Properties

Label 1840.2.i.a.1471.3
Level $1840$
Weight $2$
Character 1840.1471
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1471,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.i (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(-3.23546 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1471
Dual form 1840.2.i.a.1471.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.36943i q^{3} -1.00000i q^{5} +4.10148 q^{7} -2.61421 q^{9} +O(q^{10})\) \(q-2.36943i q^{3} -1.00000i q^{5} +4.10148 q^{7} -2.61421 q^{9} +1.58230 q^{11} +1.61421 q^{13} -2.36943 q^{15} -0.614214i q^{17} +3.28353 q^{19} -9.71819i q^{21} +(2.21968 + 4.25124i) q^{23} -1.00000 q^{25} -0.914095i q^{27} +9.48977 q^{29} +3.91336i q^{31} -3.74915i q^{33} -4.10148i q^{35} -3.87360i q^{37} -3.82477i q^{39} +3.45881 q^{41} -8.20297 q^{43} +2.61421i q^{45} -0.487628i q^{47} +9.82217 q^{49} -1.45534 q^{51} +0.645168i q^{53} -1.58230i q^{55} -7.78010i q^{57} +14.3132i q^{59} -6.94858i q^{61} -10.7222 q^{63} -1.61421i q^{65} +9.87386 q^{67} +(10.0730 - 5.25938i) q^{69} +9.75445i q^{71} -10.5918 q^{73} +2.36943i q^{75} +6.48977 q^{77} -15.7302 q^{79} -10.0085 q^{81} -9.87386 q^{83} -0.614214 q^{85} -22.4854i q^{87} +5.22843i q^{89} +6.62067 q^{91} +9.27245 q^{93} -3.28353i q^{95} -12.6958i q^{97} -4.13646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 2.36943i 1.36799i −0.729485 0.683996i \(-0.760240\pi\)
0.729485 0.683996i \(-0.239760\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 4.10148 1.55022 0.775108 0.631829i \(-0.217696\pi\)
0.775108 + 0.631829i \(0.217696\pi\)
\(8\) 0 0
\(9\) −2.61421 −0.871405
\(10\) 0 0
\(11\) 1.58230 0.477080 0.238540 0.971133i \(-0.423331\pi\)
0.238540 + 0.971133i \(0.423331\pi\)
\(12\) 0 0
\(13\) 1.61421 0.447702 0.223851 0.974623i \(-0.428137\pi\)
0.223851 + 0.974623i \(0.428137\pi\)
\(14\) 0 0
\(15\) −2.36943 −0.611785
\(16\) 0 0
\(17\) 0.614214i 0.148969i −0.997222 0.0744843i \(-0.976269\pi\)
0.997222 0.0744843i \(-0.0237311\pi\)
\(18\) 0 0
\(19\) 3.28353 0.753293 0.376647 0.926357i \(-0.377077\pi\)
0.376647 + 0.926357i \(0.377077\pi\)
\(20\) 0 0
\(21\) 9.71819i 2.12068i
\(22\) 0 0
\(23\) 2.21968 + 4.25124i 0.462835 + 0.886444i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0.914095i 0.175918i
\(28\) 0 0
\(29\) 9.48977 1.76221 0.881103 0.472925i \(-0.156802\pi\)
0.881103 + 0.472925i \(0.156802\pi\)
\(30\) 0 0
\(31\) 3.91336i 0.702861i 0.936214 + 0.351430i \(0.114305\pi\)
−0.936214 + 0.351430i \(0.885695\pi\)
\(32\) 0 0
\(33\) 3.74915i 0.652643i
\(34\) 0 0
\(35\) 4.10148i 0.693277i
\(36\) 0 0
\(37\) 3.87360i 0.636815i −0.947954 0.318408i \(-0.896852\pi\)
0.947954 0.318408i \(-0.103148\pi\)
\(38\) 0 0
\(39\) 3.82477i 0.612454i
\(40\) 0 0
\(41\) 3.45881 0.540175 0.270088 0.962836i \(-0.412947\pi\)
0.270088 + 0.962836i \(0.412947\pi\)
\(42\) 0 0
\(43\) −8.20297 −1.25094 −0.625470 0.780248i \(-0.715093\pi\)
−0.625470 + 0.780248i \(0.715093\pi\)
\(44\) 0 0
\(45\) 2.61421i 0.389704i
\(46\) 0 0
\(47\) 0.487628i 0.0711279i −0.999367 0.0355640i \(-0.988677\pi\)
0.999367 0.0355640i \(-0.0113227\pi\)
\(48\) 0 0
\(49\) 9.82217 1.40317
\(50\) 0 0
\(51\) −1.45534 −0.203788
\(52\) 0 0
\(53\) 0.645168i 0.0886206i 0.999018 + 0.0443103i \(0.0141090\pi\)
−0.999018 + 0.0443103i \(0.985891\pi\)
\(54\) 0 0
\(55\) 1.58230i 0.213357i
\(56\) 0 0
\(57\) 7.78010i 1.03050i
\(58\) 0 0
\(59\) 14.3132i 1.86342i 0.363201 + 0.931711i \(0.381684\pi\)
−0.363201 + 0.931711i \(0.618316\pi\)
\(60\) 0 0
\(61\) 6.94858i 0.889674i −0.895612 0.444837i \(-0.853262\pi\)
0.895612 0.444837i \(-0.146738\pi\)
\(62\) 0 0
\(63\) −10.7222 −1.35086
\(64\) 0 0
\(65\) 1.61421i 0.200219i
\(66\) 0 0
\(67\) 9.87386 1.20628 0.603142 0.797634i \(-0.293915\pi\)
0.603142 + 0.797634i \(0.293915\pi\)
\(68\) 0 0
\(69\) 10.0730 5.25938i 1.21265 0.633155i
\(70\) 0 0
\(71\) 9.75445i 1.15764i 0.815455 + 0.578820i \(0.196487\pi\)
−0.815455 + 0.578820i \(0.803513\pi\)
\(72\) 0 0
\(73\) −10.5918 −1.23967 −0.619837 0.784730i \(-0.712801\pi\)
−0.619837 + 0.784730i \(0.712801\pi\)
\(74\) 0 0
\(75\) 2.36943i 0.273599i
\(76\) 0 0
\(77\) 6.48977 0.739577
\(78\) 0 0
\(79\) −15.7302 −1.76978 −0.884892 0.465796i \(-0.845768\pi\)
−0.884892 + 0.465796i \(0.845768\pi\)
