Properties

Label 1216.2.s.b.31.2
Level $1216$
Weight $2$
Character 1216.31
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
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] = [1216,2,Mod(31,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.b.863.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(2.36603 - 1.36603i) q^{5} -1.26795i q^{7} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{3} +(2.36603 - 1.36603i) q^{5} -1.26795i q^{7} -4.46410 q^{11} +(-2.36603 + 4.09808i) q^{15} +(2.73205 + 4.73205i) q^{17} +(-0.500000 + 4.33013i) q^{19} +(1.09808 + 1.90192i) q^{21} +(3.63397 + 2.09808i) q^{23} +(1.23205 - 2.13397i) q^{25} -5.19615i q^{27} +(-1.09808 + 1.90192i) q^{29} +2.19615 q^{31} +(6.69615 - 3.86603i) q^{33} +(-1.73205 - 3.00000i) q^{35} +4.73205 q^{37} +(6.69615 - 3.86603i) q^{41} +(1.73205 + 3.00000i) q^{43} +(8.36603 + 4.83013i) q^{47} +5.39230 q^{49} +(-8.19615 - 4.73205i) q^{51} +(-4.73205 + 8.19615i) q^{53} +(-10.5622 + 6.09808i) q^{55} +(-3.00000 - 6.92820i) q^{57} +(0.696152 - 0.401924i) q^{59} +(-10.5622 - 6.09808i) q^{61} +(3.69615 + 2.13397i) q^{67} -7.26795 q^{69} +(1.26795 + 2.19615i) q^{71} +(2.50000 + 4.33013i) q^{73} +4.26795i q^{75} +5.66025i q^{77} +(8.19615 + 14.1962i) q^{79} +(4.50000 + 7.79423i) q^{81} +3.39230 q^{83} +(12.9282 + 7.46410i) q^{85} -3.80385i q^{87} +(-11.1962 - 6.46410i) q^{89} +(-3.29423 + 1.90192i) q^{93} +(4.73205 + 10.9282i) q^{95} +(-6.69615 + 3.86603i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 6 q^{5} - 4 q^{11} - 6 q^{15} + 4 q^{17} - 2 q^{19} - 6 q^{21} + 18 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 6 q^{33} + 12 q^{37} + 6 q^{41} + 30 q^{47} - 20 q^{49} - 12 q^{51} - 12 q^{53} - 18 q^{55} - 12 q^{57} - 18 q^{59} - 18 q^{61} - 6 q^{67} - 36 q^{69} + 12 q^{71} + 10 q^{73} + 12 q^{79} + 18 q^{81} - 28 q^{83} + 24 q^{85} - 24 q^{89} + 18 q^{93} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2.36603 1.36603i 1.05812 0.610905i 0.133207 0.991088i \(-0.457472\pi\)
0.924911 + 0.380183i \(0.124139\pi\)
\(6\) 0 0
\(7\) 1.26795i 0.479240i −0.970867 0.239620i \(-0.922977\pi\)
0.970867 0.239620i \(-0.0770228\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.46410 −1.34598 −0.672989 0.739653i \(-0.734990\pi\)
−0.672989 + 0.739653i \(0.734990\pi\)
\(12\) 0 0
\(13\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) −2.36603 + 4.09808i −0.610905 + 1.05812i
\(16\) 0 0
\(17\) 2.73205 + 4.73205i 0.662620 + 1.14769i 0.979925 + 0.199367i \(0.0638887\pi\)
−0.317305 + 0.948323i \(0.602778\pi\)
\(18\) 0 0
\(19\) −0.500000 + 4.33013i −0.114708 + 0.993399i
\(20\) 0 0
\(21\) 1.09808 + 1.90192i 0.239620 + 0.415034i
\(22\) 0 0
\(23\) 3.63397 + 2.09808i 0.757736 + 0.437479i 0.828482 0.560015i \(-0.189205\pi\)
−0.0707462 + 0.997494i \(0.522538\pi\)
\(24\) 0 0
\(25\) 1.23205 2.13397i 0.246410 0.426795i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −1.09808 + 1.90192i −0.203908 + 0.353178i −0.949784 0.312906i \(-0.898698\pi\)
0.745877 + 0.666084i \(0.232031\pi\)
\(30\) 0 0
\(31\) 2.19615 0.394441 0.197220 0.980359i \(-0.436809\pi\)
0.197220 + 0.980359i \(0.436809\pi\)
\(32\) 0 0
\(33\) 6.69615 3.86603i 1.16565 0.672989i
\(34\) 0 0
\(35\) −1.73205 3.00000i −0.292770 0.507093i
\(36\) 0 0
\(37\) 4.73205 0.777944 0.388972 0.921250i \(-0.372830\pi\)
0.388972 + 0.921250i \(0.372830\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.69615 3.86603i 1.04576 0.603772i 0.124303 0.992244i \(-0.460331\pi\)
0.921460 + 0.388473i \(0.126997\pi\)
\(42\) 0 0
\(43\) 1.73205 + 3.00000i 0.264135 + 0.457496i 0.967337 0.253495i \(-0.0815801\pi\)
−0.703201 + 0.710991i \(0.748247\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.36603 + 4.83013i 1.22031 + 0.704546i 0.964984 0.262309i \(-0.0844838\pi\)
0.255326 + 0.966855i \(0.417817\pi\)
\(48\) 0 0
\(49\) 5.39230 0.770329
\(50\) 0 0
\(51\) −8.19615 4.73205i −1.14769 0.662620i
\(52\) 0 0
\(53\) −4.73205 + 8.19615i −0.649997 + 1.12583i 0.333126 + 0.942882i \(0.391897\pi\)
−0.983123 + 0.182946i \(0.941437\pi\)
\(54\) 0 0
\(55\) −10.5622 + 6.09808i −1.42420 + 0.822264i
\(56\) 0 0
