 0.972533i \(-0.425222\pi\)
0.232766 + 0.972533i \(0.425222\pi\)
\(570\) 20.0000 0.837708
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −22.8062 −0.952745
\(574\) 0 0
\(575\) −8.70156 −0.362880
\(576\) 1.00000 0.0416667
\(577\) 23.6125 0.983001 0.491501 0.870877i \(-0.336449\pi\)
0.491501 + 0.870877i \(0.336449\pi\)
\(578\) 17.0000 0.707107
\(579\) 7.10469 0.295261
\(580\) 1.10469 0.0458696
\(581\) 0 0
\(582\) 1.10469 0.0457907
\(583\) 0 0
\(584\) −5.40312 −0.223583
\(585\) −21.1047 −0.872571
\(586\) −3.19375 −0.131933
\(587\) 1.19375 0.0492714 0.0246357 0.999696i \(-0.492157\pi\)
0.0246357 + 0.999696i \(0.492157\pi\)
\(588\) 0 0
\(589\) −32.4187 −1.33579
\(590\) −2.20937 −0.0909584
\(591\) −6.50781 −0.267696
\(592\) 3.70156 0.152133
\(593\) 41.9109 1.72108 0.860538 0.509386i \(-0.170128\pi\)
0.860538 + 0.509386i \(0.170128\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.4031 −0.712860
\(597\) 1.10469 0.0452118
\(598\) −5.70156 −0.233154
\(599\) −29.6125 −1.20993 −0.604967 0.796251i \(-0.706814\pi\)
−0.604967 + 0.796251i \(0.706814\pi\)
\(600\) 8.70156 0.355240
\(601\) −25.4031 −1.03622 −0.518108 0.855315i \(-0.673363\pi\)
−0.518108 + 0.855315i \(0.673363\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 9.10469 0.370464
\(605\) −40.7172 −1.65539
\(606\) −4.59688 −0.186735
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −5.40312 −0.219126
\(609\) 0 0
\(610\) 37.0156 1.49872
\(611\) 21.1047 0.853804
\(612\) 0 0
\(613\) −25.3141 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(614\) 5.10469 0.206008
\(615\) 28.5078 1.14955
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 17.1047 0.688051
\(619\) 30.0000 1.20580 0.602901 0.797816i \(-0.294011\pi\)
0.602901 + 0.797816i \(0.294011\pi\)
\(620\) −22.2094 −0.891950
\(621\) 1.00000 0.0401286
\(622\) −16.2094 −0.649937
\(623\) 0 0
\(624\) 5.70156 0.228245
\(625\) 7.20937 0.288375
\(626\) −4.59688 −0.183728
\(627\) 0 0
\(628\) 13.4031 0.534843
\(629\) 0 0
\(630\) 0 0
\(631\) 9.19375 0.365997 0.182999 0.983113i \(-0.441420\pi\)
0.182999 + 0.983113i \(0.441420\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) −7.70156 −0.305868
\(635\) 63.3141 2.51254
\(636\) −9.40312 −0.372858
\(637\) 0 0
\(638\) 0 0
\(639\) 7.40312 0.292863
\(640\) −3.70156 −0.146317
\(641\) −11.7016 −0.462184 −0.231092 0.972932i \(-0.574230\pi\)
−0.231092 + 0.972932i \(0.574230\pi\)
\(642\) 14.8062 0.584356
\(643\) −3.19375 −0.125949 −0.0629746 0.998015i \(-0.520059\pi\)
−0.0629746 + 0.998015i \(0.520059\pi\)
\(644\) 0 0
\(645\) −21.1047 −0.830996
\(646\) 0 0
\(647\) 4.20937 0.165488 0.0827438 0.996571i \(-0.473632\pi\)
0.0827438 + 0.996571i \(0.473632\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 49.6125 1.94596
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 47.1047 1.84335 0.921674 0.387964i \(-0.126822\pi\)