\(80\) 0 0
\(81\) −10.0085 −1.11206
\(82\) 0 0
\(83\) −9.87386 −1.08380 −0.541898 0.840444i \(-0.682294\pi\)
−0.541898 + 0.840444i \(0.682294\pi\)
\(84\) 0 0
\(85\) −0.614214 −0.0666208
\(86\) 0 0
\(87\) 22.4854i 2.41068i
\(88\) 0 0
\(89\) 5.22843i 0.554212i 0.960839 + 0.277106i \(0.0893753\pi\)
−0.960839 + 0.277106i \(0.910625\pi\)
\(90\) 0 0
\(91\) 6.62067 0.694035
\(92\) 0 0
\(93\) 9.27245 0.961509
\(94\) 0 0
\(95\) 3.28353i 0.336883i
\(96\) 0 0
\(97\) 12.6958i 1.28906i −0.764579 0.644530i \(-0.777053\pi\)
0.764579 0.644530i \(-0.222947\pi\)
\(98\) 0 0
\(99\) −4.13646 −0.415730
\(100\) 0 0
\(101\) −7.56279 −0.752526 −0.376263 0.926513i \(-0.622791\pi\)
−0.376263 + 0.926513i \(0.622791\pi\)
\(102\) 0 0
\(103\) −4.98476 −0.491163 −0.245581 0.969376i \(-0.578979\pi\)
−0.245581 + 0.969376i \(0.578979\pi\)
\(104\) 0 0
\(105\) −9.71819 −0.948398
\(106\) 0 0
\(107\) −2.64615 −0.255813 −0.127906 0.991786i \(-0.540826\pi\)
−0.127906 + 0.991786i \(0.540826\pi\)
\(108\) 0 0
\(109\) 6.97757i 0.668330i −0.942515 0.334165i \(-0.891546\pi\)
0.942515 0.334165i \(-0.108454\pi\)
\(110\) 0 0
\(111\) −9.17822 −0.871159
\(112\) 0 0
\(113\) 6.33436i 0.595887i 0.954584 + 0.297943i \(0.0963006\pi\)
−0.954584 + 0.297943i \(0.903699\pi\)
\(114\) 0 0
\(115\) 4.25124 2.21968i 0.396430 0.206986i
\(116\) 0 0
\(117\) −4.21990 −0.390130
\(118\) 0 0
\(119\) 2.51919i 0.230934i
\(120\) 0 0
\(121\) −8.49634 −0.772394
\(122\) 0 0
\(123\) 8.19542i 0.738956i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 15.6188i 1.38594i −0.720964 0.692972i \(-0.756301\pi\)
0.720964 0.692972i \(-0.243699\pi\)
\(128\) 0 0
\(129\) 19.4364i 1.71128i
\(130\) 0 0
\(131\) 4.25124i 0.371432i −0.982603 0.185716i \(-0.940539\pi\)
0.982603 0.185716i \(-0.0594605\pi\)
\(132\) 0 0
\(133\) 13.4673 1.16777
\(134\) 0 0
\(135\) −0.914095 −0.0786728
\(136\) 0 0
\(137\) 10.6668i 0.911323i 0.890153 + 0.455662i \(0.150597\pi\)
−0.890153 + 0.455662i \(0.849403\pi\)
\(138\) 0 0
\(139\) 3.49492i 0.296435i −0.988955 0.148218i \(-0.952646\pi\)
0.988955 0.148218i \(-0.0473536\pi\)
\(140\) 0 0
\(141\) −1.15540 −0.0973025
\(142\) 0 0
\(143\) 2.55417 0.213590
\(144\) 0 0
\(145\) 9.48977i 0.788082i
\(146\) 0 0
\(147\) 23.2730i 1.91952i
\(148\) 0 0
\(149\) 15.8952i 1.30219i −0.758998 0.651093i \(-0.774311\pi\)
0.758998 0.651093i \(-0.225689\pi\)
\(150\) 0 0
\(151\) 7.46945i 0.607855i 0.952695 + 0.303927i \(0.0982980\pi\)
−0.952695 + 0.303927i \(0.901702\pi\)
\(152\) 0 0
\(153\) 1.60569i 0.129812i
\(154\) 0 0
\(155\) 3.91336 0.314329
\(156\) 0 0
\(157\) 0.334362i 0.0266850i 0.999911 + 0.0133425i \(0.00424718\pi\)
−0.999911 + 0.0133425i \(0.995753\pi\)
\(158\) 0 0
\(159\) 1.52868 0.121232
\(160\) 0 0
\(161\) 9.10398 + 17.4364i 0.717494 + 1.37418i
\(162\) 0 0
\(163\) 1.52114i 0.119145i 0.998224 + 0.0595723i \(0.0189737\pi\)
−0.998224 + 0.0595723i \(0.981026\pi\)
\(164\) 0 0
\(165\) −3.74915 −0.291871
\(166\) 0 0
\(167\) 2.42721i 0.187823i 0.995581 + 0.0939114i \(0.0299371\pi\)
−0.995581 + 0.0939114i \(0.970063\pi\)
\(168\) 0 0
\(169\) −10.3943 −0.799563
\(170\) 0 0
\(171\) −8.58384 −0.656423
\(172\) 0 0
\(173\) −7.72015 −0.586952 −0.293476 0.955966i \(-0.594812\pi\)
−0.293476 + 0.955966i \(0.594812\pi\)
\(174\) 0 0
\(175\) −4.10148 −0.310043
\(176\) 0 0
\(177\) 33.9142 2.54915
\(178\) 0 0
\(179\) 12.1547i 0.908485i 0.890878 + 0.454242i \(0.150090\pi\)
−0.890878 + 0.454242i \(0.849910\pi\)
\(180\) 0 0
\(181\) 17.9571i 1.33474i −0.744726 0.667371i \(-0.767420\pi\)
0.744726 0.667371i \(-0.232580\pi\)
\(182\) 0 0
\(183\) −16.4642 −1.21707
\(184\) 0 0
\(185\) −3.87360 −0.284792
\(186\) 0 0
\(187\) 0.971868i 0.0710700i
\(188\) 0 0
\(189\) 3.74915i 0.272710i
\(190\) 0 0
\(191\) −24.9084 −1.80231 −0.901155 0.433498i \(-0.857279\pi\)
−0.901155 + 0.433498i \(0.857279\pi\)
\(192\) 0 0
\(193\) −0.383830 −0.0276287 −0.0138143 0.999905i \(-0.504397\pi\)
−0.0138143 + 0.999905i \(0.504397\pi\)
\(194\) 0 0
\(195\) −3.82477 −0.273898
\(196\) 0 0
\(197\) −25.7932 −1.83769 −0.918844 0.394622i \(-0.870876\pi\)
−0.918844 + 0.394622i \(0.870876\pi\)
\(198\) 0 0
\(199\) −15.4307 −1.09385 −0.546926 0.837181i \(-0.684202\pi\)
−0.546926 + 0.837181i \(0.684202\pi\)
\(200\) 0 0
\(201\) 23.3954i 1.65019i
\(202\) 0 0