\(57\) −3.00000 6.92820i −0.397360 0.917663i
\(58\) 0 0
\(59\) 0.696152 0.401924i 0.0906313 0.0523260i −0.453999 0.891002i \(-0.650003\pi\)
0.544631 + 0.838676i \(0.316670\pi\)
\(60\) 0 0
\(61\) −10.5622 6.09808i −1.35235 0.780779i −0.363770 0.931489i \(-0.618511\pi\)
−0.988578 + 0.150710i \(0.951844\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.69615 + 2.13397i 0.451557 + 0.260706i 0.708487 0.705723i \(-0.249378\pi\)
−0.256931 + 0.966430i \(0.582711\pi\)
\(68\) 0 0
\(69\) −7.26795 −0.874958
\(70\) 0 0
\(71\) 1.26795 + 2.19615i 0.150478 + 0.260635i 0.931403 0.363989i \(-0.118585\pi\)
−0.780925 + 0.624624i \(0.785252\pi\)
\(72\) 0 0
\(73\) 2.50000 + 4.33013i 0.292603 + 0.506803i 0.974424 0.224716i \(-0.0721453\pi\)
−0.681822 + 0.731519i \(0.738812\pi\)
\(74\) 0 0
\(75\) 4.26795i 0.492820i
\(76\) 0 0
\(77\) 5.66025i 0.645046i
\(78\) 0 0
\(79\) 8.19615 + 14.1962i 0.922139 + 1.59719i 0.796099 + 0.605167i \(0.206894\pi\)
0.126041 + 0.992025i \(0.459773\pi\)
\(80\) 0 0
\(81\) 4.50000 + 7.79423i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 3.39230 0.372354 0.186177 0.982516i \(-0.440390\pi\)
0.186177 + 0.982516i \(0.440390\pi\)
\(84\) 0 0
\(85\) 12.9282 + 7.46410i 1.40226 + 0.809595i
\(86\) 0 0
\(87\) 3.80385i 0.407815i
\(88\) 0 0
\(89\) −11.1962 6.46410i −1.18679 0.685193i −0.229214 0.973376i \(-0.573616\pi\)
−0.957575 + 0.288183i \(0.906949\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.29423 + 1.90192i −0.341596 + 0.197220i
\(94\) 0 0
\(95\) 4.73205 + 10.9282i 0.485498 + 1.12121i
\(96\) 0 0
\(97\) −6.69615 + 3.86603i −0.679891 + 0.392535i −0.799814 0.600248i \(-0.795069\pi\)
0.119923 + 0.992783i \(0.461735\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0263 + 8.09808i 1.39567 + 0.805789i 0.993935 0.109969i \(-0.0350750\pi\)
0.401732 + 0.915757i \(0.368408\pi\)
\(102\) 0 0
\(103\) −9.46410 −0.932526 −0.466263 0.884646i \(-0.654400\pi\)
−0.466263 + 0.884646i \(0.654400\pi\)
\(104\) 0 0
\(105\) 5.19615 + 3.00000i 0.507093 + 0.292770i
\(106\) 0 0
\(107\) 4.39230i 0.424620i 0.977202 + 0.212310i \(0.0680987\pi\)
−0.977202 + 0.212310i \(0.931901\pi\)
\(108\) 0 0
\(109\) 1.26795 + 2.19615i 0.121448 + 0.210353i 0.920339 0.391122i \(-0.127913\pi\)
−0.798891 + 0.601476i \(0.794580\pi\)
\(110\) 0 0
\(111\) −7.09808 + 4.09808i −0.673720 + 0.388972i
\(112\) 0 0
\(113\) 15.5885i 1.46644i −0.679992 0.733219i \(-0.738017\pi\)
0.679992 0.733219i \(-0.261983\pi\)
\(114\) 0 0
\(115\) 11.4641 1.06903
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 3.46410i 0.550019 0.317554i
\(120\) 0 0
\(121\) 8.92820 0.811655
\(122\) 0 0
\(123\) −6.69615 + 11.5981i −0.603772 + 1.04576i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 1.26795 2.19615i 0.112512 0.194877i −0.804270 0.594264i \(-0.797444\pi\)
0.916783 + 0.399387i \(0.130777\pi\)
\(128\) 0 0
\(129\) −5.19615 3.00000i −0.457496 0.264135i
\(130\) 0 0
\(131\) 0.964102 + 1.66987i 0.0842339 + 0.145897i 0.905065 0.425274i \(-0.139822\pi\)
−0.820831 + 0.571172i \(0.806489\pi\)
\(132\) 0 0
\(133\) 5.49038 + 0.633975i 0.476076 + 0.0549726i
\(134\) 0 0
\(135\) −7.09808 12.2942i −0.610905 1.05812i
\(136\) 0 0
\(137\) −8.96410 + 15.5263i −0.765855 + 1.32650i 0.173939 + 0.984757i \(0.444351\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 2.69615 4.66987i 0.228685 0.396093i −0.728734 0.684797i \(-0.759891\pi\)
0.957419 + 0.288704i \(0.0932242\pi\)
\(140\) 0 0
\(141\) −16.7321 −1.40909
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000i 0.498273i
\(146\) 0 0
\(147\) −8.08846 + 4.66987i −0.667125 + 0.385165i
\(148\) 0 0
\(149\) 8.36603 4.83013i 0.685372 0.395699i −0.116504 0.993190i \(-0.537169\pi\)
0.801876 + 0.597491i \(0.203836\pi\)
\(150\) 0 0
\(151\) −16.7321 −1.36163 −0.680817 0.732453i \(-0.738375\pi\)
−0.680817 + 0.732453i \(0.738375\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.19615 3.00000i 0.417365 0.240966i
\(156\) 0 0
\(157\) 0.339746 0.196152i 0.0271147 0.0156547i −0.486381 0.873747i \(-0.661683\pi\)
0.513496 + 0.858092i \(0.328350\pi\)
\(158\) 0 0
\(159\) 16.3923i 1.29999i
\(160\) 0 0
\(161\) 2.66025 4.60770i 0.209657 0.363137i