0.921674 + 0.387964i \(0.126822\pi\)
\(654\) 15.7016 0.613980
\(655\) −57.0156 −2.22778
\(656\) −7.70156 −0.300695
\(657\) 5.40312 0.210796
\(658\) 0 0
\(659\) 25.6125 0.997721 0.498861 0.866682i \(-0.333752\pi\)
0.498861 + 0.866682i \(0.333752\pi\)
\(660\) 0 0
\(661\) 20.8062 0.809269 0.404635 0.914478i \(-0.367399\pi\)
0.404635 + 0.914478i \(0.367399\pi\)
\(662\) 17.6125 0.684529
\(663\) 0 0
\(664\) −2.59688 −0.100778
\(665\) 0 0
\(666\) −3.70156 −0.143433
\(667\) −0.298438 −0.0115556
\(668\) 21.4031 0.828112
\(669\) −20.8062 −0.804416
\(670\) −14.8062 −0.572015
\(671\) 0 0
\(672\) 0 0
\(673\) −8.29844 −0.319881 −0.159941 0.987127i \(-0.551130\pi\)
−0.159941 + 0.987127i \(0.551130\pi\)
\(674\) −17.4031 −0.670343
\(675\) −8.70156 −0.334923
\(676\) 19.5078 0.750300
\(677\) 3.19375 0.122746 0.0613729 0.998115i \(-0.480452\pi\)
0.0613729 + 0.998115i \(0.480452\pi\)
\(678\) −4.29844 −0.165081
\(679\) 0 0
\(680\) 0 0
\(681\) −26.5078 −1.01578
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 5.40312 0.206594
\(685\) 3.31406 0.126624
\(686\) 0 0
\(687\) 28.2094 1.07625
\(688\) 5.70156 0.217370
\(689\) −53.6125 −2.04247
\(690\) −3.70156 −0.140916
\(691\) 41.7016 1.58640 0.793201 0.608960i \(-0.208413\pi\)
0.793201 + 0.608960i \(0.208413\pi\)
\(692\) 3.40312 0.129367
\(693\) 0 0
\(694\) −20.5078 −0.778466
\(695\) 78.1203 2.96327
\(696\) 0.298438 0.0113123
\(697\) 0 0
\(698\) 34.2094 1.29484
\(699\) −28.8062 −1.08955
\(700\) 0 0
\(701\) 14.5969 0.551316 0.275658 0.961256i \(-0.411104\pi\)
0.275658 + 0.961256i \(0.411104\pi\)
\(702\) −5.70156 −0.215192
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 13.7016 0.516031
\(706\) −33.3141 −1.25379
\(707\) 0 0
\(708\) −0.596876 −0.0224320
\(709\) −15.1938 −0.570613 −0.285307 0.958436i \(-0.592095\pi\)
−0.285307 + 0.958436i \(0.592095\pi\)
\(710\) −27.4031 −1.02842
\(711\) 0 0
\(712\) −15.4031 −0.577256
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 5.70156 0.213077
\(717\) 18.8062 0.702332
\(718\) −9.70156 −0.362059
\(719\) −14.5078 −0.541050 −0.270525 0.962713i \(-0.587197\pi\)
−0.270525 + 0.962713i \(0.587197\pi\)
\(720\) 3.70156 0.137949
\(721\) 0 0
\(722\) −10.1938 −0.379372
\(723\) 5.10469 0.189845
\(724\) −16.8062 −0.624599
\(725\) 2.59688 0.0964455
\(726\) −11.0000 −0.408248
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) −31.0156 −1.14559 −0.572794 0.819699i \(-0.694141\pi\)
−0.572794 + 0.819699i \(0.694141\pi\)
\(734\) 15.9109 0.587283
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 7.70156 0.283498
\(739\) 38.2094 1.40555 0.702777 0.711410i \(-0.251943\pi\)
0.702777 + 0.711410i \(0.251943\pi\)
\(740\) 13.7016 0.503679
\(741\) 30.8062 1.13170
\(742\) 0 0
\(743\) −21.6125 −0.792886 −0.396443 0.918059i \(-0.629755\pi\)
−0.396443 + 0.918059i \(0.629755\pi\)
\(744\) −6.00000 −0.219971
\(745\) −64.4187 −2.36012
\(746\) 4.80625 0.175969