\(203\) 38.9221 2.73180
\(204\) 0 0
\(205\) 3.45881i 0.241574i
\(206\) 0 0
\(207\) −5.80272 11.1136i −0.403317 0.772452i
\(208\) 0 0
\(209\) 5.19552 0.359381
\(210\) 0 0
\(211\) 15.5880i 1.07312i 0.843862 + 0.536560i \(0.180277\pi\)
−0.843862 + 0.536560i \(0.819723\pi\)
\(212\) 0 0
\(213\) 23.1125 1.58364
\(214\) 0 0
\(215\) 8.20297i 0.559438i
\(216\) 0 0
\(217\) 16.0506i 1.08959i
\(218\) 0 0
\(219\) 25.0965i 1.69587i
\(220\) 0 0
\(221\) 0.991472i 0.0666936i
\(222\) 0 0
\(223\) 0.975257i 0.0653080i −0.999467 0.0326540i \(-0.989604\pi\)
0.999467 0.0326540i \(-0.0103959\pi\)
\(224\) 0 0
\(225\) 2.61421 0.174281
\(226\) 0 0
\(227\) 6.62870 0.439962 0.219981 0.975504i \(-0.429400\pi\)
0.219981 + 0.975504i \(0.429400\pi\)
\(228\) 0 0
\(229\) 8.97953i 0.593384i 0.954973 + 0.296692i \(0.0958835\pi\)
−0.954973 + 0.296692i \(0.904117\pi\)
\(230\) 0 0
\(231\) 15.3771i 1.01174i
\(232\) 0 0
\(233\) 5.88212 0.385351 0.192675 0.981263i \(-0.438284\pi\)
0.192675 + 0.981263i \(0.438284\pi\)
\(234\) 0 0
\(235\) −0.487628 −0.0318094
\(236\) 0 0
\(237\) 37.2716i 2.42105i
\(238\) 0 0
\(239\) 3.19541i 0.206694i 0.994645 + 0.103347i \(0.0329552\pi\)
−0.994645 + 0.103347i \(0.967045\pi\)
\(240\) 0 0
\(241\) 7.17043i 0.461888i −0.972967 0.230944i \(-0.925819\pi\)
0.972967 0.230944i \(-0.0741814\pi\)
\(242\) 0 0
\(243\) 20.9723i 1.34537i
\(244\) 0 0
\(245\) 9.82217i 0.627515i
\(246\) 0 0
\(247\) 5.30032 0.337251
\(248\) 0 0
\(249\) 23.3954i 1.48263i
\(250\) 0 0
\(251\) 24.6055 1.55309 0.776543 0.630064i \(-0.216971\pi\)
0.776543 + 0.630064i \(0.216971\pi\)
\(252\) 0 0
\(253\) 3.51219 + 6.72672i 0.220810 + 0.422905i
\(254\) 0 0
\(255\) 1.45534i 0.0911368i
\(256\) 0 0
\(257\) 6.38383 0.398212 0.199106 0.979978i \(-0.436196\pi\)
0.199106 + 0.979978i \(0.436196\pi\)
\(258\) 0 0
\(259\) 15.8875i 0.987200i
\(260\) 0 0
\(261\) −24.8083 −1.53559
\(262\) 0 0
\(263\) 14.3668 0.885897 0.442948 0.896547i \(-0.353933\pi\)
0.442948 + 0.896547i \(0.353933\pi\)
\(264\) 0 0
\(265\) 0.645168 0.0396324
\(266\) 0 0
\(267\) 12.3884 0.758158
\(268\) 0 0
\(269\) 8.00853 0.488289 0.244144 0.969739i \(-0.421493\pi\)
0.244144 + 0.969739i \(0.421493\pi\)
\(270\) 0 0
\(271\) 3.92430i 0.238384i −0.992871 0.119192i \(-0.961970\pi\)
0.992871 0.119192i \(-0.0380304\pi\)
\(272\) 0 0
\(273\) 15.6872i 0.949435i
\(274\) 0 0
\(275\) −1.58230 −0.0954161
\(276\) 0 0
\(277\) 7.67417 0.461096 0.230548 0.973061i \(-0.425948\pi\)
0.230548 + 0.973061i \(0.425948\pi\)
\(278\) 0 0
\(279\) 10.2304i 0.612476i
\(280\) 0 0
\(281\) 14.0619i 0.838863i 0.907787 + 0.419432i \(0.137771\pi\)
−0.907787 + 0.419432i \(0.862229\pi\)
\(282\) 0 0
\(283\) 30.9570 1.84020 0.920102 0.391679i \(-0.128106\pi\)
0.920102 + 0.391679i \(0.128106\pi\)
\(284\) 0 0
\(285\) −7.78010 −0.460853
\(286\) 0 0
\(287\) 14.1863 0.837388
\(288\) 0 0
\(289\) 16.6227 0.977808
\(290\) 0 0
\(291\) −30.0818 −1.76342
\(292\) 0 0
\(293\) 25.2520i 1.47524i −0.675218 0.737618i \(-0.735950\pi\)
0.675218 0.737618i \(-0.264050\pi\)
\(294\) 0 0
\(295\) 14.3132 0.833348
\(296\) 0 0
\(297\) 1.44637i 0.0839269i
\(298\) 0 0
\(299\) 3.58304 + 6.86241i 0.207212 + 0.396863i
\(300\) 0 0
\(301\) −33.6443 −1.93923
\(302\) 0 0
\(303\) 17.9195i 1.02945i
\(304\) 0 0
\(305\) −6.94858 −0.397874
\(306\) 0 0
\(307\) 27.4053i 1.56410i 0.623214 + 0.782051i \(0.285826\pi\)
−0.623214 + 0.782051i \(0.714174\pi\)
\(308\) 0 0
\(309\) 11.8111i 0.671907i
\(310\) 0 0
\(311\) 23.0844i 1.30900i 0.756064 + 0.654498i \(0.227120\pi\)
−0.756064 + 0.654498i \(0.772880\pi\)
\(312\) 0 0
\(313\) 22.9972i 1.29988i 0.759986 + 0.649940i \(0.225206\pi\)
−0.759986 + 0.649940i \(0.774794\pi\)
\(314\) 0 0
\(315\) 10.7222i 0.604125i
\(316\) 0 0
\(317\) 0.312762 0.0175665 0.00878324 0.999961i \(-0.497204\pi\)
0.00878324 + 0.999961i \(0.497204\pi\)
\(318\) 0 0
\(319\) 15.0156 0.840714
\(320\) 0 0
\(321\) 6.26987i 0.349950i
\(322\) 0 0
\(323\) 2.01679i 0.112217i
\(324\) 0 0
\(325\) −1.61421 −0.0895405
\(326\) 0 0
\(327\) −16.5329 −0.914271
\(328\) 0 0
\(329\) 2.00000i 0.110264i
\(330\) 0 0
\(331\) 21.9669i 1.20741i 0.797207 + 0.603705i \(0.206310\pi\)
−0.797207 + 0.603705i \(0.793690\pi\)
\(332\) 0 0
\(333\) 10.1264i 0.554924i
\(334\) 0 0
\(335\) 9.87386i 0.539466i
\(336\) 0 0
