Properties

Label 1380.2.f.b.829.6
Level $1380$
Weight $2$
Character 1380.829
Analytic conductor $11.019$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,2,Mod(829,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.f (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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 829.6
Root \(0.0729221i\) of defining polynomial
Character \(\chi\) \(=\) 1380.829
Dual form 1380.2.f.b.829.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(1.54426 - 1.61718i) q^{5} -3.99468i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(1.54426 - 1.61718i) q^{5} -3.99468i q^{7} -1.00000 q^{9} +2.31084 q^{11} -5.60596i q^{13} +(-1.61718 - 1.54426i) q^{15} +3.27777i q^{17} -1.49731 q^{19} -3.99468 q^{21} +1.00000i q^{23} +(-0.230539 - 4.99468i) q^{25} +1.00000i q^{27} +4.51213 q^{29} -2.38895 q^{31} -2.31084i q^{33} +(-6.46012 - 6.16882i) q^{35} +6.66237i q^{37} -5.60596 q^{39} +1.07468 q^{41} +12.5260i q^{43} +(-1.54426 + 1.61718i) q^{45} -4.43129i q^{47} -8.95749 q^{49} +3.27777 q^{51} -2.01735i q^{53} +(3.56853 - 3.73705i) q^{55} +1.49731i q^{57} -13.1502 q^{59} +11.1266 q^{61} +3.99468i q^{63} +(-9.06585 - 8.65705i) q^{65} -9.91149i q^{67} +1.00000 q^{69} +4.94360 q^{71} +4.38340i q^{73} +(-4.99468 + 0.230539i) q^{75} -9.23108i q^{77} -9.35481 q^{79} +1.00000 q^{81} -3.95659i q^{83} +(5.30074 + 5.06172i) q^{85} -4.51213i q^{87} +11.6996 q^{89} -22.3940 q^{91} +2.38895i q^{93} +(-2.31224 + 2.42142i) q^{95} +16.4888i q^{97} -2.31084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{9} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 6 q^{25} - 30 q^{29} + 6 q^{31} - 14 q^{35} + 4 q^{39} + 46 q^{41} - 20 q^{49} + 2 q^{51} - 16 q^{55} - 10 q^{59} + 64 q^{61} - 36 q^{65} + 14 q^{69} + 42 q^{71} - 16 q^{75} - 32 q^{79} + 14 q^{81} - 42 q^{85} - 52 q^{89} + 28 q^{91} - 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.54426 1.61718i 0.690613 0.723225i
\(6\) 0 0
\(7\) 3.99468i 1.50985i −0.655812 0.754924i \(-0.727674\pi\)
0.655812 0.754924i \(-0.272326\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.31084 0.696745 0.348372 0.937356i \(-0.386734\pi\)
0.348372 + 0.937356i \(0.386734\pi\)
\(12\) 0 0
\(13\) 5.60596i 1.55481i −0.628998 0.777407i \(-0.716535\pi\)
0.628998 0.777407i \(-0.283465\pi\)
\(14\) 0 0
\(15\) −1.61718 1.54426i −0.417554 0.398726i
\(16\) 0 0
\(17\) 3.27777i 0.794976i 0.917607 + 0.397488i \(0.130118\pi\)
−0.917607 + 0.397488i \(0.869882\pi\)
\(18\) 0 0
\(19\) −1.49731 −0.343507 −0.171754 0.985140i \(-0.554943\pi\)
−0.171754 + 0.985140i \(0.554943\pi\)
\(20\) 0 0
\(21\) −3.99468 −0.871711
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) −0.230539 4.99468i −0.0461077 0.998936i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.51213 0.837882 0.418941 0.908014i \(-0.362401\pi\)
0.418941 + 0.908014i \(0.362401\pi\)
\(30\) 0 0
\(31\) −2.38895 −0.429069 −0.214534 0.976716i \(-0.568823\pi\)
−0.214534 + 0.976716i \(0.568823\pi\)
\(32\) 0 0
\(33\) 2.31084i 0.402266i
\(34\) 0 0
\(35\) −6.46012 6.16882i −1.09196 1.04272i
\(36\) 0 0
\(37\) 6.66237i 1.09529i 0.836712 + 0.547643i \(0.184475\pi\)
−0.836712 + 0.547643i \(0.815525\pi\)
\(38\) 0 0
\(39\) −5.60596 −0.897672
\(40\) 0 0
\(41\) 1.07468 0.167837 0.0839184 0.996473i \(-0.473257\pi\)
0.0839184 + 0.996473i \(0.473257\pi\)
\(42\) 0 0
\(43\) 12.5260i 1.91019i 0.296297 + 0.955096i \(0.404248\pi\)
−0.296297 + 0.955096i \(0.595752\pi\)
\(44\) 0 0
\(45\) −1.54426 + 1.61718i −0.230204 + 0.241075i
\(46\) 0 0
\(47\) 4.43129i 0.646370i −0.946336 0.323185i \(-0.895246\pi\)
0.946336 0.323185i \(-0.104754\pi\)
\(48\) 0 0
\(49\) −8.95749 −1.27964
\(50\) 0 0
\(51\) 3.27777 0.458980
\(52\) 0 0
\(53\) 2.01735i 0.277104i −0.990355 0.138552i \(-0.955755\pi\)
0.990355 0.138552i \(-0.0442449\pi\)
\(54\) 0 0
\(55\) 3.56853 3.73705i 0.481181 0.503903i
\(56\) 0 0
\(57\) 1.49731i 0.198324i
\(58\) 0 0
\(59\) −13.1502 −1.71202 −0.856008 0.516963i \(-0.827062\pi\)
−0.856008 + 0.516963i \(0.827062\pi\)
\(60\) 0 0
\(61\) 11.1266 1.42462 0.712308 0.701867i \(-0.247650\pi\)
0.712308 + 0.701867i \(0.247650\pi\)
\(62\) 0 0
\(63\) 3.99468i 0.503283i
\(64\) 0 0
\(65\) −9.06585 8.65705i −1.12448 1.07377i
\(66\) 0 0
\(67\) 9.91149i 1.21088i −0.795891 0.605440i \(-0.792997\pi\)
0.795891 0.605440i \(-0.207003\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.94360 0.586697 0.293348 0.956006i \(-0.405230\pi\)
0.293348 + 0.956006i \(0.405230\pi\)
\(72\) 0 0