\(162\) 0 0
\(163\) −9.39230 −0.735662 −0.367831 0.929893i \(-0.619900\pi\)
−0.367831 + 0.929893i \(0.619900\pi\)
\(164\) 0 0
\(165\) 10.5622 18.2942i 0.822264 1.42420i
\(166\) 0 0
\(167\) 2.53590 4.39230i 0.196234 0.339887i −0.751071 0.660222i \(-0.770462\pi\)
0.947304 + 0.320335i \(0.103796\pi\)
\(168\) 0 0
\(169\) 6.50000 + 11.2583i 0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.9282 22.3923i −0.982913 1.70246i −0.650866 0.759193i \(-0.725594\pi\)
−0.332047 0.943263i \(-0.607739\pi\)
\(174\) 0 0
\(175\) −2.70577 1.56218i −0.204537 0.118090i
\(176\) 0 0
\(177\) −0.696152 + 1.20577i −0.0523260 + 0.0906313i
\(178\) 0 0
\(179\) 5.19615i 0.388379i 0.980964 + 0.194189i \(0.0622076\pi\)
−0.980964 + 0.194189i \(0.937792\pi\)
\(180\) 0 0
\(181\) 2.36603 4.09808i 0.175865 0.304608i −0.764595 0.644511i \(-0.777061\pi\)
0.940460 + 0.339903i \(0.110394\pi\)
\(182\) 0 0
\(183\) 21.1244 1.56156
\(184\) 0 0
\(185\) 11.1962 6.46410i 0.823157 0.475250i
\(186\) 0 0
\(187\) −12.1962 21.1244i −0.891871 1.54477i
\(188\) 0 0
\(189\) −6.58846 −0.479240
\(190\) 0 0
\(191\) 17.8564i 1.29204i −0.763319 0.646022i \(-0.776431\pi\)
0.763319 0.646022i \(-0.223569\pi\)
\(192\) 0 0
\(193\) −21.5885 + 12.4641i −1.55397 + 0.897186i −0.556159 + 0.831076i \(0.687725\pi\)
−0.997812 + 0.0661096i \(0.978941\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.80385i 0.556001i 0.960581 + 0.278001i \(0.0896717\pi\)
−0.960581 + 0.278001i \(0.910328\pi\)
\(198\) 0 0
\(199\) 10.7321 + 6.19615i 0.760775 + 0.439234i 0.829574 0.558397i \(-0.188583\pi\)
−0.0687990 + 0.997631i \(0.521917\pi\)
\(200\) 0 0
\(201\) −7.39230 −0.521413
\(202\) 0 0
\(203\) 2.41154 + 1.39230i 0.169257 + 0.0977206i
\(204\) 0 0
\(205\) 10.5622 18.2942i 0.737694 1.27772i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.23205 19.3301i 0.154394 1.33709i
\(210\) 0 0
\(211\) 15.0000 8.66025i 1.03264 0.596196i 0.114902 0.993377i \(-0.463345\pi\)
0.917741 + 0.397180i \(0.130011\pi\)
\(212\) 0 0
\(213\) −3.80385 2.19615i −0.260635 0.150478i
\(214\) 0 0
\(215\) 8.19615 + 4.73205i 0.558973 + 0.322723i
\(216\) 0 0
\(217\) 2.78461i 0.189032i
\(218\) 0 0
\(219\) −7.50000 4.33013i −0.506803 0.292603i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.7583 22.0981i −0.854361 1.47980i −0.877237 0.480058i \(-0.840615\pi\)
0.0228756 0.999738i \(-0.492718\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.1244i 1.20296i −0.798889 0.601478i \(-0.794579\pi\)
0.798889 0.601478i \(-0.205421\pi\)
\(228\) 0 0
\(229\) 12.7846i 0.844831i 0.906402 + 0.422415i \(0.138818\pi\)
−0.906402 + 0.422415i \(0.861182\pi\)
\(230\) 0 0
\(231\) −4.90192 8.49038i −0.322523 0.558626i
\(232\) 0 0
\(233\) −3.42820 5.93782i −0.224589 0.389000i 0.731607 0.681727i \(-0.238771\pi\)
−0.956196 + 0.292727i \(0.905437\pi\)
\(234\) 0 0
\(235\) 26.3923 1.72164
\(236\) 0 0
\(237\) −24.5885 14.1962i −1.59719 0.922139i
\(238\) 0 0
\(239\) 5.85641i 0.378819i 0.981898 + 0.189410i \(0.0606574\pi\)
−0.981898 + 0.189410i \(0.939343\pi\)
\(240\) 0 0
\(241\) −22.2846 12.8660i −1.43548 0.828774i −0.437947 0.899001i \(-0.644294\pi\)
−0.997531 + 0.0702273i \(0.977628\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.7583 7.36603i 0.815100 0.470598i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.08846 + 2.93782i −0.322468 + 0.186177i
\(250\) 0 0
\(251\) −8.16025 + 14.1340i −0.515071 + 0.892129i 0.484776 + 0.874638i \(0.338901\pi\)
−0.999847 + 0.0174904i \(0.994432\pi\)
\(252\) 0 0
\(253\) −16.2224 9.36603i −1.01990 0.588837i
\(254\) 0 0
\(255\) −25.8564 −1.61919
\(256\) 0 0
\(257\) 11.3038 + 6.52628i 0.705115 + 0.407098i 0.809250 0.587465i \(-0.199874\pi\)
−0.104135 + 0.994563i \(0.533207\pi\)
\(258\) 0 0
\(259\) 6.00000i 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.8301 6.83013i 0.729477 0.421164i −0.0887539 0.996054i \(-0.528288\pi\)
0.818231 + 0.574890i \(0.194955\pi\)
\(264\) 0 0
\(265\) 25.8564i 1.58835i
\(266\) 0 0
\(267\) 22.3923 1.37039
\(268\) 0 0
\(269\) 10.7321 + 18.5885i 0.654345 + 1.13336i 0.982058 + 0.188581i \(0.0603888\pi\)
−0.327713 + 0.944777i \(0.606278\pi\)