\(747\) 2.59688 0.0950147
\(748\) 0 0
\(749\) 0 0
\(750\) 13.7016 0.500310
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −3.70156 −0.134982
\(753\) 13.9109 0.506943
\(754\) 1.70156 0.0619672
\(755\) 33.7016 1.22653
\(756\) 0 0
\(757\) 7.19375 0.261461 0.130731 0.991418i \(-0.458268\pi\)
0.130731 + 0.991418i \(0.458268\pi\)
\(758\) 17.7016 0.642950
\(759\) 0 0
\(760\) −20.0000 −0.725476
\(761\) −15.6125 −0.565953 −0.282976 0.959127i \(-0.591322\pi\)
−0.282976 + 0.959127i \(0.591322\pi\)
\(762\) 17.1047 0.619637
\(763\) 0 0
\(764\) 22.8062 0.825101
\(765\) 0 0
\(766\) 6.80625 0.245920
\(767\) −3.40312 −0.122880
\(768\) −1.00000 −0.0360844
\(769\) 29.1047 1.04954 0.524771 0.851243i \(-0.324151\pi\)
0.524771 + 0.851243i \(0.324151\pi\)
\(770\) 0 0
\(771\) 23.6125 0.850383
\(772\) −7.10469 −0.255703
\(773\) −44.7172 −1.60837 −0.804183 0.594382i \(-0.797397\pi\)
−0.804183 + 0.594382i \(0.797397\pi\)
\(774\) −5.70156 −0.204938
\(775\) −52.2094 −1.87542
\(776\) −1.10469 −0.0396559
\(777\) 0 0
\(778\) −36.8062 −1.31957
\(779\) −41.6125 −1.49092
\(780\) 21.1047 0.755669
\(781\) 0 0
\(782\) 0 0
\(783\) −0.298438 −0.0106653
\(784\) 0 0
\(785\) 49.6125 1.77075
\(786\) −15.4031 −0.549411
\(787\) −49.8219 −1.77596 −0.887979 0.459884i \(-0.847891\pi\)
−0.887979 + 0.459884i \(0.847891\pi\)
\(788\) 6.50781 0.231831
\(789\) 16.5078 0.587694
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 57.0156 2.02468
\(794\) 6.20937 0.220362
\(795\) −34.8062 −1.23445
\(796\) −1.10469 −0.0391546
\(797\) 31.7016 1.12293 0.561463 0.827502i \(-0.310239\pi\)
0.561463 + 0.827502i \(0.310239\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −8.70156 −0.307647
\(801\) 15.4031 0.544243
\(802\) −28.8062 −1.01718
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −34.2094 −1.20497
\(807\) −3.40312 −0.119796
\(808\) 4.59688 0.161718
\(809\) −21.4031 −0.752494 −0.376247 0.926519i \(-0.622786\pi\)
−0.376247 + 0.926519i \(0.622786\pi\)
\(810\) −3.70156 −0.130060
\(811\) −11.3141 −0.397290 −0.198645 0.980071i \(-0.563654\pi\)
−0.198645 + 0.980071i \(0.563654\pi\)
\(812\) 0 0
\(813\) 17.4031 0.610354
\(814\) 0 0
\(815\) −44.4187 −1.55592
\(816\) 0 0
\(817\) 30.8062 1.07777
\(818\) −32.2094 −1.12617
\(819\) 0 0
\(820\) −28.5078 −0.995536
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0.895314 0.0312276
\(823\) 46.7172 1.62846 0.814229 0.580543i \(-0.197160\pi\)
0.814229 + 0.580543i \(0.197160\pi\)
\(824\) −17.1047 −0.595870
\(825\) 0 0
\(826\) 0 0
\(827\) −33.0156 −1.14807 −0.574033 0.818832i \(-0.694622\pi\)
−0.574033 + 0.818832i \(0.694622\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 30.2094 1.04921 0.524607 0.851344i \(-0.324212\pi\)
0.524607 + 0.851344i \(0.324212\pi\)
\(830\) −9.61250 −0.333655
\(831\) 13.4031 0.464949
\(832\) −5.70156 −0.197666
\(833\) 0 0
\(834\) 21.1047 0.730796