\(73\) 4.38340i 0.513038i 0.966539 + 0.256519i \(0.0825757\pi\)
−0.966539 + 0.256519i \(0.917424\pi\)
\(74\) 0 0
\(75\) −4.99468 + 0.230539i −0.576736 + 0.0266203i
\(76\) 0 0
\(77\) 9.23108i 1.05198i
\(78\) 0 0
\(79\) −9.35481 −1.05250 −0.526249 0.850331i \(-0.676402\pi\)
−0.526249 + 0.850331i \(0.676402\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.95659i 0.434292i −0.976139 0.217146i \(-0.930325\pi\)
0.976139 0.217146i \(-0.0696748\pi\)
\(84\) 0 0
\(85\) 5.30074 + 5.06172i 0.574946 + 0.549021i
\(86\) 0 0
\(87\) 4.51213i 0.483751i
\(88\) 0 0
\(89\) 11.6996 1.24015 0.620076 0.784542i \(-0.287102\pi\)
0.620076 + 0.784542i \(0.287102\pi\)
\(90\) 0 0
\(91\) −22.3940 −2.34753
\(92\) 0 0
\(93\) 2.38895i 0.247723i
\(94\) 0 0
\(95\) −2.31224 + 2.42142i −0.237231 + 0.248433i
\(96\) 0 0
\(97\) 16.4888i 1.67418i 0.547064 + 0.837090i \(0.315745\pi\)
−0.547064 + 0.837090i \(0.684255\pi\)
\(98\) 0 0
\(99\) −2.31084 −0.232248
\(100\) 0 0
\(101\) 10.3994 1.03478 0.517390 0.855750i \(-0.326904\pi\)
0.517390 + 0.855750i \(0.326904\pi\)
\(102\) 0 0
\(103\) 12.4044i 1.22225i −0.791536 0.611123i \(-0.790718\pi\)
0.791536 0.611123i \(-0.209282\pi\)
\(104\) 0 0
\(105\) −6.16882 + 6.46012i −0.602015 + 0.630443i
\(106\) 0 0
\(107\) 1.51748i 0.146700i 0.997306 + 0.0733502i \(0.0233691\pi\)
−0.997306 + 0.0733502i \(0.976631\pi\)
\(108\) 0 0
\(109\) −4.86276 −0.465767 −0.232884 0.972505i \(-0.574816\pi\)
−0.232884 + 0.972505i \(0.574816\pi\)
\(110\) 0 0
\(111\) 6.66237 0.632364
\(112\) 0 0
\(113\) 1.72566i 0.162337i −0.996700 0.0811683i \(-0.974135\pi\)
0.996700 0.0811683i \(-0.0258651\pi\)
\(114\) 0 0
\(115\) 1.61718 + 1.54426i 0.150803 + 0.144003i
\(116\) 0 0
\(117\) 5.60596i 0.518271i
\(118\) 0 0
\(119\) 13.0937 1.20029
\(120\) 0 0
\(121\) −5.66001 −0.514546
\(122\) 0 0
\(123\) 1.07468i 0.0969006i
\(124\) 0 0
\(125\) −8.43331 7.34025i −0.754298 0.656532i
\(126\) 0 0
\(127\) 14.1962i 1.25971i −0.776713 0.629855i \(-0.783114\pi\)
0.776713 0.629855i \(-0.216886\pi\)
\(128\) 0 0
\(129\) 12.5260 1.10285
\(130\) 0 0
\(131\) −5.34301 −0.466821 −0.233410 0.972378i \(-0.574989\pi\)
−0.233410 + 0.972378i \(0.574989\pi\)
\(132\) 0 0
\(133\) 5.98129i 0.518644i
\(134\) 0 0
\(135\) 1.61718 + 1.54426i 0.139185 + 0.132909i
\(136\) 0 0
\(137\) 18.1842i 1.55358i −0.629759 0.776791i \(-0.716846\pi\)
0.629759 0.776791i \(-0.283154\pi\)
\(138\) 0 0
\(139\) −1.86064 −0.157817 −0.0789086 0.996882i \(-0.525144\pi\)
−0.0789086 + 0.996882i \(0.525144\pi\)
\(140\) 0 0
\(141\) −4.43129 −0.373182
\(142\) 0 0
\(143\) 12.9545i 1.08331i
\(144\) 0 0
\(145\) 6.96789 7.29692i 0.578652 0.605977i
\(146\) 0 0
\(147\) 8.95749i 0.738801i
\(148\) 0 0
\(149\) −6.33848 −0.519268 −0.259634 0.965707i \(-0.583602\pi\)
−0.259634 + 0.965707i \(0.583602\pi\)
\(150\) 0 0
\(151\) 7.44809 0.606117 0.303058 0.952972i \(-0.401992\pi\)
0.303058 + 0.952972i \(0.401992\pi\)
\(152\) 0 0
\(153\) 3.27777i 0.264992i
\(154\) 0 0
\(155\) −3.68916 + 3.86337i −0.296320 + 0.310313i
\(156\) 0 0
\(157\) 5.10147i 0.407142i 0.979060 + 0.203571i \(0.0652547\pi\)
−0.979060 + 0.203571i \(0.934745\pi\)
\(158\) 0 0
\(159\) −2.01735 −0.159986
\(160\) 0 0
\(161\) 3.99468 0.314825
\(162\) 0 0
\(163\) 4.41321i 0.345669i 0.984951 + 0.172835i \(0.0552927\pi\)
−0.984951 + 0.172835i \(0.944707\pi\)
\(164\) 0 0
\(165\) −3.73705 3.56853i −0.290929 0.277810i
\(166\) 0 0
\(167\) 10.3349i 0.799739i −0.916572 0.399869i \(-0.869055\pi\)
0.916572 0.399869i \(-0.130945\pi\)
\(168\) 0 0
\(169\) −18.4268 −1.41745
\(170\) 0 0
\(171\) 1.49731 0.114502
\(172\) 0 0
\(173\) 9.56659i 0.727335i −0.931529 0.363667i \(-0.881525\pi\)
0.931529 0.363667i \(-0.118475\pi\)
\(174\) 0 0
\(175\) −19.9522 + 0.920929i −1.50824 + 0.0696157i
\(176\) 0 0
\(177\) 13.1502i 0.988432i
\(178\) 0 0
\(179\) 18.9657 1.41757 0.708783 0.705427i \(-0.249245\pi\)
0.708783 + 0.705427i \(0.249245\pi\)
\(180\) 0 0
\(181\) 24.4964 1.82080 0.910401 0.413727i \(-0.135773\pi\)
0.910401 + 0.413727i \(0.135773\pi\)
\(182\) 0 0
\(183\) 11.1266i 0.822503i
\(184\) 0 0
\(185\) 10.7742 + 10.2884i 0.792138 + 0.756419i
\(186\) 0 0
\(187\) 7.57441i 0.553896i
\(188\) 0 0
\(189\) 3.99468 0.290570
\(190\) 0 0
\(191\) 15.3403 1.10998 0.554992 0.831856i \(-0.312721\pi\)
0.554992 + 0.831856i \(0.312721\pi\)
\(192\) 0 0
\(193\) 11.4740i 0.825915i 0.910750 + 0.412958i \(0.135504\pi\)
−0.910750 + 0.412958i \(0.864496\pi\)
\(194\) 0 0
\(195\) −8.65705 + 9.06585i −0.619944 + 0.649219i
\(196\) 0 0
\(197\) 18.8530i 1.34322i 0.740904 + 0.671611i \(0.234397\pi\)