\(337\) 9.12362i 0.496995i −0.968632 0.248498i \(-0.920063\pi\)
0.968632 0.248498i \(-0.0799368\pi\)
\(338\) 0 0
\(339\) 15.0088 0.815169
\(340\) 0 0
\(341\) 6.19210i 0.335321i
\(342\) 0 0
\(343\) 11.5751 0.624996
\(344\) 0 0
\(345\) −5.25938 10.0730i −0.283156 0.542313i
\(346\) 0 0
\(347\) 21.0761i 1.13142i −0.824603 0.565712i \(-0.808601\pi\)
0.824603 0.565712i \(-0.191399\pi\)
\(348\) 0 0
\(349\) −11.4279 −0.611719 −0.305860 0.952077i \(-0.598944\pi\)
−0.305860 + 0.952077i \(0.598944\pi\)
\(350\) 0 0
\(351\) 1.47554i 0.0787588i
\(352\) 0 0
\(353\) −13.2435 −0.704878 −0.352439 0.935835i \(-0.614648\pi\)
−0.352439 + 0.935835i \(0.614648\pi\)
\(354\) 0 0
\(355\) 9.75445 0.517712
\(356\) 0 0
\(357\) −5.96905 −0.315915
\(358\) 0 0
\(359\) 30.5609 1.61294 0.806471 0.591273i \(-0.201375\pi\)
0.806471 + 0.591273i \(0.201375\pi\)
\(360\) 0 0
\(361\) −8.21844 −0.432550
\(362\) 0 0
\(363\) 20.1315i 1.05663i
\(364\) 0 0
\(365\) 10.5918i 0.554399i
\(366\) 0 0
\(367\) 24.7046 1.28957 0.644784 0.764365i \(-0.276947\pi\)
0.644784 + 0.764365i \(0.276947\pi\)
\(368\) 0 0
\(369\) −9.04207 −0.470711
\(370\) 0 0
\(371\) 2.64615i 0.137381i
\(372\) 0 0
\(373\) 23.8972i 1.23735i −0.785648 0.618674i \(-0.787670\pi\)
0.785648 0.618674i \(-0.212330\pi\)
\(374\) 0 0
\(375\) 2.36943 0.122357
\(376\) 0 0
\(377\) 15.3185 0.788943
\(378\) 0 0
\(379\) −21.5130 −1.10505 −0.552525 0.833496i \(-0.686336\pi\)
−0.552525 + 0.833496i \(0.686336\pi\)
\(380\) 0 0
\(381\) −37.0077 −1.89596
\(382\) 0 0
\(383\) −10.5952 −0.541389 −0.270695 0.962665i \(-0.587253\pi\)
−0.270695 + 0.962665i \(0.587253\pi\)
\(384\) 0 0
\(385\) 6.48977i 0.330749i
\(386\) 0 0
\(387\) 21.4443 1.09008
\(388\) 0 0
\(389\) 2.63386i 0.133542i 0.997768 + 0.0667709i \(0.0212697\pi\)
−0.997768 + 0.0667709i \(0.978730\pi\)
\(390\) 0 0
\(391\) 2.61117 1.36336i 0.132052 0.0689479i
\(392\) 0 0
\(393\) −10.0730 −0.508117
\(394\) 0 0
\(395\) 15.7302i 0.791472i
\(396\) 0 0
\(397\) 32.1433 1.61322 0.806612 0.591081i \(-0.201299\pi\)
0.806612 + 0.591081i \(0.201299\pi\)
\(398\) 0 0
\(399\) 31.9100i 1.59750i
\(400\) 0 0
\(401\) 3.28642i 0.164116i 0.996628 + 0.0820581i \(0.0261493\pi\)
−0.996628 + 0.0820581i \(0.973851\pi\)
\(402\) 0 0
\(403\) 6.31700i 0.314672i
\(404\) 0 0
\(405\) 10.0085i 0.497328i
\(406\) 0 0
\(407\) 6.12918i 0.303812i
\(408\) 0 0
\(409\) 22.9242 1.13353 0.566764 0.823880i \(-0.308195\pi\)
0.566764 + 0.823880i \(0.308195\pi\)
\(410\) 0 0
\(411\) 25.2742 1.24668
\(412\) 0 0
\(413\) 58.7054i 2.88871i
\(414\) 0 0
\(415\) 9.87386i 0.484689i
\(416\) 0 0
\(417\) −8.28098 −0.405521
\(418\) 0 0
\(419\) −3.52574 −0.172244 −0.0861218 0.996285i \(-0.527447\pi\)
−0.0861218 + 0.996285i \(0.527447\pi\)
\(420\) 0 0
\(421\) 23.7491i 1.15746i 0.815518 + 0.578731i \(0.196452\pi\)
−0.815518 + 0.578731i \(0.803548\pi\)
\(422\) 0 0
\(423\) 1.27476i 0.0619812i
\(424\) 0 0
\(425\) 0.614214i 0.0297937i
\(426\) 0 0
\(427\) 28.4995i 1.37919i
\(428\) 0 0
\(429\) 6.05192i 0.292190i
\(430\) 0 0
\(431\) −12.2812 −0.591564 −0.295782 0.955256i \(-0.595580\pi\)
−0.295782 + 0.955256i \(0.595580\pi\)
\(432\) 0 0
\(433\) 26.5984i 1.27824i −0.769109 0.639118i \(-0.779300\pi\)
0.769109 0.639118i \(-0.220700\pi\)
\(434\) 0 0
\(435\) −22.4854 −1.07809
\(436\) 0 0
\(437\) 7.28838 + 13.9591i 0.348650 + 0.667752i
\(438\) 0 0
\(439\) 2.24247i 0.107027i −0.998567 0.0535137i \(-0.982958\pi\)
0.998567 0.0535137i \(-0.0170421\pi\)
\(440\) 0 0
\(441\) −25.6773 −1.22273
\(442\) 0 0
\(443\) 34.8704i 1.65674i −0.560180 0.828371i \(-0.689268\pi\)
0.560180 0.828371i \(-0.310732\pi\)
\(444\) 0 0
\(445\) 5.22843 0.247851
\(446\) 0 0
\(447\) −37.6626 −1.78138
\(448\) 0 0
\(449\) −19.7069 −0.930025 −0.465013 0.885304i \(-0.653950\pi\)
−0.465013 + 0.885304i \(0.653950\pi\)
\(450\) 0 0
\(451\) 5.47287 0.257707
\(452\) 0 0
\(453\) 17.6984 0.831541
\(454\) 0 0
\(455\) 6.62067i 0.310382i
\(456\) 0 0
\(457\) 23.3719i 1.09329i −0.837364 0.546645i \(-0.815905\pi\)
0.837364 0.546645i \(-0.184095\pi\)
\(458\) 0 0
\(459\) −0.561450 −0.0262062
\(460\) 0 0
\(461\) −20.0730 −0.934894 −0.467447 0.884021i \(-0.654826\pi\)
−0.467447 + 0.884021i \(0.654826\pi\)
\(462\) 0 0
\(463\) 23.3958i 1.08729i −0.839314 0.543647i \(-0.817043\pi\)