\(270\) 0 0
\(271\) 24.7583 14.2942i 1.50396 0.868313i 0.503972 0.863720i \(-0.331871\pi\)
0.999989 0.00459256i \(-0.00146186\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.50000 + 9.52628i −0.331662 + 0.574456i
\(276\) 0 0
\(277\) 4.19615i 0.252122i −0.992022 0.126061i \(-0.959766\pi\)
0.992022 0.126061i \(-0.0402336\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.8923 8.59808i −0.888400 0.512918i −0.0149815 0.999888i \(-0.504769\pi\)
−0.873419 + 0.486970i \(0.838102\pi\)
\(282\) 0 0
\(283\) −9.23205 15.9904i −0.548788 0.950529i −0.998358 0.0572841i \(-0.981756\pi\)
0.449569 0.893245i \(-0.351577\pi\)
\(284\) 0 0
\(285\) −16.5622 12.2942i −0.981059 0.728247i
\(286\) 0 0
\(287\) −4.90192 8.49038i −0.289351 0.501171i
\(288\) 0 0
\(289\) −6.42820 + 11.1340i −0.378130 + 0.654940i
\(290\) 0 0
\(291\) 6.69615 11.5981i 0.392535 0.679891i
\(292\) 0 0
\(293\) −4.73205 −0.276449 −0.138225 0.990401i \(-0.544140\pi\)
−0.138225 + 0.990401i \(0.544140\pi\)
\(294\) 0 0
\(295\) 1.09808 1.90192i 0.0639325 0.110734i
\(296\) 0 0
\(297\) 23.1962i 1.34598i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.80385 2.19615i 0.219250 0.126584i
\(302\) 0 0
\(303\) −28.0526 −1.61158
\(304\) 0 0
\(305\) −33.3205 −1.90793
\(306\) 0 0
\(307\) 11.0885 6.40192i 0.632852 0.365377i −0.149004 0.988837i \(-0.547607\pi\)
0.781856 + 0.623460i \(0.214273\pi\)
\(308\) 0 0
\(309\) 14.1962 8.19615i 0.807591 0.466263i
\(310\) 0 0
\(311\) 18.7321i 1.06220i 0.847310 + 0.531099i \(0.178221\pi\)
−0.847310 + 0.531099i \(0.821779\pi\)
\(312\) 0 0
\(313\) 9.69615 16.7942i 0.548059 0.949266i −0.450349 0.892853i \(-0.648700\pi\)
0.998408 0.0564131i \(-0.0179664\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.1244 26.1962i 0.849468 1.47132i −0.0322149 0.999481i \(-0.510256\pi\)
0.881683 0.471842i \(-0.156411\pi\)
\(318\) 0 0
\(319\) 4.90192 8.49038i 0.274455 0.475370i
\(320\) 0 0
\(321\) −3.80385 6.58846i −0.212310 0.367732i
\(322\) 0 0
\(323\) −21.8564 + 9.46410i −1.21612 + 0.526597i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.80385 2.19615i −0.210353 0.121448i
\(328\) 0 0
\(329\) 6.12436 10.6077i 0.337647 0.584821i
\(330\) 0 0
\(331\) 31.7321i 1.74415i 0.489371 + 0.872076i \(0.337226\pi\)
−0.489371 + 0.872076i \(0.662774\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.6603 0.637068
\(336\) 0 0
\(337\) −5.89230 + 3.40192i −0.320974 + 0.185315i −0.651827 0.758368i \(-0.725997\pi\)
0.330852 + 0.943682i \(0.392664\pi\)
\(338\) 0 0
\(339\) 13.5000 + 23.3827i 0.733219 + 1.26997i
\(340\) 0 0
\(341\) −9.80385 −0.530908
\(342\) 0 0
\(343\) 15.7128i 0.848412i
\(344\) 0 0
\(345\) −17.1962 + 9.92820i −0.925810 + 0.534516i
\(346\) 0 0
\(347\) −1.23205 2.13397i −0.0661400 0.114558i 0.831059 0.556184i \(-0.187735\pi\)
−0.897199 + 0.441626i \(0.854402\pi\)
\(348\) 0 0
\(349\) 34.6410i 1.85429i 0.374701 + 0.927146i \(0.377745\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 25.7846 1.37238 0.686188 0.727424i \(-0.259283\pi\)
0.686188 + 0.727424i \(0.259283\pi\)
\(354\) 0 0
\(355\) 6.00000 + 3.46410i 0.318447 + 0.183855i
\(356\) 0 0
\(357\) −6.00000 + 10.3923i −0.317554 + 0.550019i
\(358\) 0 0
\(359\) 2.02628 1.16987i 0.106943 0.0617435i −0.445575 0.895245i \(-0.647001\pi\)
0.552517 + 0.833501i \(0.313667\pi\)
\(360\) 0 0
\(361\) −18.5000 4.33013i −0.973684 0.227901i
\(362\) 0 0
\(363\) −13.3923 + 7.73205i −0.702914 + 0.405827i
\(364\) 0 0
\(365\) 11.8301 + 6.83013i 0.619217 + 0.357505i
\(366\) 0 0
\(367\) −24.7583 14.2942i −1.29237 0.746153i −0.313300 0.949654i \(-0.601434\pi\)
−0.979075 + 0.203502i \(0.934768\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.3923 + 6.00000i 0.539542 + 0.311504i
\(372\) 0 0
\(373\) 6.92820 0.358729 0.179364 0.983783i \(-0.442596\pi\)
0.179364 + 0.983783i \(0.442596\pi\)
\(374\) 0 0
\(375\) −6.00000 10.3923i −0.309839 0.536656i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 17.3205i 0.889695i −0.895606 0.444847i \(-0.853258\pi\)
0.895606 0.444847i \(-0.146742\pi\)
\(380\) 0 0
\(381\) 4.39230i 0.225025i
\(382\) 0 0
\(383\) 8.36603 + 14.4904i 0.427484 + 0.740424i 0.996649 0.0817995i \(-0.0260667\pi\)