−0.740904 + 0.671611i \(0.765603\pi\)
\(198\) 0 0
\(199\) −20.3152 −1.44011 −0.720053 0.693919i \(-0.755883\pi\)
−0.720053 + 0.693919i \(0.755883\pi\)
\(200\) 0 0
\(201\) −9.91149 −0.699102
\(202\) 0 0
\(203\) 18.0245i 1.26507i
\(204\) 0 0
\(205\) 1.65958 1.73795i 0.115910 0.121384i
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) −3.46005 −0.239337
\(210\) 0 0
\(211\) −0.0136020 −0.000936399 −0.000468199 1.00000i \(-0.500149\pi\)
−0.000468199 1.00000i \(0.500149\pi\)
\(212\) 0 0
\(213\) 4.94360i 0.338730i
\(214\) 0 0
\(215\) 20.2567 + 19.3433i 1.38150 + 1.31920i
\(216\) 0 0
\(217\) 9.54311i 0.647828i
\(218\) 0 0
\(219\) 4.38340 0.296203
\(220\) 0 0
\(221\) 18.3751 1.23604
\(222\) 0 0
\(223\) 8.61622i 0.576985i 0.957482 + 0.288493i \(0.0931540\pi\)
−0.957482 + 0.288493i \(0.906846\pi\)
\(224\) 0 0
\(225\) 0.230539 + 4.99468i 0.0153692 + 0.332979i
\(226\) 0 0
\(227\) 16.2173i 1.07638i 0.842824 + 0.538190i \(0.180892\pi\)
−0.842824 + 0.538190i \(0.819108\pi\)
\(228\) 0 0
\(229\) −11.2202 −0.741450 −0.370725 0.928743i \(-0.620891\pi\)
−0.370725 + 0.928743i \(0.620891\pi\)
\(230\) 0 0
\(231\) −9.23108 −0.607360
\(232\) 0 0
\(233\) 1.24723i 0.0817090i −0.999165 0.0408545i \(-0.986992\pi\)
0.999165 0.0408545i \(-0.0130080\pi\)
\(234\) 0 0
\(235\) −7.16619 6.84305i −0.467471 0.446391i
\(236\) 0 0
\(237\) 9.35481i 0.607660i
\(238\) 0 0
\(239\) −13.8930 −0.898663 −0.449332 0.893365i \(-0.648338\pi\)
−0.449332 + 0.893365i \(0.648338\pi\)
\(240\) 0 0
\(241\) 16.1962 1.04329 0.521645 0.853163i \(-0.325319\pi\)
0.521645 + 0.853163i \(0.325319\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −13.8327 + 14.4859i −0.883737 + 0.925468i
\(246\) 0 0
\(247\) 8.39388i 0.534090i
\(248\) 0 0
\(249\) −3.95659 −0.250739
\(250\) 0 0
\(251\) −0.451519 −0.0284996 −0.0142498 0.999898i \(-0.504536\pi\)
−0.0142498 + 0.999898i \(0.504536\pi\)
\(252\) 0 0
\(253\) 2.31084i 0.145281i
\(254\) 0 0
\(255\) 5.06172 5.30074i 0.316977 0.331945i
\(256\) 0 0
\(257\) 3.94554i 0.246116i 0.992399 + 0.123058i \(0.0392701\pi\)
−0.992399 + 0.123058i \(0.960730\pi\)
\(258\) 0 0
\(259\) 26.6140 1.65372
\(260\) 0 0
\(261\) −4.51213 −0.279294
\(262\) 0 0
\(263\) 21.7884i 1.34353i 0.740765 + 0.671764i \(0.234463\pi\)
−0.740765 + 0.671764i \(0.765537\pi\)
\(264\) 0 0
\(265\) −3.26242 3.11531i −0.200409 0.191372i
\(266\) 0 0
\(267\) 11.6996i 0.716002i
\(268\) 0 0
\(269\) 29.3071 1.78688 0.893442 0.449179i \(-0.148283\pi\)
0.893442 + 0.449179i \(0.148283\pi\)
\(270\) 0 0
\(271\) 15.6009 0.947685 0.473842 0.880610i \(-0.342867\pi\)
0.473842 + 0.880610i \(0.342867\pi\)
\(272\) 0 0
\(273\) 22.3940i 1.35535i
\(274\) 0 0
\(275\) −0.532739 11.5419i −0.0321253 0.696004i
\(276\) 0 0
\(277\) 1.76382i 0.105977i 0.998595 + 0.0529887i \(0.0168747\pi\)
−0.998595 + 0.0529887i \(0.983125\pi\)
\(278\) 0 0
\(279\) 2.38895 0.143023
\(280\) 0 0
\(281\) 28.6750 1.71061 0.855305 0.518125i \(-0.173370\pi\)
0.855305 + 0.518125i \(0.173370\pi\)
\(282\) 0 0
\(283\) 23.6822i 1.40776i −0.710319 0.703880i \(-0.751449\pi\)
0.710319 0.703880i \(-0.248551\pi\)
\(284\) 0 0
\(285\) 2.42142 + 2.31224i 0.143433 + 0.136965i
\(286\) 0 0
\(287\) 4.29300i 0.253408i
\(288\) 0 0
\(289\) 6.25621 0.368013
\(290\) 0 0
\(291\) 16.4888 0.966589
\(292\) 0 0
\(293\) 32.7897i 1.91559i −0.287444 0.957797i \(-0.592805\pi\)
0.287444 0.957797i \(-0.407195\pi\)
\(294\) 0 0
\(295\) −20.3073 + 21.2663i −1.18234 + 1.23817i
\(296\) 0 0
\(297\) 2.31084i 0.134089i
\(298\) 0 0
\(299\) 5.60596 0.324201
\(300\) 0 0
\(301\) 50.0373 2.88410
\(302\) 0 0
\(303\) 10.3994i 0.597431i
\(304\) 0 0
\(305\) 17.1823 17.9937i 0.983858 1.03032i
\(306\) 0 0
\(307\) 2.50214i 0.142804i 0.997448 + 0.0714022i \(0.0227474\pi\)
−0.997448 + 0.0714022i \(0.977253\pi\)
\(308\) 0 0
\(309\) −12.4044 −0.705664
\(310\) 0 0
\(311\) 33.4641 1.89757 0.948787 0.315918i \(-0.102312\pi\)
0.948787 + 0.315918i \(0.102312\pi\)
\(312\) 0 0
\(313\) 24.5389i 1.38702i 0.720446 + 0.693511i \(0.243937\pi\)
−0.720446 + 0.693511i \(0.756063\pi\)
\(314\) 0 0
\(315\) 6.46012 + 6.16882i 0.363986 + 0.347573i
\(316\) 0 0
\(317\) 24.5232i 1.37736i 0.725066 + 0.688680i \(0.241809\pi\)
−0.725066 + 0.688680i \(0.758191\pi\)
\(318\) 0 0
\(319\) 10.4268 0.583790
\(320\) 0 0
\(321\) 1.51748 0.0846975
\(322\) 0 0
\(323\) 4.90785i 0.273080i
\(324\) 0 0
\(325\) −28.0000 + 1.29239i −1.55316 + 0.0716890i
\(326\) 0 0
\(327\) 4.86276i 0.268911i
\(328\) 0 0
\(329\) −17.7016 −0.975920
\(330\) 0 0
\(331\) −3.00518 −0.165179 −0.0825897 0.996584i \(-0.526319\pi\)