0.839314 0.543647i \(-0.182957\pi\)
\(464\) 0 0
\(465\) 9.27245i 0.430000i
\(466\) 0 0
\(467\) 2.70683 0.125257 0.0626286 0.998037i \(-0.480052\pi\)
0.0626286 + 0.998037i \(0.480052\pi\)
\(468\) 0 0
\(469\) 40.4975 1.87000
\(470\) 0 0
\(471\) 0.792249 0.0365049
\(472\) 0 0
\(473\) −12.9795 −0.596799
\(474\) 0 0
\(475\) −3.28353 −0.150659
\(476\) 0 0
\(477\) 1.68661i 0.0772244i
\(478\) 0 0
\(479\) 37.9119 1.73224 0.866119 0.499838i \(-0.166607\pi\)
0.866119 + 0.499838i \(0.166607\pi\)
\(480\) 0 0
\(481\) 6.25281i 0.285104i
\(482\) 0 0
\(483\) 41.3143 21.5713i 1.87987 0.981527i
\(484\) 0 0
\(485\) −12.6958 −0.576485
\(486\) 0 0
\(487\) 29.4903i 1.33633i 0.744012 + 0.668167i \(0.232921\pi\)
−0.744012 + 0.668167i \(0.767079\pi\)
\(488\) 0 0
\(489\) 3.60423 0.162989
\(490\) 0 0
\(491\) 27.4432i 1.23849i 0.785197 + 0.619246i \(0.212562\pi\)
−0.785197 + 0.619246i \(0.787438\pi\)
\(492\) 0 0
\(493\) 5.82874i 0.262513i
\(494\) 0 0
\(495\) 4.13646i 0.185920i
\(496\) 0 0
\(497\) 40.0077i 1.79459i
\(498\) 0 0
\(499\) 22.4504i 1.00502i −0.864572 0.502509i \(-0.832410\pi\)
0.864572 0.502509i \(-0.167590\pi\)
\(500\) 0 0
\(501\) 5.75110 0.256940
\(502\) 0 0
\(503\) −13.9555 −0.622243 −0.311122 0.950370i \(-0.600705\pi\)
−0.311122 + 0.950370i \(0.600705\pi\)
\(504\) 0 0
\(505\) 7.56279i 0.336540i
\(506\) 0 0
\(507\) 24.6286i 1.09380i
\(508\) 0 0
\(509\) −25.9702 −1.15111 −0.575554 0.817764i \(-0.695214\pi\)
−0.575554 + 0.817764i \(0.695214\pi\)
\(510\) 0 0
\(511\) −43.4420 −1.92176
\(512\) 0 0
\(513\) 3.00146i 0.132518i
\(514\) 0 0
\(515\) 4.98476i 0.219655i
\(516\) 0 0
\(517\) 0.771573i 0.0339337i
\(518\) 0 0
\(519\) 18.2924i 0.802946i
\(520\) 0 0
\(521\) 14.1500i 0.619921i −0.950749 0.309961i \(-0.899684\pi\)
0.950749 0.309961i \(-0.100316\pi\)
\(522\) 0 0
\(523\) −19.5089 −0.853064 −0.426532 0.904472i \(-0.640265\pi\)
−0.426532 + 0.904472i \(0.640265\pi\)
\(524\) 0 0
\(525\) 9.71819i 0.424137i
\(526\) 0 0
\(527\) 2.40364 0.104704
\(528\) 0 0
\(529\) −13.1460 + 18.8728i −0.571567 + 0.820555i
\(530\) 0 0
\(531\) 37.4178i 1.62379i
\(532\) 0 0
\(533\) 5.58326 0.241838
\(534\) 0 0
\(535\) 2.64615i 0.114403i
\(536\) 0 0
\(537\) 28.7997 1.24280
\(538\) 0 0
\(539\) 15.5416 0.669424
\(540\) 0 0
\(541\) 16.2191 0.697312 0.348656 0.937251i \(-0.386638\pi\)
0.348656 + 0.937251i \(0.386638\pi\)
\(542\) 0 0
\(543\) −42.5482 −1.82592
\(544\) 0 0
\(545\) −6.97757 −0.298886
\(546\) 0 0
\(547\) 30.6301i 1.30965i 0.755782 + 0.654824i \(0.227257\pi\)
−0.755782 + 0.654824i \(0.772743\pi\)
\(548\) 0 0
\(549\) 18.1651i 0.775266i
\(550\) 0 0
\(551\) 31.1599 1.32746
\(552\) 0 0
\(553\) −64.5171 −2.74355
\(554\) 0 0
\(555\) 9.17822i 0.389594i
\(556\) 0 0
\(557\) 27.4167i 1.16168i 0.814016 + 0.580842i \(0.197277\pi\)
−0.814016 + 0.580842i \(0.802723\pi\)
\(558\) 0 0
\(559\) −13.2413 −0.560049
\(560\) 0 0
\(561\) −2.30278 −0.0972233
\(562\) 0 0
\(563\) 25.7424 1.08491 0.542457 0.840084i \(-0.317494\pi\)
0.542457 + 0.840084i \(0.317494\pi\)
\(564\) 0 0
\(565\) 6.33436 0.266489
\(566\) 0 0
\(567\) −41.0498 −1.72393
\(568\) 0 0
\(569\) 27.0414i 1.13364i 0.823843 + 0.566818i \(0.191826\pi\)
−0.823843 + 0.566818i \(0.808174\pi\)
\(570\) 0 0
\(571\) −9.72458 −0.406961 −0.203481 0.979079i \(-0.565225\pi\)
−0.203481 + 0.979079i \(0.565225\pi\)
\(572\) 0 0
\(573\) 59.0188i 2.46555i
\(574\) 0 0
\(575\) −2.21968 4.25124i −0.0925670 0.177289i
\(576\) 0 0
\(577\) 36.6969 1.52771 0.763856 0.645387i \(-0.223304\pi\)
0.763856 + 0.645387i \(0.223304\pi\)
\(578\) 0 0
\(579\) 0.909460i 0.0377959i
\(580\) 0 0
\(581\) −40.4975 −1.68012
\(582\) 0 0
\(583\) 1.02085i 0.0422792i
\(584\) 0 0
\(585\) 4.21990i 0.174471i
\(586\) 0 0
\(587\) 5.91027i 0.243943i 0.992534 + 0.121971i \(0.0389216\pi\)
−0.992534 + 0.121971i \(0.961078\pi\)
\(588\) 0 0
\(589\) 12.8496i 0.529460i
\(590\) 0 0
\(591\) 61.1152i 2.51394i
\(592\) 0 0
\(593\) 2.62552 0.107817 0.0539087 0.998546i \(-0.482832\pi\)
0.0539087 + 0.998546i \(0.482832\pi\)
\(594\) 0 0
\(595\) −2.51919 −0.103277
\(596\) 0 0
\(597\) 36.5620i 1.49638i
\(598\) 0 0
\(599\) 44.2263i 1.80704i −0.428550 0.903518i \(-0.640975\pi\)
0.428550 0.903518i \(-0.359025\pi\)
\(600\) 0 0
\(601\) 21.6378 0.882623 0.441312 0.897354i \(-0.354513\pi\)