−0.569165 + 0.822223i \(0.692733\pi\)
\(384\) 0 0
\(385\) 7.73205 + 13.3923i 0.394062 + 0.682535i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.0526 + 12.7321i 1.11811 + 0.645541i 0.940918 0.338636i \(-0.109965\pi\)
0.177192 + 0.984176i \(0.443299\pi\)
\(390\) 0 0
\(391\) 22.9282i 1.15953i
\(392\) 0 0
\(393\) −2.89230 1.66987i −0.145897 0.0842339i
\(394\) 0 0
\(395\) 38.7846 + 22.3923i 1.95147 + 1.12668i
\(396\) 0 0
\(397\) −30.0788 + 17.3660i −1.50961 + 0.871576i −0.509676 + 0.860366i \(0.670235\pi\)
−0.999937 + 0.0112097i \(0.996432\pi\)
\(398\) 0 0
\(399\) −8.78461 + 3.80385i −0.439781 + 0.190431i
\(400\) 0 0
\(401\) 17.0885 9.86603i 0.853357 0.492686i −0.00842522 0.999965i \(-0.502682\pi\)
0.861782 + 0.507279i \(0.169349\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 21.2942 + 12.2942i 1.05812 + 0.610905i
\(406\) 0 0
\(407\) −21.1244 −1.04710
\(408\) 0 0
\(409\) −14.3038 8.25833i −0.707280 0.408348i 0.102773 0.994705i \(-0.467228\pi\)
−0.810053 + 0.586357i \(0.800562\pi\)
\(410\) 0 0
\(411\) 31.0526i 1.53171i
\(412\) 0 0
\(413\) −0.509619 0.882686i −0.0250767 0.0434341i
\(414\) 0 0
\(415\) 8.02628 4.63397i 0.393995 0.227473i
\(416\) 0 0
\(417\) 9.33975i 0.457369i
\(418\) 0 0
\(419\) 35.1769 1.71850 0.859252 0.511552i \(-0.170929\pi\)
0.859252 + 0.511552i \(0.170929\pi\)
\(420\) 0 0
\(421\) −1.09808 1.90192i −0.0535170 0.0926941i 0.838026 0.545631i \(-0.183710\pi\)
−0.891543 + 0.452936i \(0.850376\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.4641 0.653105
\(426\) 0 0
\(427\) −7.73205 + 13.3923i −0.374180 + 0.648099i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.7321 18.5885i 0.516945 0.895374i −0.482862 0.875697i \(-0.660403\pi\)
0.999806 0.0196778i \(-0.00626406\pi\)
\(432\) 0 0
\(433\) −6.80385 3.92820i −0.326972 0.188777i 0.327524 0.944843i \(-0.393786\pi\)
−0.654496 + 0.756065i \(0.727119\pi\)
\(434\) 0 0
\(435\) −5.19615 9.00000i −0.249136 0.431517i
\(436\) 0 0
\(437\) −10.9019 + 14.6865i −0.521510 + 0.702552i
\(438\) 0 0
\(439\) 9.29423 + 16.0981i 0.443589 + 0.768319i 0.997953 0.0639555i \(-0.0203716\pi\)
−0.554363 + 0.832275i \(0.687038\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.23205 2.13397i 0.0585365 0.101388i −0.835272 0.549837i \(-0.814690\pi\)
0.893809 + 0.448449i \(0.148023\pi\)
\(444\) 0 0
\(445\) −35.3205 −1.67435
\(446\) 0 0
\(447\) −8.36603 + 14.4904i −0.395699 + 0.685372i
\(448\) 0 0
\(449\) 10.5167i 0.496312i 0.968720 + 0.248156i \(0.0798245\pi\)
−0.968720 + 0.248156i \(0.920175\pi\)
\(450\) 0 0
\(451\) −29.8923 + 17.2583i −1.40757 + 0.812663i
\(452\) 0 0
\(453\) 25.0981 14.4904i 1.17921 0.680817i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.1769 −1.13095 −0.565474 0.824766i \(-0.691307\pi\)
−0.565474 + 0.824766i \(0.691307\pi\)
\(458\) 0 0
\(459\) 24.5885 14.1962i 1.14769 0.662620i
\(460\) 0 0
\(461\) 11.6603 6.73205i 0.543072 0.313543i −0.203251 0.979127i \(-0.565151\pi\)
0.746323 + 0.665584i \(0.231817\pi\)
\(462\) 0 0
\(463\) 16.7846i 0.780047i 0.920805 + 0.390023i \(0.127533\pi\)
−0.920805 + 0.390023i \(0.872467\pi\)
\(464\) 0 0
\(465\) −5.19615 + 9.00000i −0.240966 + 0.417365i
\(466\) 0 0
\(467\) −4.46410 −0.206574 −0.103287 0.994652i \(-0.532936\pi\)
−0.103287 + 0.994652i \(0.532936\pi\)
\(468\) 0 0
\(469\) 2.70577 4.68653i 0.124941 0.216404i
\(470\) 0 0
\(471\) −0.339746 + 0.588457i −0.0156547 + 0.0271147i
\(472\) 0 0
\(473\) −7.73205 13.3923i −0.355520 0.615779i
\(474\) 0 0
\(475\) 8.62436 + 6.40192i 0.395713 + 0.293740i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.2679 + 14.5885i 1.15452 + 0.666564i 0.949985 0.312295i \(-0.101098\pi\)
0.204537 + 0.978859i \(0.434431\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 9.21539i 0.419315i
\(484\) 0 0
\(485\) −10.5622 + 18.2942i −0.479604 + 0.830698i
\(486\) 0 0
\(487\) −33.1244 −1.50101 −0.750504 0.660866i \(-0.770189\pi\)
−0.750504 + 0.660866i \(0.770189\pi\)
\(488\) 0 0
\(489\) 14.0885 8.13397i 0.637102 0.367831i
\(490\) 0 0
\(491\) −11.5885 20.0718i −0.522980 0.905828i −0.999642 0.0267412i \(-0.991487\pi\)