−0.0825897 + 0.996584i \(0.526319\pi\)
\(332\) 0 0
\(333\) 6.66237i 0.365095i
\(334\) 0 0
\(335\) −16.0287 15.3059i −0.875739 0.836250i
\(336\) 0 0
\(337\) 1.83422i 0.0999162i −0.998751 0.0499581i \(-0.984091\pi\)
0.998751 0.0499581i \(-0.0159088\pi\)
\(338\) 0 0
\(339\) −1.72566 −0.0937250
\(340\) 0 0
\(341\) −5.52049 −0.298951
\(342\) 0 0
\(343\) 7.81954i 0.422215i
\(344\) 0 0
\(345\) 1.54426 1.61718i 0.0831400 0.0870660i
\(346\) 0 0
\(347\) 19.7001i 1.05756i 0.848759 + 0.528779i \(0.177350\pi\)
−0.848759 + 0.528779i \(0.822650\pi\)
\(348\) 0 0
\(349\) −19.1625 −1.02575 −0.512874 0.858464i \(-0.671419\pi\)
−0.512874 + 0.858464i \(0.671419\pi\)
\(350\) 0 0
\(351\) 5.60596 0.299224
\(352\) 0 0
\(353\) 21.9329i 1.16737i −0.811979 0.583686i \(-0.801610\pi\)
0.811979 0.583686i \(-0.198390\pi\)
\(354\) 0 0
\(355\) 7.63418 7.99468i 0.405180 0.424314i
\(356\) 0 0
\(357\) 13.0937i 0.692990i
\(358\) 0 0
\(359\) 12.0257 0.634693 0.317347 0.948310i \(-0.397208\pi\)
0.317347 + 0.948310i \(0.397208\pi\)
\(360\) 0 0
\(361\) −16.7581 −0.882003
\(362\) 0 0
\(363\) 5.66001i 0.297074i
\(364\) 0 0
\(365\) 7.08875 + 6.76910i 0.371042 + 0.354311i
\(366\) 0 0
\(367\) 3.39198i 0.177060i 0.996074 + 0.0885298i \(0.0282169\pi\)
−0.996074 + 0.0885298i \(0.971783\pi\)
\(368\) 0 0
\(369\) −1.07468 −0.0559456
\(370\) 0 0
\(371\) −8.05867 −0.418385
\(372\) 0 0
\(373\) 14.4907i 0.750299i 0.926964 + 0.375150i \(0.122409\pi\)
−0.926964 + 0.375150i \(0.877591\pi\)
\(374\) 0 0
\(375\) −7.34025 + 8.43331i −0.379049 + 0.435494i
\(376\) 0 0
\(377\) 25.2948i 1.30275i
\(378\) 0 0
\(379\) 10.2473 0.526366 0.263183 0.964746i \(-0.415228\pi\)
0.263183 + 0.964746i \(0.415228\pi\)
\(380\) 0 0
\(381\) −14.1962 −0.727294
\(382\) 0 0
\(383\) 21.0326i 1.07471i 0.843355 + 0.537357i \(0.180577\pi\)
−0.843355 + 0.537357i \(0.819423\pi\)
\(384\) 0 0
\(385\) −14.9283 14.2552i −0.760817 0.726510i
\(386\) 0 0
\(387\) 12.5260i 0.636731i
\(388\) 0 0
\(389\) −19.4421 −0.985754 −0.492877 0.870099i \(-0.664055\pi\)
−0.492877 + 0.870099i \(0.664055\pi\)
\(390\) 0 0
\(391\) −3.27777 −0.165764
\(392\) 0 0
\(393\) 5.34301i 0.269519i
\(394\) 0 0
\(395\) −14.4462 + 15.1284i −0.726869 + 0.761192i
\(396\) 0 0
\(397\) 4.53486i 0.227598i 0.993504 + 0.113799i \(0.0363020\pi\)
−0.993504 + 0.113799i \(0.963698\pi\)
\(398\) 0 0
\(399\) 5.98129 0.299439
\(400\) 0 0
\(401\) 26.8076 1.33871 0.669353 0.742944i \(-0.266571\pi\)
0.669353 + 0.742944i \(0.266571\pi\)
\(402\) 0 0
\(403\) 13.3924i 0.667122i
\(404\) 0 0
\(405\) 1.54426 1.61718i 0.0767348 0.0803583i
\(406\) 0 0
\(407\) 15.3957i 0.763135i
\(408\) 0 0
\(409\) 25.5992 1.26580 0.632899 0.774234i \(-0.281865\pi\)
0.632899 + 0.774234i \(0.281865\pi\)
\(410\) 0 0
\(411\) −18.1842 −0.896961
\(412\) 0 0
\(413\) 52.5310i 2.58488i
\(414\) 0 0
\(415\) −6.39851 6.10999i −0.314091 0.299928i
\(416\) 0 0
\(417\) 1.86064i 0.0911158i
\(418\) 0 0
\(419\) −36.9334 −1.80431 −0.902157 0.431407i \(-0.858017\pi\)
−0.902157 + 0.431407i \(0.858017\pi\)
\(420\) 0 0
\(421\) 18.4129 0.897391 0.448696 0.893685i \(-0.351889\pi\)
0.448696 + 0.893685i \(0.351889\pi\)
\(422\) 0 0
\(423\) 4.43129i 0.215457i
\(424\) 0 0
\(425\) 16.3714 0.755653i 0.794131 0.0366546i
\(426\) 0 0
\(427\) 44.4473i 2.15095i
\(428\) 0 0
\(429\) −12.9545 −0.625449
\(430\) 0 0
\(431\) −32.2149 −1.55174 −0.775869 0.630894i \(-0.782688\pi\)
−0.775869 + 0.630894i \(0.782688\pi\)
\(432\) 0 0
\(433\) 38.7313i 1.86131i 0.365900 + 0.930654i \(0.380761\pi\)
−0.365900 + 0.930654i \(0.619239\pi\)
\(434\) 0 0
\(435\) −7.29692 6.96789i −0.349861 0.334085i
\(436\) 0 0
\(437\) 1.49731i 0.0716262i
\(438\) 0 0
\(439\) −12.4945 −0.596329 −0.298165 0.954514i \(-0.596374\pi\)
−0.298165 + 0.954514i \(0.596374\pi\)
\(440\) 0 0
\(441\) 8.95749 0.426547
\(442\) 0 0
\(443\) 24.8438i 1.18036i −0.807270 0.590182i \(-0.799056\pi\)
0.807270 0.590182i \(-0.200944\pi\)
\(444\) 0 0
\(445\) 18.0671 18.9203i 0.856464 0.896908i
\(446\) 0 0
\(447\) 6.33848i 0.299800i
\(448\) 0 0
\(449\) 14.3084 0.675254 0.337627 0.941280i \(-0.390376\pi\)
0.337627 + 0.941280i \(0.390376\pi\)
\(450\) 0 0
\(451\) 2.48341 0.116939
\(452\) 0 0
\(453\) 7.44809i 0.349942i
\(454\) 0 0
\(455\) −34.5822 + 36.2152i −1.62124 + 1.69779i
\(456\) 0 0
\(457\) 30.0544i 1.40589i −0.711247 0.702943i \(-0.751869\pi\)
0.711247 0.702943i \(-0.248131\pi\)
\(458\) 0 0
\(459\) −3.27777 −0.152993
\(460\) 0 0
\(461\) −4.27280 −0.199004 −0.0995021 0.995037i \(-0.531725\pi\)
−0.0995021 + 0.995037i \(0.531725\pi\)