\(835\) 79.2250 2.74169
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) −2.59688 −0.0897076
\(839\) 19.4031 0.669870 0.334935 0.942241i \(-0.391286\pi\)
0.334935 + 0.942241i \(0.391286\pi\)
\(840\) 0 0
\(841\) −28.9109 −0.996929
\(842\) −15.7016 −0.541112
\(843\) −4.29844 −0.148046
\(844\) −12.0000 −0.413057
\(845\) 72.2094 2.48408
\(846\) 3.70156 0.127262
\(847\) 0 0
\(848\) 9.40312 0.322905
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −3.70156 −0.126888
\(852\) −7.40312 −0.253627
\(853\) −41.1047 −1.40740 −0.703699 0.710498i \(-0.748470\pi\)
−0.703699 + 0.710498i \(0.748470\pi\)
\(854\) 0 0
\(855\) 20.0000 0.683986
\(856\) −14.8062 −0.506067
\(857\) −2.08907 −0.0713611 −0.0356806 0.999363i \(-0.511360\pi\)
−0.0356806 + 0.999363i \(0.511360\pi\)
\(858\) 0 0
\(859\) 3.91093 0.133439 0.0667197 0.997772i \(-0.478747\pi\)
0.0667197 + 0.997772i \(0.478747\pi\)
\(860\) 21.1047 0.719664
\(861\) 0 0
\(862\) −5.10469 −0.173866
\(863\) −10.8062 −0.367849 −0.183924 0.982940i \(-0.558880\pi\)
−0.183924 + 0.982940i \(0.558880\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.5969 0.428307
\(866\) 17.1047 0.581241
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) 1.10469 0.0374524
\(871\) −22.8062 −0.772760
\(872\) −15.7016 −0.531722
\(873\) 1.10469 0.0373880
\(874\) 5.40312 0.182763
\(875\) 0 0
\(876\) −5.40312 −0.182555
\(877\) −48.2094 −1.62791 −0.813957 0.580925i \(-0.802691\pi\)
−0.813957 + 0.580925i \(0.802691\pi\)
\(878\) 3.19375 0.107784
\(879\) −3.19375 −0.107723
\(880\) 0 0
\(881\) 10.2094 0.343963 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(882\) 0 0
\(883\) 16.5969 0.558529 0.279265 0.960214i \(-0.409909\pi\)
0.279265 + 0.960214i \(0.409909\pi\)
\(884\) 0 0
\(885\) −2.20937 −0.0742673
\(886\) −17.1047 −0.574643
\(887\) 52.2094 1.75302 0.876510 0.481384i \(-0.159866\pi\)
0.876510 + 0.481384i \(0.159866\pi\)
\(888\) 3.70156 0.124216
\(889\) 0 0
\(890\) −57.0156 −1.91117
\(891\) 0 0
\(892\) 20.8062 0.696645
\(893\) −20.0000 −0.669274
\(894\) −17.4031 −0.582048
\(895\) 21.1047 0.705452
\(896\) 0 0
\(897\) −5.70156 −0.190370
\(898\) −27.6125 −0.921441
\(899\) −1.79063 −0.0597208
\(900\) 8.70156 0.290052
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 4.29844 0.142964
\(905\) −62.2094 −2.06791
\(906\) 9.10469 0.302483
\(907\) −2.89531 −0.0961373 −0.0480687 0.998844i \(-0.515307\pi\)
−0.0480687 + 0.998844i \(0.515307\pi\)
\(908\) 26.5078 0.879693
\(909\) −4.59688 −0.152469
\(910\) 0 0
\(911\) 1.70156 0.0563753 0.0281876 0.999603i \(-0.491026\pi\)
0.0281876 + 0.999603i \(0.491026\pi\)
\(912\) −5.40312 −0.178915
\(913\) 0 0
\(914\) 16.2094 0.536158
\(915\) 37.0156 1.22370
\(916\) −28.2094 −0.932064
\(917\) 0 0
\(918\) 0 0
\(919\) 1.19375 0.0393782 0.0196891 0.999806i \(-0.493732\pi\)
0.0196891 + 0.999806i \(0.493732\pi\)
\(920\) 3.70156 0.122037
\(921\) 5.10469 0.168205