0.476663 0.879086i \(-0.341846\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.78461 1.60770i 0.124907 0.0721150i
\(498\) 0 0
\(499\) −15.0885 26.1340i −0.675452 1.16992i −0.976337 0.216257i \(-0.930615\pi\)
0.300884 0.953661i \(-0.402718\pi\)
\(500\) 0 0
\(501\) 8.78461i 0.392467i
\(502\) 0 0
\(503\) 25.4378 + 14.6865i 1.13422 + 0.654840i 0.944992 0.327094i \(-0.106069\pi\)
0.189225 + 0.981934i \(0.439403\pi\)
\(504\) 0 0
\(505\) 44.2487 1.96904
\(506\) 0 0
\(507\) −19.5000 11.2583i −0.866025 0.500000i
\(508\) 0 0
\(509\) 13.8564 24.0000i 0.614174 1.06378i −0.376354 0.926476i \(-0.622822\pi\)
0.990529 0.137305i \(-0.0438442\pi\)
\(510\) 0 0
\(511\) 5.49038 3.16987i 0.242880 0.140227i
\(512\) 0 0
\(513\) 22.5000 + 2.59808i 0.993399 + 0.114708i
\(514\) 0 0
\(515\) −22.3923 + 12.9282i −0.986723 + 0.569685i
\(516\) 0 0
\(517\) −37.3468 21.5622i −1.64251 0.948303i
\(518\) 0 0
\(519\) 38.7846 + 22.3923i 1.70246 + 0.982913i
\(520\) 0 0
\(521\) 19.0526i 0.834708i −0.908744 0.417354i \(-0.862958\pi\)
0.908744 0.417354i \(-0.137042\pi\)
\(522\) 0 0
\(523\) −16.9808 9.80385i −0.742517 0.428692i 0.0804668 0.996757i \(-0.474359\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(524\) 0 0
\(525\) 5.41154 0.236179
\(526\) 0 0
\(527\) 6.00000 + 10.3923i 0.261364 + 0.452696i
\(528\) 0 0
\(529\) −2.69615 4.66987i −0.117224 0.203038i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 + 10.3923i 0.259403 + 0.449299i
\(536\) 0 0
\(537\) −4.50000 7.79423i −0.194189 0.336346i
\(538\) 0 0
\(539\) −24.0718 −1.03685
\(540\) 0 0
\(541\) 0.588457 + 0.339746i 0.0252998 + 0.0146068i 0.512597 0.858630i \(-0.328684\pi\)
−0.487297 + 0.873236i \(0.662017\pi\)
\(542\) 0 0
\(543\) 8.19615i 0.351731i
\(544\) 0 0
\(545\) 6.00000 + 3.46410i 0.257012 + 0.148386i
\(546\) 0 0
\(547\) −14.1962 8.19615i −0.606984 0.350442i 0.164800 0.986327i \(-0.447302\pi\)
−0.771784 + 0.635885i \(0.780635\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.68653 5.70577i −0.327457 0.243074i
\(552\) 0 0
\(553\) 18.0000 10.3923i 0.765438 0.441926i
\(554\) 0 0
\(555\) −11.1962 + 19.3923i −0.475250 + 0.823157i
\(556\) 0 0
\(557\) 8.53590 + 4.92820i 0.361678 + 0.208815i 0.669816 0.742527i \(-0.266373\pi\)
−0.308139 + 0.951341i \(0.599706\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 36.5885 + 21.1244i 1.54477 + 0.891871i
\(562\) 0 0
\(563\) 35.1962i 1.48334i 0.670764 + 0.741670i \(0.265966\pi\)
−0.670764 + 0.741670i \(0.734034\pi\)
\(564\) 0 0
\(565\) −21.2942 36.8827i −0.895855 1.55167i
\(566\) 0 0
\(567\) 9.88269 5.70577i 0.415034 0.239620i
\(568\) 0 0
\(569\) 0.928203i 0.0389123i 0.999811 + 0.0194562i \(0.00619348\pi\)
−0.999811 + 0.0194562i \(0.993807\pi\)
\(570\) 0 0
\(571\) −4.60770 −0.192826 −0.0964130 0.995341i \(-0.530737\pi\)
−0.0964130 + 0.995341i \(0.530737\pi\)
\(572\) 0 0
\(573\) 15.4641 + 26.7846i 0.646022 + 1.11894i
\(574\) 0 0
\(575\) 8.95448 5.16987i 0.373428 0.215599i
\(576\) 0 0
\(577\) 10.8564 0.451958 0.225979 0.974132i \(-0.427442\pi\)
0.225979 + 0.974132i \(0.427442\pi\)
\(578\) 0 0
\(579\) 21.5885 37.3923i 0.897186 1.55397i
\(580\) 0 0
\(581\) 4.30127i 0.178447i
\(582\) 0 0
\(583\) 21.1244 36.5885i 0.874881 1.51534i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.26795 + 3.92820i 0.0936083 + 0.162134i 0.909027 0.416737i \(-0.136827\pi\)
−0.815419 + 0.578872i \(0.803493\pi\)
\(588\) 0 0
\(589\) −1.09808 + 9.50962i −0.0452454 + 0.391837i
\(590\) 0 0
\(591\) −6.75833 11.7058i −0.278001 0.481511i
\(592\) 0 0
\(593\) 20.6962 35.8468i 0.849889 1.47205i −0.0314175 0.999506i \(-0.510002\pi\)
0.881307 0.472545i \(-0.156665\pi\)
\(594\) 0 0
\(595\) 9.46410 16.3923i 0.387990 0.672019i
\(596\) 0 0
\(597\) −21.4641 −0.878467
\(598\) 0 0
\(599\) −3.29423 + 5.70577i −0.134599 + 0.233131i −0.925444 0.378884i \(-0.876308\pi\)
0.790845 + 0.612016i \(0.209641\pi\)
\(600\) 0 0
\(601\) 13.7321i 0.560142i −0.959979 0.280071i \(-0.909642\pi\)
0.959979 0.280071i \(-0.0903580\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.1244 12.1962i 0.858827 0.495844i
\(606\) 0 0
\(607\) −42.5885 −1.72861 −0.864306 0.502966i \(-0.832242\pi\)