\(462\) 0 0
\(463\) 36.5765i 1.69986i 0.526899 + 0.849928i \(0.323354\pi\)
−0.526899 + 0.849928i \(0.676646\pi\)
\(464\) 0 0
\(465\) 3.86337 + 3.68916i 0.179159 + 0.171081i
\(466\) 0 0
\(467\) 34.4252i 1.59301i −0.604634 0.796503i \(-0.706681\pi\)
0.604634 0.796503i \(-0.293319\pi\)
\(468\) 0 0
\(469\) −39.5932 −1.82825
\(470\) 0 0
\(471\) 5.10147 0.235063
\(472\) 0 0
\(473\) 28.9455i 1.33092i
\(474\) 0 0
\(475\) 0.345189 + 7.47860i 0.0158383 + 0.343142i
\(476\) 0 0
\(477\) 2.01735i 0.0923681i
\(478\) 0 0
\(479\) 9.31790 0.425746 0.212873 0.977080i \(-0.431718\pi\)
0.212873 + 0.977080i \(0.431718\pi\)
\(480\) 0 0
\(481\) 37.3490 1.70297
\(482\) 0 0
\(483\) 3.99468i 0.181764i
\(484\) 0 0
\(485\) 26.6653 + 25.4629i 1.21081 + 1.15621i
\(486\) 0 0
\(487\) 5.49269i 0.248897i −0.992226 0.124449i \(-0.960284\pi\)
0.992226 0.124449i \(-0.0397162\pi\)
\(488\) 0 0
\(489\) 4.41321 0.199572
\(490\) 0 0
\(491\) 2.97963 0.134469 0.0672344 0.997737i \(-0.478582\pi\)
0.0672344 + 0.997737i \(0.478582\pi\)
\(492\) 0 0
\(493\) 14.7897i 0.666096i
\(494\) 0 0
\(495\) −3.56853 + 3.73705i −0.160394 + 0.167968i
\(496\) 0 0
\(497\) 19.7481i 0.885823i
\(498\) 0 0
\(499\) −2.38895 −0.106944 −0.0534721 0.998569i \(-0.517029\pi\)
−0.0534721 + 0.998569i \(0.517029\pi\)
\(500\) 0 0
\(501\) −10.3349 −0.461730
\(502\) 0 0
\(503\) 25.4790i 1.13605i −0.823010 0.568027i \(-0.807707\pi\)
0.823010 0.568027i \(-0.192293\pi\)
\(504\) 0 0
\(505\) 16.0594 16.8177i 0.714632 0.748378i
\(506\) 0 0
\(507\) 18.4268i 0.818364i
\(508\) 0 0
\(509\) 11.3695 0.503942 0.251971 0.967735i \(-0.418921\pi\)
0.251971 + 0.967735i \(0.418921\pi\)
\(510\) 0 0
\(511\) 17.5103 0.774610
\(512\) 0 0
\(513\) 1.49731i 0.0661080i
\(514\) 0 0
\(515\) −20.0602 19.1556i −0.883958 0.844099i
\(516\) 0 0
\(517\) 10.2400i 0.450355i
\(518\) 0 0
\(519\) −9.56659 −0.419927
\(520\) 0 0
\(521\) 17.6336 0.772543 0.386271 0.922385i \(-0.373763\pi\)
0.386271 + 0.922385i \(0.373763\pi\)
\(522\) 0 0
\(523\) 3.47209i 0.151824i 0.997115 + 0.0759120i \(0.0241868\pi\)
−0.997115 + 0.0759120i \(0.975813\pi\)
\(524\) 0 0
\(525\) 0.920929 + 19.9522i 0.0401926 + 0.870784i
\(526\) 0 0
\(527\) 7.83044i 0.341099i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 13.1502 0.570672
\(532\) 0 0
\(533\) 6.02461i 0.260955i
\(534\) 0 0
\(535\) 2.45404 + 2.34338i 0.106097 + 0.101313i
\(536\) 0 0
\(537\) 18.9657i 0.818432i
\(538\) 0 0
\(539\) −20.6993 −0.891583
\(540\) 0 0
\(541\) −43.0785 −1.85209 −0.926046 0.377411i \(-0.876814\pi\)
−0.926046 + 0.377411i \(0.876814\pi\)
\(542\) 0 0
\(543\) 24.4964i 1.05124i
\(544\) 0 0
\(545\) −7.50935 + 7.86395i −0.321665 + 0.336855i
\(546\) 0 0
\(547\) 3.12643i 0.133677i 0.997764 + 0.0668383i \(0.0212912\pi\)
−0.997764 + 0.0668383i \(0.978709\pi\)
\(548\) 0 0
\(549\) −11.1266 −0.474872
\(550\) 0 0
\(551\) −6.75607 −0.287818
\(552\) 0 0
\(553\) 37.3695i 1.58911i
\(554\) 0 0
\(555\) 10.2884 10.7742i 0.436719 0.457341i
\(556\) 0 0
\(557\) 32.9391i 1.39567i −0.716257 0.697837i \(-0.754146\pi\)
0.716257 0.697837i \(-0.245854\pi\)
\(558\) 0 0
\(559\) 70.2201 2.96999
\(560\) 0 0
\(561\) 7.57441 0.319792
\(562\) 0 0
\(563\) 24.4548i 1.03065i −0.856996 0.515324i \(-0.827672\pi\)
0.856996 0.515324i \(-0.172328\pi\)
\(564\) 0 0
\(565\) −2.79070 2.66486i −0.117406 0.112112i
\(566\) 0 0
\(567\) 3.99468i 0.167761i
\(568\) 0 0
\(569\) 5.16894 0.216693 0.108347 0.994113i \(-0.465444\pi\)
0.108347 + 0.994113i \(0.465444\pi\)
\(570\) 0 0
\(571\) −17.0392 −0.713070 −0.356535 0.934282i \(-0.616042\pi\)
−0.356535 + 0.934282i \(0.616042\pi\)
\(572\) 0 0
\(573\) 15.3403i 0.640849i
\(574\) 0 0
\(575\) 4.99468 0.230539i 0.208293 0.00961413i
\(576\) 0 0
\(577\) 38.2584i 1.59272i −0.604824 0.796359i \(-0.706757\pi\)
0.604824 0.796359i \(-0.293243\pi\)
\(578\) 0 0
\(579\) 11.4740 0.476842
\(580\) 0 0
\(581\) −15.8053 −0.655715
\(582\) 0 0
\(583\) 4.66178i 0.193071i
\(584\) 0 0
\(585\) 9.06585 + 8.65705i 0.374827 + 0.357925i
\(586\) 0 0
\(587\) 25.6094i 1.05701i 0.848929 + 0.528506i \(0.177248\pi\)
−0.848929 + 0.528506i \(0.822752\pi\)
\(588\) 0 0
\(589\) 3.57701 0.147388
\(590\) 0 0
\(591\) 18.8530 0.775510
\(592\) 0 0
\(593\) 5.83920i 0.239787i 0.992787 + 0.119894i \(0.0382553\pi\)
−0.992787 + 0.119894i \(0.961745\pi\)
\(594\) 0 0
\(595\) 20.2200 21.1748i 0.828938 0.868082i
\(596\) 0 0
\(597\) 20.3152i 0.831446i
\(598\) 0 0
\(599\) −43.4013 −1.77333 −0.886665 0.462413i \(-0.846984\pi\)
−0.886665 + 0.462413i \(0.846984\pi\)
\(600\) 0 0