\(922\) 27.4031 0.902474
\(923\) −42.2094 −1.38934
\(924\) 0 0
\(925\) 32.2094 1.05904
\(926\) 13.1047 0.430647
\(927\) 17.1047 0.561792
\(928\) −0.298438 −0.00979670
\(929\) −23.7016 −0.777623 −0.388812 0.921317i \(-0.627114\pi\)
−0.388812 + 0.921317i \(0.627114\pi\)
\(930\) −22.2094 −0.728274
\(931\) 0 0
\(932\) 28.8062 0.943580
\(933\) −16.2094 −0.530671
\(934\) −8.29844 −0.271533
\(935\) 0 0
\(936\) 5.70156 0.186361
\(937\) −47.3141 −1.54568 −0.772841 0.634599i \(-0.781165\pi\)
−0.772841 + 0.634599i \(0.781165\pi\)
\(938\) 0 0
\(939\) −4.59688 −0.150013
\(940\) −13.7016 −0.446896
\(941\) 9.31406 0.303630 0.151815 0.988409i \(-0.451488\pi\)
0.151815 + 0.988409i \(0.451488\pi\)
\(942\) 13.4031 0.436697
\(943\) 7.70156 0.250797
\(944\) 0.596876 0.0194267
\(945\) 0 0
\(946\) 0 0
\(947\) 3.49219 0.113481 0.0567405 0.998389i \(-0.481929\pi\)
0.0567405 + 0.998389i \(0.481929\pi\)
\(948\) 0 0
\(949\) −30.8062 −1.00001
\(950\) −47.0156 −1.52539
\(951\) −7.70156 −0.249740
\(952\) 0 0
\(953\) −20.8062 −0.673980 −0.336990 0.941508i \(-0.609409\pi\)
−0.336990 + 0.941508i \(0.609409\pi\)
\(954\) −9.40312 −0.304437
\(955\) 84.4187 2.73173
\(956\) −18.8062 −0.608238
\(957\) 0 0
\(958\) 7.40312 0.239184
\(959\) 0 0
\(960\) −3.70156 −0.119467
\(961\) 5.00000 0.161290
\(962\) 21.1047 0.680442
\(963\) 14.8062 0.477125
\(964\) −5.10469 −0.164411
\(965\) −26.2984 −0.846577
\(966\) 0 0
\(967\) 20.4187 0.656623 0.328311 0.944570i \(-0.393521\pi\)
0.328311 + 0.944570i \(0.393521\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) −4.08907 −0.131292
\(971\) 33.4031 1.07196 0.535979 0.844232i \(-0.319943\pi\)
0.535979 + 0.844232i \(0.319943\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 13.1047 0.419901
\(975\) 49.6125 1.58887
\(976\) −10.0000 −0.320092
\(977\) −40.2984 −1.28926 −0.644631 0.764494i \(-0.722989\pi\)
−0.644631 + 0.764494i \(0.722989\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) 15.7016 0.501312
\(982\) 17.6125 0.562037
\(983\) 55.8219 1.78044 0.890221 0.455530i \(-0.150550\pi\)
0.890221 + 0.455530i \(0.150550\pi\)
\(984\) −7.70156 −0.245517
\(985\) 24.0891 0.767541
\(986\) 0 0
\(987\) 0 0
\(988\) −30.8062 −0.980077
\(989\) −5.70156 −0.181299
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 6.00000 0.190500
\(993\) 17.6125 0.558916
\(994\) 0 0
\(995\) −4.08907 −0.129632
\(996\) −2.59688 −0.0822852
\(997\) −48.5969 −1.53908 −0.769539 0.638600i \(-0.779514\pi\)
−0.769539 + 0.638600i \(0.779514\pi\)
\(998\) −37.0156 −1.17171
\(999\) −3.70156 −0.117112
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 6762.2.a.bq.1.2 2
7.6 odd 2 966.2.a.m.1.1 2
21.20 even 2 2898.2.a.bc.1.2 2
28.27 even 2 7728.2.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.m.1.1 2 7.6 odd 2
2898.2.a.bc.1.2 2 21.20 even 2
6762.2.a.bq.1.2 2 1.1 even 1 trivial
7728.2.a.z.1.1 2 28.27 even 2