−0.864306 + 0.502966i \(0.832242\pi\)
\(608\) 0 0
\(609\) −4.82309 −0.195441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17.3205 10.0000i 0.699569 0.403896i −0.107618 0.994192i \(-0.534322\pi\)
0.807187 + 0.590296i \(0.200989\pi\)
\(614\) 0 0
\(615\) 36.5885i 1.47539i
\(616\) 0 0
\(617\) 5.16025 8.93782i 0.207744 0.359823i −0.743260 0.669003i \(-0.766721\pi\)
0.951004 + 0.309180i \(0.100055\pi\)
\(618\) 0 0
\(619\) −30.3923 −1.22157 −0.610785 0.791797i \(-0.709146\pi\)
−0.610785 + 0.791797i \(0.709146\pi\)
\(620\) 0 0
\(621\) 10.9019 18.8827i 0.437479 0.757736i
\(622\) 0 0
\(623\) −8.19615 + 14.1962i −0.328372 + 0.568757i
\(624\) 0 0
\(625\) 15.6244 + 27.0622i 0.624974 + 1.08249i
\(626\) 0 0
\(627\) 13.3923 + 30.9282i 0.534837 + 1.23515i
\(628\) 0 0
\(629\) 12.9282 + 22.3923i 0.515481 + 0.892840i
\(630\) 0 0
\(631\) −32.2750 18.6340i −1.28485 0.741807i −0.307117 0.951672i \(-0.599364\pi\)
−0.977730 + 0.209865i \(0.932698\pi\)
\(632\) 0 0
\(633\) −15.0000 + 25.9808i −0.596196 + 1.03264i
\(634\) 0 0
\(635\) 6.92820i 0.274937i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31.5000 18.1865i 1.24418 0.718325i 0.274234 0.961663i \(-0.411576\pi\)
0.969942 + 0.243338i \(0.0782425\pi\)
\(642\) 0 0
\(643\) −19.9641 34.5788i −0.787307 1.36366i −0.927611 0.373548i \(-0.878141\pi\)
0.140304 0.990109i \(-0.455192\pi\)
\(644\) 0 0
\(645\) −16.3923 −0.645446
\(646\) 0 0
\(647\) 33.8564i 1.33103i 0.746383 + 0.665516i \(0.231789\pi\)
−0.746383 + 0.665516i \(0.768211\pi\)
\(648\) 0 0
\(649\) −3.10770 + 1.79423i −0.121988 + 0.0704296i
\(650\) 0 0
\(651\) 2.41154 + 4.17691i 0.0945158 + 0.163706i
\(652\) 0 0
\(653\) 3.32051i 0.129942i −0.997887 0.0649708i \(-0.979305\pi\)
0.997887 0.0649708i \(-0.0206954\pi\)
\(654\) 0 0
\(655\) 4.56218 + 2.63397i 0.178259 + 0.102918i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −33.3731 19.2679i −1.30003 0.750573i −0.319621 0.947545i \(-0.603556\pi\)
−0.980409 + 0.196973i \(0.936889\pi\)
\(660\) 0 0
\(661\) −18.7583 + 32.4904i −0.729614 + 1.26373i 0.227432 + 0.973794i \(0.426967\pi\)
−0.957046 + 0.289935i \(0.906366\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.8564 6.00000i 0.537328 0.232670i
\(666\) 0 0
\(667\) −7.98076 + 4.60770i −0.309016 + 0.178411i
\(668\) 0 0
\(669\) 38.2750 + 22.0981i 1.47980 + 0.854361i
\(670\) 0 0
\(671\) 47.1506 + 27.2224i 1.82023 + 1.05091i
\(672\) 0 0
\(673\) 0.928203i 0.0357796i 0.999840 + 0.0178898i \(0.00569480\pi\)
−0.999840 + 0.0178898i \(0.994305\pi\)
\(674\) 0 0
\(675\) −11.0885 6.40192i −0.426795 0.246410i
\(676\) 0 0
\(677\) 3.21539 0.123577 0.0617887 0.998089i \(-0.480319\pi\)
0.0617887 + 0.998089i \(0.480319\pi\)
\(678\) 0 0
\(679\) 4.90192 + 8.49038i 0.188119 + 0.325831i
\(680\) 0 0
\(681\) 15.6962 + 27.1865i 0.601478 + 1.04179i
\(682\) 0 0
\(683\) 51.0333i 1.95274i 0.216115 + 0.976368i \(0.430661\pi\)
−0.216115 + 0.976368i \(0.569339\pi\)
\(684\) 0 0
\(685\) 48.9808i 1.87146i
\(686\) 0 0
\(687\) −11.0718 19.1769i −0.422415 0.731645i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −17.3205 −0.658903 −0.329452 0.944172i \(-0.606864\pi\)
−0.329452 + 0.944172i \(0.606864\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.7321i 0.558819i
\(696\) 0 0
\(697\) 36.5885 + 21.1244i 1.38589 + 0.800142i
\(698\) 0 0
\(699\) 10.2846 + 5.93782i 0.389000 + 0.224589i
\(700\) 0 0
\(701\) 24.4186 14.0981i 0.922277 0.532477i 0.0379164 0.999281i \(-0.487928\pi\)
0.884361 + 0.466804i \(0.154595\pi\)
\(702\) 0 0
\(703\) −2.36603 + 20.4904i −0.0892363 + 0.772809i
\(704\) 0 0
\(705\) −39.5885 + 22.8564i −1.49099 + 0.860822i
\(706\) 0 0
\(707\) 10.2679 17.7846i 0.386166 0.668859i
\(708\) 0 0
\(709\) 31.6865 + 18.2942i 1.19001 + 0.687054i 0.958309 0.285732i \(-0.0922368\pi\)
0.231703 + 0.972787i \(0.425570\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.98076 + 4.60770i 0.298882 + 0.172560i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.07180 8.78461i −0.189410 0.328067i
\(718\) 0 0
\(719\) 19.8564 11.4641i 0.740519 0.427539i −0.0817390 0.996654i \(-0.526047\pi\)
0.822258 + 0.569115i \(0.192714\pi\)
\(720\) 0 0