\(601\) −35.3983 −1.44393 −0.721964 0.691931i \(-0.756760\pi\)
−0.721964 + 0.691931i \(0.756760\pi\)
\(602\) 0 0
\(603\) 9.91149i 0.403627i
\(604\) 0 0
\(605\) −8.74051 + 9.15325i −0.355352 + 0.372133i
\(606\) 0 0
\(607\) 35.9548i 1.45936i −0.683788 0.729681i \(-0.739669\pi\)
0.683788 0.729681i \(-0.260331\pi\)
\(608\) 0 0
\(609\) −18.0245 −0.730391
\(610\) 0 0
\(611\) −24.8416 −1.00499
\(612\) 0 0
\(613\) 6.70382i 0.270765i −0.990793 0.135382i \(-0.956774\pi\)
0.990793 0.135382i \(-0.0432263\pi\)
\(614\) 0 0
\(615\) −1.73795 1.65958i −0.0700809 0.0669208i
\(616\) 0 0
\(617\) 12.0722i 0.486010i 0.970025 + 0.243005i \(0.0781332\pi\)
−0.970025 + 0.243005i \(0.921867\pi\)
\(618\) 0 0
\(619\) 13.3418 0.536254 0.268127 0.963384i \(-0.413595\pi\)
0.268127 + 0.963384i \(0.413595\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 46.7360i 1.87244i
\(624\) 0 0
\(625\) −24.8937 + 2.30294i −0.995748 + 0.0921174i
\(626\) 0 0
\(627\) 3.46005i 0.138181i
\(628\) 0 0
\(629\) −21.8377 −0.870727
\(630\) 0 0
\(631\) 30.3182 1.20695 0.603474 0.797383i \(-0.293783\pi\)
0.603474 + 0.797383i \(0.293783\pi\)
\(632\) 0 0
\(633\) 0.0136020i 0.000540630i
\(634\) 0 0
\(635\) −22.9578 21.9226i −0.911053 0.869972i
\(636\) 0 0
\(637\) 50.2153i 1.98960i
\(638\) 0 0
\(639\) −4.94360 −0.195566
\(640\) 0 0
\(641\) −29.3840 −1.16060 −0.580299 0.814403i \(-0.697064\pi\)
−0.580299 + 0.814403i \(0.697064\pi\)
\(642\) 0 0
\(643\) 33.3568i 1.31546i −0.753252 0.657732i \(-0.771516\pi\)
0.753252 0.657732i \(-0.228484\pi\)
\(644\) 0 0
\(645\) 19.3433 20.2567i 0.761642 0.797608i
\(646\) 0 0
\(647\) 29.5633i 1.16225i 0.813814 + 0.581126i \(0.197388\pi\)
−0.813814 + 0.581126i \(0.802612\pi\)
\(648\) 0 0
\(649\) −30.3881 −1.19284
\(650\) 0 0
\(651\) 9.54311 0.374024
\(652\) 0 0
\(653\) 26.4562i 1.03531i 0.855589 + 0.517655i \(0.173195\pi\)
−0.855589 + 0.517655i \(0.826805\pi\)
\(654\) 0 0
\(655\) −8.25098 + 8.64060i −0.322392 + 0.337616i
\(656\) 0 0
\(657\) 4.38340i 0.171013i
\(658\) 0 0
\(659\) −31.9043 −1.24282 −0.621408 0.783487i \(-0.713439\pi\)
−0.621408 + 0.783487i \(0.713439\pi\)
\(660\) 0 0
\(661\) 11.3266 0.440553 0.220277 0.975437i \(-0.429304\pi\)
0.220277 + 0.975437i \(0.429304\pi\)
\(662\) 0 0
\(663\) 18.3751i 0.713628i
\(664\) 0 0
\(665\) 9.67282 + 9.23665i 0.375096 + 0.358182i
\(666\) 0 0
\(667\) 4.51213i 0.174710i
\(668\) 0 0
\(669\) 8.61622 0.333122
\(670\) 0 0
\(671\) 25.7118 0.992594
\(672\) 0 0
\(673\) 15.1178i 0.582749i 0.956609 + 0.291375i \(0.0941126\pi\)
−0.956609 + 0.291375i \(0.905887\pi\)
\(674\) 0 0
\(675\) 4.99468 0.230539i 0.192245 0.00887344i
\(676\) 0 0
\(677\) 9.77610i 0.375726i −0.982195 0.187863i \(-0.939844\pi\)
0.982195 0.187863i \(-0.0601561\pi\)
\(678\) 0 0
\(679\) 65.8674 2.52776
\(680\) 0 0
\(681\) 16.2173 0.621448
\(682\) 0 0
\(683\) 2.52943i 0.0967861i 0.998828 + 0.0483931i \(0.0154100\pi\)
−0.998828 + 0.0483931i \(0.984590\pi\)
\(684\) 0 0
\(685\) −29.4071 28.0811i −1.12359 1.07292i
\(686\) 0 0
\(687\) 11.2202i 0.428076i
\(688\) 0 0
\(689\) −11.3092 −0.430846
\(690\) 0 0
\(691\) −5.95336 −0.226476 −0.113238 0.993568i \(-0.536122\pi\)
−0.113238 + 0.993568i \(0.536122\pi\)
\(692\) 0 0
\(693\) 9.23108i 0.350660i
\(694\) 0 0
\(695\) −2.87330 + 3.00898i −0.108991 + 0.114137i
\(696\) 0 0
\(697\) 3.52255i 0.133426i
\(698\) 0 0
\(699\) −1.24723 −0.0471747
\(700\) 0 0
\(701\) −12.0260 −0.454215 −0.227108 0.973870i \(-0.572927\pi\)
−0.227108 + 0.973870i \(0.572927\pi\)
\(702\) 0 0
\(703\) 9.97565i 0.376239i
\(704\) 0 0
\(705\) −6.84305 + 7.16619i −0.257724 + 0.269894i
\(706\) 0 0
\(707\) 41.5423i 1.56236i
\(708\) 0 0
\(709\) −7.49518 −0.281488 −0.140744 0.990046i \(-0.544949\pi\)
−0.140744 + 0.990046i \(0.544949\pi\)
\(710\) 0 0
\(711\) 9.35481 0.350833
\(712\) 0 0
\(713\) 2.38895i 0.0894670i
\(714\) 0 0
\(715\) −20.9497 20.0051i −0.783476 0.748147i
\(716\) 0 0
\(717\) 13.8930i 0.518844i
\(718\) 0 0
\(719\) 37.2523 1.38928 0.694639 0.719359i \(-0.255564\pi\)
0.694639 + 0.719359i \(0.255564\pi\)
\(720\) 0 0
\(721\) −49.5518 −1.84541
\(722\) 0 0
\(723\) 16.1962i 0.602343i
\(724\) 0 0
\(725\) −1.04022 22.5367i −0.0386328 0.836990i
\(726\) 0 0
\(727\) 31.5588i 1.17045i −0.810870 0.585226i \(-0.801006\pi\)
0.810870 0.585226i \(-0.198994\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −41.0573 −1.51856
\(732\) 0 0
\(733\) 3.54807i 0.131051i 0.997851 + 0.0655254i \(0.0208723\pi\)
−0.997851 + 0.0655254i \(0.979128\pi\)
\(734\) 0 0
\(735\) 14.4859 + 13.8327i 0.534319 + 0.510226i
\(736\) 0 0