0.441312 + 0.897354i \(0.354513\pi\)
\(602\) 0 0
\(603\) −25.8124 −1.05116
\(604\) 0 0
\(605\) 8.49634i 0.345425i
\(606\) 0 0
\(607\) 11.7278i 0.476015i −0.971263 0.238007i \(-0.923506\pi\)
0.971263 0.238007i \(-0.0764943\pi\)
\(608\) 0 0
\(609\) 92.2234i 3.73708i
\(610\) 0 0
\(611\) 0.787136i 0.0318441i
\(612\) 0 0
\(613\) 18.5636i 0.749777i 0.927070 + 0.374889i \(0.122319\pi\)
−0.927070 + 0.374889i \(0.877681\pi\)
\(614\) 0 0
\(615\) −8.19542 −0.330471
\(616\) 0 0
\(617\) 43.5040i 1.75141i 0.482850 + 0.875703i \(0.339602\pi\)
−0.482850 + 0.875703i \(0.660398\pi\)
\(618\) 0 0
\(619\) 3.64467 0.146492 0.0732459 0.997314i \(-0.476664\pi\)
0.0732459 + 0.997314i \(0.476664\pi\)
\(620\) 0 0
\(621\) 3.88604 2.02900i 0.155941 0.0814209i
\(622\) 0 0
\(623\) 21.4443i 0.859148i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 12.3104i 0.491631i
\(628\) 0 0
\(629\) −2.37921 −0.0948655
\(630\) 0 0
\(631\) −48.9799 −1.94986 −0.974930 0.222511i \(-0.928575\pi\)
−0.974930 + 0.222511i \(0.928575\pi\)
\(632\) 0 0
\(633\) 36.9347 1.46802
\(634\) 0 0
\(635\) −15.6188 −0.619813
\(636\) 0 0
\(637\) 15.8551 0.628201
\(638\) 0 0
\(639\) 25.5002i 1.00877i
\(640\) 0 0
\(641\) 19.0637i 0.752970i −0.926423 0.376485i \(-0.877133\pi\)
0.926423 0.376485i \(-0.122867\pi\)
\(642\) 0 0
\(643\) −38.3231 −1.51132 −0.755658 0.654967i \(-0.772683\pi\)
−0.755658 + 0.654967i \(0.772683\pi\)
\(644\) 0 0
\(645\) 19.4364 0.765307
\(646\) 0 0
\(647\) 19.7426i 0.776162i 0.921625 + 0.388081i \(0.126862\pi\)
−0.921625 + 0.388081i \(0.873138\pi\)
\(648\) 0 0
\(649\) 22.6478i 0.889002i
\(650\) 0 0
\(651\) 38.0308 1.49055
\(652\) 0 0
\(653\) −10.3858 −0.406427 −0.203214 0.979134i \(-0.565139\pi\)
−0.203214 + 0.979134i \(0.565139\pi\)
\(654\) 0 0
\(655\) −4.25124 −0.166110
\(656\) 0 0
\(657\) 27.6892 1.08026
\(658\) 0 0
\(659\) −3.77870 −0.147197 −0.0735987 0.997288i \(-0.523448\pi\)
−0.0735987 + 0.997288i \(0.523448\pi\)
\(660\) 0 0
\(661\) 42.7970i 1.66461i −0.554319 0.832304i \(-0.687021\pi\)
0.554319 0.832304i \(-0.312979\pi\)
\(662\) 0 0
\(663\) −2.34923 −0.0912364
\(664\) 0 0
\(665\) 13.4673i 0.522241i
\(666\) 0 0
\(667\) 21.0642 + 40.3432i 0.815610 + 1.56210i
\(668\) 0 0
\(669\) −2.31081 −0.0893409
\(670\) 0 0
\(671\) 10.9947i 0.424446i
\(672\) 0 0
\(673\) −20.3877 −0.785890 −0.392945 0.919562i \(-0.628544\pi\)
−0.392945 + 0.919562i \(0.628544\pi\)
\(674\) 0 0
\(675\) 0.914095i 0.0351835i
\(676\) 0 0
\(677\) 10.1844i 0.391418i −0.980662 0.195709i \(-0.937299\pi\)
0.980662 0.195709i \(-0.0627009\pi\)
\(678\) 0 0
\(679\) 52.0715i 1.99832i
\(680\) 0 0
\(681\) 15.7063i 0.601865i
\(682\) 0 0
\(683\) 26.4295i 1.01130i −0.862739 0.505649i \(-0.831253\pi\)
0.862739 0.505649i \(-0.168747\pi\)
\(684\) 0 0
\(685\) 10.6668 0.407556
\(686\) 0 0
\(687\) 21.2764 0.811745
\(688\) 0 0
\(689\) 1.04144i 0.0396757i
\(690\) 0 0
\(691\) 3.46410i 0.131781i −0.997827 0.0658903i \(-0.979011\pi\)
0.997827 0.0658903i \(-0.0209887\pi\)
\(692\) 0 0
\(693\) −16.9656 −0.644471
\(694\) 0 0
\(695\) −3.49492 −0.132570
\(696\) 0 0
\(697\) 2.12445i 0.0804692i
\(698\) 0 0
\(699\) 13.9373i 0.527157i
\(700\) 0 0
\(701\) 50.6219i 1.91196i −0.293426 0.955982i \(-0.594796\pi\)
0.293426 0.955982i \(-0.405204\pi\)
\(702\) 0 0
\(703\) 12.7191i 0.479708i
\(704\) 0 0
\(705\) 1.15540i 0.0435150i
\(706\) 0 0
\(707\) −31.0187 −1.16658
\(708\) 0 0
\(709\) 51.1958i 1.92270i 0.275330 + 0.961350i \(0.411213\pi\)
−0.275330 + 0.961350i \(0.588787\pi\)
\(710\) 0 0
\(711\) 41.1221 1.54220
\(712\) 0 0
\(713\) −16.6366 + 8.68641i −0.623047 + 0.325309i
\(714\) 0 0
\(715\) 2.55417i 0.0955204i
\(716\) 0 0
\(717\) 7.57132 0.282756
\(718\) 0 0
\(719\) 11.8826i 0.443147i −0.975144 0.221573i \(-0.928881\pi\)
0.975144 0.221573i \(-0.0711193\pi\)
\(720\) 0 0
\(721\) −20.4449 −0.761408
\(722\) 0 0
\(723\) −16.9899 −0.631860
\(724\) 0 0
\(725\) −9.48977 −0.352441
\(726\) 0 0
\(727\) 19.4203 0.720259 0.360130 0.932902i \(-0.382733\pi\)
0.360130 + 0.932902i \(0.382733\pi\)
\(728\) 0 0
\(729\) 19.6668 0.728399
\(730\) 0 0
\(731\) 5.03837i 0.186351i
\(732\) 0 0
\(733\) 50.5014i 1.86531i −0.360768 0.932656i \(-0.617485\pi\)
0.360768 0.932656i \(-0.382515\pi\)
\(734\) 0 0
\(735\) −23.2730 −0.858437
\(736\) 0 0
\(737\) 15.6234 0.575494
\(738\) 0 0