\(737\) 22.9039i 0.843675i
\(738\) 0 0
\(739\) −4.75173 −0.174795 −0.0873976 0.996174i \(-0.527855\pi\)
−0.0873976 + 0.996174i \(0.527855\pi\)
\(740\) 0 0
\(741\) 8.39388 0.308357
\(742\) 0 0
\(743\) 31.7958i 1.16647i 0.812302 + 0.583237i \(0.198214\pi\)
−0.812302 + 0.583237i \(0.801786\pi\)
\(744\) 0 0
\(745\) −9.78824 + 10.2505i −0.358613 + 0.375548i
\(746\) 0 0
\(747\) 3.95659i 0.144764i
\(748\) 0 0
\(749\) 6.06186 0.221495
\(750\) 0 0
\(751\) −15.7363 −0.574225 −0.287113 0.957897i \(-0.592695\pi\)
−0.287113 + 0.957897i \(0.592695\pi\)
\(752\) 0 0
\(753\) 0.451519i 0.0164543i
\(754\) 0 0
\(755\) 11.5018 12.0449i 0.418592 0.438359i
\(756\) 0 0
\(757\) 26.2259i 0.953198i −0.879121 0.476599i \(-0.841869\pi\)
0.879121 0.476599i \(-0.158131\pi\)
\(758\) 0 0
\(759\) 2.31084 0.0838782
\(760\) 0 0
\(761\) 37.1436 1.34646 0.673228 0.739435i \(-0.264907\pi\)
0.673228 + 0.739435i \(0.264907\pi\)
\(762\) 0 0
\(763\) 19.4252i 0.703238i
\(764\) 0 0
\(765\) −5.30074 5.06172i −0.191649 0.183007i
\(766\) 0 0
\(767\) 73.7197i 2.66187i
\(768\) 0 0
\(769\) 21.4226 0.772519 0.386259 0.922390i \(-0.373767\pi\)
0.386259 + 0.922390i \(0.373767\pi\)
\(770\) 0 0
\(771\) 3.94554 0.142095
\(772\) 0 0
\(773\) 52.0313i 1.87143i −0.352751 0.935717i \(-0.614754\pi\)
0.352751 0.935717i \(-0.385246\pi\)
\(774\) 0 0
\(775\) 0.550746 + 11.9321i 0.0197834 + 0.428612i
\(776\) 0 0
\(777\) 26.6140i 0.954773i
\(778\) 0 0
\(779\) −1.60913 −0.0576531
\(780\) 0 0
\(781\) 11.4239 0.408778
\(782\) 0 0
\(783\) 4.51213i 0.161250i
\(784\) 0 0
\(785\) 8.25000 + 7.87799i 0.294455 + 0.281177i
\(786\) 0 0
\(787\) 29.3028i 1.04453i −0.852782 0.522266i \(-0.825087\pi\)
0.852782 0.522266i \(-0.174913\pi\)
\(788\) 0 0
\(789\) 21.7884 0.775686
\(790\) 0 0
\(791\) −6.89347 −0.245103
\(792\) 0 0
\(793\) 62.3754i 2.21501i
\(794\) 0 0
\(795\) −3.11531 + 3.26242i −0.110489 + 0.115706i
\(796\) 0 0
\(797\) 25.3855i 0.899200i 0.893230 + 0.449600i \(0.148434\pi\)
−0.893230 + 0.449600i \(0.851566\pi\)
\(798\) 0 0
\(799\) 14.5248 0.513849
\(800\) 0 0
\(801\) −11.6996 −0.413384
\(802\) 0 0
\(803\) 10.1293i 0.357457i
\(804\) 0 0
\(805\) 6.16882 6.46012i 0.217422 0.227689i
\(806\) 0 0
\(807\) 29.3071i 1.03166i
\(808\) 0 0
\(809\) −43.0003 −1.51181 −0.755905 0.654681i \(-0.772803\pi\)
−0.755905 + 0.654681i \(0.772803\pi\)
\(810\) 0 0
\(811\) −16.0750 −0.564470 −0.282235 0.959345i \(-0.591076\pi\)
−0.282235 + 0.959345i \(0.591076\pi\)
\(812\) 0 0
\(813\) 15.6009i 0.547146i
\(814\) 0 0
\(815\) 7.13696 + 6.81514i 0.249997 + 0.238724i
\(816\) 0 0
\(817\) 18.7553i 0.656165i
\(818\) 0 0
\(819\) 22.3940 0.782511
\(820\) 0 0
\(821\) −19.9668 −0.696846 −0.348423 0.937337i \(-0.613283\pi\)
−0.348423 + 0.937337i \(0.613283\pi\)
\(822\) 0 0
\(823\) 22.9199i 0.798939i 0.916747 + 0.399469i \(0.130806\pi\)
−0.916747 + 0.399469i \(0.869194\pi\)
\(824\) 0 0
\(825\) −11.5419 + 0.532739i −0.401838 + 0.0185476i
\(826\) 0 0
\(827\) 20.2103i 0.702779i 0.936229 + 0.351390i \(0.114291\pi\)
−0.936229 + 0.351390i \(0.885709\pi\)
\(828\) 0 0
\(829\) −38.7839 −1.34702 −0.673510 0.739178i \(-0.735214\pi\)
−0.673510 + 0.739178i \(0.735214\pi\)
\(830\) 0 0
\(831\) 1.76382 0.0611861
\(832\) 0 0
\(833\) 29.3606i 1.01728i
\(834\) 0 0
\(835\) −16.7134 15.9598i −0.578391 0.552310i
\(836\) 0 0
\(837\) 2.38895i 0.0825743i
\(838\) 0 0
\(839\) 46.6947 1.61208 0.806040 0.591861i \(-0.201607\pi\)
0.806040 + 0.591861i \(0.201607\pi\)
\(840\) 0 0
\(841\) −8.64068 −0.297954
\(842\) 0 0
\(843\) 28.6750i 0.987621i
\(844\) 0 0
\(845\) −28.4557 + 29.7995i −0.978907 + 1.02513i
\(846\) 0 0
\(847\) 22.6099i 0.776887i
\(848\) 0 0
\(849\) −23.6822 −0.812771
\(850\) 0 0
\(851\) −6.66237 −0.228383
\(852\) 0 0
\(853\) 27.3238i 0.935549i −0.883848 0.467775i \(-0.845056\pi\)
0.883848 0.467775i \(-0.154944\pi\)
\(854\) 0 0
\(855\) 2.31224 2.42142i 0.0790768 0.0828110i
\(856\) 0 0
\(857\) 24.8111i 0.847530i −0.905772 0.423765i \(-0.860708\pi\)
0.905772 0.423765i \(-0.139292\pi\)
\(858\) 0 0
\(859\) −34.1139 −1.16395 −0.581975 0.813207i \(-0.697720\pi\)
−0.581975 + 0.813207i \(0.697720\pi\)
\(860\) 0 0
\(861\) −4.29300 −0.146305
\(862\) 0 0
\(863\) 25.8989i 0.881609i 0.897603 + 0.440805i \(0.145307\pi\)
−0.897603 + 0.440805i \(0.854693\pi\)
\(864\) 0 0
\(865\) −15.4709 14.7733i −0.526026 0.502307i
\(866\) 0 0
\(867\) 6.25621i 0.212472i
\(868\) 0 0
\(869\) −21.6175 −0.733323
\(870\) 0 0
\(871\) −55.5634 −1.88269
\(872\) 0 0
\(873\) 16.4888i 0.558060i
\(874\) 0 0
\(875\) −29.3220 + 33.6884i −0.991264 + 1.13888i
\(876\) 0 0
\(877\) 1.59319i 0.0537980i 0.999638 + 0.0268990i \(0.00856326\pi\)
−0.999638 + 0.0268990i \(0.991437\pi\)
\(878\) 0 0
\(879\) −32.7897 −1.10597
\(880\) 0 0
\(881\) 45.7424 1.54110 0.770551 0.637379i \(-0.219981\pi\)
0.770551 + 0.637379i \(0.219981\pi\)
\(882\) 0 0
\(883\) 38.8682i 1.30802i 0.756487 + 0.654009i \(0.226914\pi\)
−0.756487 + 0.654009i \(0.773086\pi\)
\(884\) 0 0
\(885\) 21.2663 + 20.3073i 0.714859 + 0.682624i
\(886\) 0 0
\(887\) 56.7934i 1.90694i 0.301495 + 0.953468i \(0.402514\pi\)
−0.301495 + 0.953468i \(0.597486\pi\)
\(888\) 0 0
\(889\) −56.7093 −1.90197
\(890\) 0 0
\(891\) 2.31084 0.0774161
\(892\) 0 0
\(893\) 6.63503i 0.222033i
\(894\) 0 0
\(895\) 29.2880 30.6710i 0.978989 1.02522i
\(896\) 0 0
\(897\) 5.60596i 0.187178i
\(898\) 0 0
\(899\) −10.7793 −0.359509
\(900\) 0 0
\(901\) 6.61241 0.220291
\(902\) 0 0
\(903\) 50.0373i 1.66514i
\(904\) 0 0
\(905\) 37.8287 39.6151i 1.25747 1.31685i
\(906\) 0 0
\(907\) 38.7967i 1.28822i 0.764931 + 0.644112i \(0.222773\pi\)
−0.764931 + 0.644112i \(0.777227\pi\)
\(908\) 0 0
\(909\) −10.3994 −0.344927
\(910\) 0 0
\(911\) 31.2655 1.03587 0.517936 0.855419i \(-0.326700\pi\)
0.517936 + 0.855419i \(0.326700\pi\)
\(912\) 0 0
\(913\) 9.14305i 0.302591i
\(914\) 0 0
\(915\) −17.9937 17.1823i −0.594854 0.568031i
\(916\) 0 0
\(917\) 21.3436i 0.704828i
\(918\) 0 0
\(919\) 29.1623 0.961976 0.480988 0.876727i \(-0.340278\pi\)
0.480988 + 0.876727i \(0.340278\pi\)
\(920\) 0 0
\(921\) 2.50214 0.0824482
\(922\) 0 0
\(923\) 27.7136i 0.912205i
\(924\) 0 0
\(925\) 33.2764 1.53593i 1.09412 0.0505012i
\(926\) 0 0
\(927\) 12.4044i 0.407415i
\(928\) 0 0
\(929\) 32.4054 1.06319 0.531593 0.847000i \(-0.321594\pi\)
0.531593 + 0.847000i \(0.321594\pi\)
\(930\) 0 0
\(931\) 13.4122 0.439566
\(932\) 0 0
\(933\) 33.4641i 1.09556i
\(934\) 0 0
\(935\) 12.2492 + 11.6968i 0.400591 + 0.382528i
\(936\) 0 0
\(937\) 35.1464i 1.14818i 0.818792 + 0.574091i \(0.194644\pi\)
−0.818792 + 0.574091i \(0.805356\pi\)
\(938\) 0 0
\(939\) 24.5389 0.800798
\(940\) 0 0
\(941\) 10.1574 0.331121 0.165561 0.986200i \(-0.447057\pi\)
0.165561 + 0.986200i \(0.447057\pi\)
\(942\) 0 0
\(943\) 1.07468i 0.0349964i
\(944\) 0 0
\(945\) 6.16882 6.46012i 0.200672 0.210148i
\(946\) 0 0
\(947\) 14.8674i 0.483126i −0.970385 0.241563i \(-0.922340\pi\)
0.970385 0.241563i \(-0.0776601\pi\)
\(948\) 0 0
\(949\) 24.5732 0.797679
\(950\) 0 0
\(951\) 24.5232 0.795219
\(952\) 0 0
\(953\) 3.40621i 0.110338i 0.998477 + 0.0551690i \(0.0175698\pi\)
−0.998477 + 0.0551690i \(0.982430\pi\)
\(954\) 0 0
\(955\) 23.6893 24.8080i 0.766569 0.802767i
\(956\) 0 0
\(957\) 10.4268i 0.337051i
\(958\) 0 0
\(959\) −72.6401 −2.34567
\(960\) 0 0
\(961\) −25.2929 −0.815900
\(962\) 0 0
\(963\) 1.51748i 0.0489001i
\(964\) 0 0
\(965\) 18.5555 + 17.7188i 0.597322 + 0.570388i
\(966\) 0 0
\(967\) 26.5021i 0.852251i −0.904664 0.426126i \(-0.859878\pi\)
0.904664 0.426126i \(-0.140122\pi\)
\(968\) 0 0
\(969\) −4.90785 −0.157663
\(970\) 0 0
\(971\) −15.8259 −0.507878 −0.253939 0.967220i \(-0.581726\pi\)
−0.253939 + 0.967220i \(0.581726\pi\)
\(972\) 0 0
\(973\) 7.43265i 0.238280i
\(974\) 0 0
\(975\) 1.29239 + 28.0000i 0.0413897 + 0.896718i
\(976\) 0 0
\(977\) 18.0343i 0.576968i −0.957485 0.288484i \(-0.906849\pi\)
0.957485 0.288484i \(-0.0931512\pi\)
\(978\) 0 0
\(979\) 27.0358 0.864069
\(980\) 0 0
\(981\) 4.86276 0.155256
\(982\) 0 0
\(983\) 42.4751i 1.35475i 0.735640 + 0.677373i \(0.236882\pi\)
−0.735640 + 0.677373i \(0.763118\pi\)
\(984\) 0 0
\(985\) 30.4887 + 29.1139i 0.971451 + 0.927646i
\(986\) 0 0
\(987\) 17.7016i 0.563448i
\(988\) 0 0
\(989\) −12.5260 −0.398303
\(990\) 0 0
\(991\) −32.8613 −1.04387 −0.521936 0.852985i \(-0.674790\pi\)
−0.521936 + 0.852985i \(0.674790\pi\)
\(992\) 0 0
\(993\) 3.00518i 0.0953664i
\(994\) 0 0
\(995\) −31.3719 + 32.8533i −0.994556 + 1.04152i
\(996\) 0 0
\(997\) 17.7690i 0.562750i −0.959598 0.281375i \(-0.909210\pi\)
0.959598 0.281375i \(-0.0907904\pi\)
\(998\) 0 0
\(999\) −6.66237 −0.210788
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 1380.2.f.b.829.6 14
3.2 odd 2 4140.2.f.c.829.4 14
5.2 odd 4 6900.2.a.bc.1.7 7
5.3 odd 4 6900.2.a.bd.1.1 7
5.4 even 2 inner 1380.2.f.b.829.13 yes 14
15.14 odd 2 4140.2.f.c.829.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.f.b.829.6 14 1.1 even 1 trivial
1380.2.f.b.829.13 yes 14 5.4 even 2 inner
4140.2.f.c.829.3 14 15.14 odd 2
4140.2.f.c.829.4 14 3.2 odd 2
6900.2.a.bc.1.7 7 5.2 odd 4
6900.2.a.bd.1.1 7 5.3 odd 4