\(739\) 27.9655i 1.02873i 0.857573 + 0.514363i \(0.171972\pi\)
−0.857573 + 0.514363i \(0.828028\pi\)
\(740\) 0 0
\(741\) 12.5587i 0.461357i
\(742\) 0 0
\(743\) 17.1088 0.627661 0.313830 0.949479i \(-0.398388\pi\)
0.313830 + 0.949479i \(0.398388\pi\)
\(744\) 0 0
\(745\) −15.8952 −0.582355
\(746\) 0 0
\(747\) 25.8124 0.944425
\(748\) 0 0
\(749\) −10.8531 −0.396565
\(750\) 0 0
\(751\) −0.859711 −0.0313713 −0.0156856 0.999877i \(-0.504993\pi\)
−0.0156856 + 0.999877i \(0.504993\pi\)
\(752\) 0 0
\(753\) 58.3011i 2.12461i
\(754\) 0 0
\(755\) 7.46945 0.271841
\(756\) 0 0
\(757\) 29.7707i 1.08204i −0.841011 0.541018i \(-0.818039\pi\)
0.841011 0.541018i \(-0.181961\pi\)
\(758\) 0 0
\(759\) 15.9385 8.32190i 0.578531 0.302066i
\(760\) 0 0
\(761\) −13.6509 −0.494845 −0.247423 0.968908i \(-0.579584\pi\)
−0.247423 + 0.968908i \(0.579584\pi\)
\(762\) 0 0
\(763\) 28.6184i 1.03606i
\(764\) 0 0
\(765\) 1.60569 0.0580537
\(766\) 0 0
\(767\) 23.1046i 0.834258i
\(768\) 0 0
\(769\) 12.9625i 0.467439i 0.972304 + 0.233719i \(0.0750897\pi\)
−0.972304 + 0.233719i \(0.924910\pi\)
\(770\) 0 0
\(771\) 15.1261i 0.544752i
\(772\) 0 0
\(773\) 7.18749i 0.258516i 0.991611 + 0.129258i \(0.0412595\pi\)
−0.991611 + 0.129258i \(0.958740\pi\)
\(774\) 0 0
\(775\) 3.91336i 0.140572i
\(776\) 0 0
\(777\) −37.6443 −1.35048
\(778\) 0 0
\(779\) 11.3571 0.406910
\(780\) 0 0
\(781\) 15.4344i 0.552287i
\(782\) 0 0
\(783\) 8.67455i 0.310003i
\(784\) 0 0
\(785\) 0.334362 0.0119339
\(786\) 0 0
\(787\) −21.7181 −0.774167 −0.387084 0.922045i \(-0.626517\pi\)
−0.387084 + 0.922045i \(0.626517\pi\)
\(788\) 0 0
\(789\) 34.0412i 1.21190i
\(790\) 0 0
\(791\) 25.9803i 0.923753i
\(792\) 0 0
\(793\) 11.2165i 0.398309i
\(794\) 0 0
\(795\) 1.52868i 0.0542168i
\(796\) 0 0
\(797\) 30.8963i 1.09440i −0.837001 0.547202i \(-0.815693\pi\)
0.837001 0.547202i \(-0.184307\pi\)
\(798\) 0 0
\(799\) −0.299508 −0.0105958
\(800\) 0 0
\(801\) 13.6682i 0.482943i
\(802\) 0 0
\(803\) −16.7594 −0.591425
\(804\) 0 0
\(805\) 17.4364 9.10398i 0.614552 0.320873i
\(806\) 0 0
\(807\) 18.9757i 0.667975i
\(808\) 0 0
\(809\) 12.3034 0.432565 0.216282 0.976331i \(-0.430607\pi\)
0.216282 + 0.976331i \(0.430607\pi\)
\(810\) 0 0
\(811\) 20.0232i 0.703109i 0.936167 + 0.351554i \(0.114347\pi\)
−0.936167 + 0.351554i \(0.885653\pi\)
\(812\) 0 0
\(813\) −9.29836 −0.326108
\(814\) 0 0
\(815\) 1.52114 0.0532830
\(816\) 0 0
\(817\) −26.9347 −0.942325
\(818\) 0 0
\(819\) −17.3078 −0.604785
\(820\) 0 0
\(821\) −38.2682 −1.33557 −0.667785 0.744354i \(-0.732757\pi\)
−0.667785 + 0.744354i \(0.732757\pi\)
\(822\) 0 0
\(823\) 18.7368i 0.653125i −0.945176 0.326563i \(-0.894110\pi\)
0.945176 0.326563i \(-0.105890\pi\)
\(824\) 0 0
\(825\) 3.74915i 0.130529i
\(826\) 0 0
\(827\) −24.1370 −0.839326 −0.419663 0.907680i \(-0.637852\pi\)
−0.419663 + 0.907680i \(0.637852\pi\)
\(828\) 0 0
\(829\) −28.7912 −0.999960 −0.499980 0.866037i \(-0.666659\pi\)
−0.499980 + 0.866037i \(0.666659\pi\)
\(830\) 0 0
\(831\) 18.1834i 0.630776i
\(832\) 0 0
\(833\) 6.03291i 0.209028i
\(834\) 0 0
\(835\) 2.42721 0.0839969
\(836\) 0 0
\(837\) 3.57719 0.123646
\(838\) 0 0
\(839\) 28.6881 0.990422 0.495211 0.868773i \(-0.335091\pi\)
0.495211 + 0.868773i \(0.335091\pi\)
\(840\) 0 0
\(841\) 61.0556 2.10537
\(842\) 0 0
\(843\) 33.3188 1.14756
\(844\) 0 0
\(845\) 10.3943i 0.357575i
\(846\) 0 0
\(847\) −34.8476 −1.19738
\(848\) 0 0
\(849\) 73.3506i 2.51739i
\(850\) 0 0
\(851\) 16.4676 8.59814i 0.564501 0.294740i
\(852\) 0 0
\(853\) 53.8952 1.84534 0.922668 0.385595i \(-0.126004\pi\)
0.922668 + 0.385595i \(0.126004\pi\)
\(854\) 0 0
\(855\) 8.58384i 0.293561i
\(856\) 0 0
\(857\) 56.2532 1.92157 0.960786 0.277291i \(-0.0894365\pi\)
0.960786 + 0.277291i \(0.0894365\pi\)
\(858\) 0 0
\(859\) 4.58572i 0.156463i −0.996935 0.0782314i \(-0.975073\pi\)
0.996935 0.0782314i \(-0.0249273\pi\)
\(860\) 0 0
\(861\) 33.6134i 1.14554i
\(862\) 0 0
\(863\) 21.6941i 0.738475i −0.929335 0.369237i \(-0.879619\pi\)
0.929335 0.369237i \(-0.120381\pi\)
\(864\) 0 0
\(865\) 7.72015i 0.262493i
\(866\) 0 0
\(867\) 39.3865i 1.33763i
\(868\) 0 0
\(869\) −24.8898 −0.844329
\(870\) 0 0
\(871\) 15.9385 0.540056
\(872\) 0 0
\(873\) 33.1894i 1.12329i
\(874\) 0 0
\(875\) 4.10148i 0.138655i
\(876\) 0 0
\(877\) 19.7201 0.665902 0.332951 0.942944i \(-0.391956\pi\)
0.332951 + 0.942944i \(0.391956\pi\)
\(878\) 0 0
\(879\) −59.8329 −2.01811
\(880\) 0 0
\(881\) 39.2755i 1.32323i 0.749845 + 0.661613i \(0.230128\pi\)
−0.749845 + 0.661613i \(0.769872\pi\)
\(882\) 0 0
\(883\) 46.2498i 1.55643i −0.627997 0.778215i \(-0.716125\pi\)
0.627997 0.778215i \(-0.283875\pi\)
\(884\) 0 0
\(885\) 33.9142i 1.14001i
\(886\) 0 0
\(887\) 4.78957i 0.160818i −0.996762 0.0804090i \(-0.974377\pi\)
0.996762 0.0804090i \(-0.0256226\pi\)
\(888\) 0 0
\(889\) 64.0603i 2.14851i
\(890\) 0 0
\(891\) −15.8365 −0.530541
\(892\) 0 0
\(893\) 1.60114i 0.0535802i
\(894\) 0 0
\(895\) 12.1547 0.406287
\(896\) 0 0
\(897\) 16.2600 8.48977i 0.542906 0.283465i
\(898\) 0 0
\(899\) 37.1369i 1.23858i
\(900\) 0 0
\(901\) 0.396271 0.0132017
\(902\) 0 0
\(903\) 79.7180i 2.65285i
\(904\) 0 0
\(905\) −17.9571 −0.596914
\(906\) 0 0
\(907\) 9.61899 0.319393 0.159697 0.987166i \(-0.448948\pi\)
0.159697 + 0.987166i \(0.448948\pi\)
\(908\) 0 0
\(909\) 19.7707 0.655754
\(910\) 0 0
\(911\) 40.7144 1.34893 0.674464 0.738308i \(-0.264375\pi\)
0.674464 + 0.738308i \(0.264375\pi\)
\(912\) 0 0
\(913\) −15.6234 −0.517058
\(914\) 0 0
\(915\) 16.4642i 0.544289i
\(916\) 0 0
\(917\) 17.4364i 0.575800i
\(918\) 0 0
\(919\) −8.61898 −0.284314 −0.142157 0.989844i \(-0.545404\pi\)
−0.142157 + 0.989844i \(0.545404\pi\)
\(920\) 0 0
\(921\) 64.9350 2.13968
\(922\) 0 0
\(923\) 15.7458i 0.518278i
\(924\) 0 0
\(925\) 3.87360i 0.127363i
\(926\) 0 0
\(927\) 13.0312 0.428002
\(928\) 0 0
\(929\) −2.57214 −0.0843893 −0.0421946 0.999109i \(-0.513435\pi\)
−0.0421946 + 0.999109i \(0.513435\pi\)
\(930\) 0 0
\(931\) 32.2514 1.05700
\(932\) 0 0
\(933\) 54.6969 1.79070
\(934\) 0 0
\(935\) −0.971868 −0.0317835
\(936\) 0 0
\(937\) 37.8055i 1.23505i −0.786551 0.617526i \(-0.788135\pi\)
0.786551 0.617526i \(-0.211865\pi\)
\(938\) 0 0
\(939\) 54.4904 1.77823
\(940\) 0 0
\(941\) 40.1941i 1.31029i 0.755504 + 0.655144i \(0.227392\pi\)
−0.755504 + 0.655144i \(0.772608\pi\)
\(942\) 0 0
\(943\) 7.67745 + 14.7042i 0.250012 + 0.478836i
\(944\) 0 0
\(945\) −3.74915 −0.121960
\(946\) 0 0
\(947\) 18.5480i 0.602728i −0.953509 0.301364i \(-0.902558\pi\)
0.953509 0.301364i \(-0.0974419\pi\)
\(948\) 0 0
\(949\) −17.0974 −0.555005
\(950\) 0 0
\(951\) 0.741069i 0.0240308i
\(952\) 0 0
\(953\) 11.5741i 0.374922i 0.982272 + 0.187461i \(0.0600258\pi\)
−0.982272 + 0.187461i \(0.939974\pi\)
\(954\) 0 0
\(955\) 24.9084i 0.806017i
\(956\) 0 0
\(957\) 35.5785i 1.15009i
\(958\) 0 0
\(959\) 43.7496i 1.41275i
\(960\) 0 0
\(961\) 15.6856 0.505987
\(962\) 0 0
\(963\) 6.91759 0.222916
\(964\) 0 0
\(965\) 0.383830i 0.0123559i
\(966\) 0 0
\(967\) 40.9423i 1.31661i 0.752749 + 0.658307i \(0.228727\pi\)
−0.752749 + 0.658307i \(0.771273\pi\)
\(968\) 0 0
\(969\) −4.77864 −0.153512
\(970\) 0 0
\(971\) −34.0340 −1.09220 −0.546101 0.837719i \(-0.683889\pi\)
−0.546101 + 0.837719i \(0.683889\pi\)
\(972\) 0 0
\(973\) 14.3344i 0.459539i
\(974\) 0 0
\(975\) 3.82477i 0.122491i
\(976\) 0 0
\(977\) 24.4336i 0.781700i 0.920454 + 0.390850i \(0.127819\pi\)
−0.920454 + 0.390850i \(0.872181\pi\)
\(978\) 0 0
\(979\) 8.27292i 0.264404i
\(980\) 0 0
\(981\) 18.2409i 0.582386i
\(982\) 0 0
\(983\) −7.76928 −0.247801 −0.123901 0.992295i \(-0.539540\pi\)
−0.123901 + 0.992295i \(0.539540\pi\)
\(984\) 0 0
\(985\) 25.7932i 0.821839i
\(986\) 0 0
\(987\) −4.73887 −0.150840
\(988\) 0 0
\(989\) −18.2080 34.8728i −0.578979 1.10889i
\(990\) 0 0
\(991\) 30.2428i 0.960693i 0.877079 + 0.480346i \(0.159489\pi\)
−0.877079 + 0.480346i \(0.840511\pi\)
\(992\) 0 0
\(993\) 52.0491 1.65173
\(994\) 0 0
\(995\) 15.4307i 0.489185i
\(996\) 0 0
\(997\) 16.2040 0.513187 0.256594 0.966519i \(-0.417400\pi\)
0.256594 + 0.966519i \(0.417400\pi\)
\(998\) 0 0
\(999\) −3.54083 −0.112027
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 1840.2.i.a.1471.3 16
4.3 odd 2 inner 1840.2.i.a.1471.13 yes 16
23.22 odd 2 inner 1840.2.i.a.1471.4 yes 16
92.91 even 2 inner 1840.2.i.a.1471.14 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.i.a.1471.3 16 1.1 even 1 trivial
1840.2.i.a.1471.4 yes 16 23.22 odd 2 inner
1840.2.i.a.1471.13 yes 16 4.3 odd 2 inner
1840.2.i.a.1471.14 yes 16 92.91 even 2 inner