\(721\) 12.0000i 0.446903i
\(722\) 0 0
\(723\) 44.5692 1.65755
\(724\) 0 0
\(725\) 2.70577 + 4.68653i 0.100490 + 0.174053i
\(726\) 0 0
\(727\) −0.679492 + 0.392305i −0.0252010 + 0.0145498i −0.512548 0.858659i \(-0.671298\pi\)
0.487347 + 0.873209i \(0.337965\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −9.46410 + 16.3923i −0.350042 + 0.606291i
\(732\) 0 0
\(733\) 24.5885i 0.908195i −0.890952 0.454098i \(-0.849962\pi\)
0.890952 0.454098i \(-0.150038\pi\)
\(734\) 0 0
\(735\) −12.7583 + 22.0981i −0.470598 + 0.815100i
\(736\) 0 0
\(737\) −16.5000 9.52628i −0.607785 0.350905i
\(738\) 0 0
\(739\) 21.8923 + 37.9186i 0.805321 + 1.39486i 0.916074 + 0.401010i \(0.131341\pi\)
−0.110752 + 0.993848i \(0.535326\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.22243 7.31347i −0.154906 0.268305i 0.778119 0.628117i \(-0.216174\pi\)
−0.933025 + 0.359812i \(0.882841\pi\)
\(744\) 0 0
\(745\) 13.1962 22.8564i 0.483470 0.837394i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.56922 0.203495
\(750\) 0 0
\(751\) 0.928203 1.60770i 0.0338706 0.0586656i −0.848593 0.529046i \(-0.822550\pi\)
0.882464 + 0.470380i \(0.155883\pi\)
\(752\) 0 0
\(753\) 28.2679i 1.03014i
\(754\) 0 0
\(755\) −39.5885 + 22.8564i −1.44077 + 0.831830i
\(756\) 0 0
\(757\) 14.5359 8.39230i 0.528316 0.305024i −0.212014 0.977267i \(-0.568002\pi\)
0.740331 + 0.672243i \(0.234669\pi\)
\(758\) 0 0
\(759\) 32.4449 1.17767
\(760\) 0 0
\(761\) −6.85641 −0.248545 −0.124272 0.992248i \(-0.539660\pi\)
−0.124272 + 0.992248i \(0.539660\pi\)
\(762\) 0 0
\(763\) 2.78461 1.60770i 0.100810 0.0582025i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 21.1244 36.5885i 0.761764 1.31941i −0.180177 0.983634i \(-0.557667\pi\)
0.941941 0.335779i \(-0.109000\pi\)
\(770\) 0 0
\(771\) −22.6077 −0.814196
\(772\) 0 0
\(773\) −12.7583 + 22.0981i −0.458885 + 0.794813i −0.998902 0.0468415i \(-0.985084\pi\)
0.540017 + 0.841654i \(0.318418\pi\)
\(774\) 0 0
\(775\) 2.70577 4.68653i 0.0971942 0.168345i
\(776\) 0 0
\(777\) 5.19615 + 9.00000i 0.186411 + 0.322873i
\(778\) 0 0
\(779\) 13.3923 + 30.9282i 0.479829 + 1.10812i
\(780\) 0 0
\(781\) −5.66025 9.80385i −0.202540 0.350809i
\(782\) 0 0
\(783\) 9.88269 + 5.70577i 0.353178 + 0.203908i
\(784\) 0 0
\(785\) 0.535898 0.928203i 0.0191270 0.0331290i
\(786\) 0 0
\(787\) 30.8038i 1.09804i 0.835810 + 0.549019i \(0.184999\pi\)
−0.835810 + 0.549019i \(0.815001\pi\)
\(788\) 0 0
\(789\) −11.8301 + 20.4904i −0.421164 + 0.729477i
\(790\) 0 0
\(791\) −19.7654 −0.702776
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −22.3923 38.7846i −0.794173 1.37555i
\(796\) 0 0
\(797\) −28.7321 −1.01774 −0.508871 0.860843i \(-0.669937\pi\)
−0.508871 + 0.860843i \(0.669937\pi\)
\(798\) 0 0
\(799\) 52.7846i 1.86739i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.1603 19.3301i −0.393837 0.682145i
\(804\) 0 0
\(805\) 14.5359i 0.512323i
\(806\) 0 0
\(807\) −32.1962 18.5885i −1.13336 0.654345i
\(808\) 0 0
\(809\) −0.320508 −0.0112685 −0.00563423 0.999984i \(-0.501793\pi\)
−0.00563423 + 0.999984i \(0.501793\pi\)
\(810\) 0 0
\(811\) 40.9808 + 23.6603i 1.43903 + 0.830824i 0.997782 0.0665619i \(-0.0212030\pi\)
0.441247 + 0.897386i \(0.354536\pi\)
\(812\) 0 0
\(813\) −24.7583 + 42.8827i −0.868313 + 1.50396i
\(814\) 0 0
\(815\) −22.2224 + 12.8301i −0.778418 + 0.449420i
\(816\) 0 0
\(817\) −13.8564 + 6.00000i −0.484774 + 0.209913i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.8372 21.2679i −1.28563 0.742257i −0.307755 0.951466i \(-0.599578\pi\)
−0.977871 + 0.209209i \(0.932911\pi\)
\(822\) 0 0
\(823\) −46.9808 27.1244i −1.63765 0.945496i −0.981641 0.190740i \(-0.938911\pi\)
−0.656006 0.754756i \(-0.727755\pi\)
\(824\) 0 0
\(825\) 19.0526i 0.663325i
\(826\) 0 0
\(827\) −27.6962 15.9904i −0.963090 0.556040i −0.0659670 0.997822i \(-0.521013\pi\)
−0.897123 + 0.441782i \(0.854347\pi\)
\(828\) 0 0
\(829\) 40.3923 1.40288 0.701441 0.712727i \(-0.252540\pi\)
0.701441 + 0.712727i \(0.252540\pi\)
\(830\) 0 0
\(831\) 3.63397 + 6.29423i 0.126061 + 0.218344i
\(832\) 0 0
\(833\) 14.7321 + 25.5167i 0.510435 + 0.884100i
\(834\) 0 0
\(835\) 13.8564i 0.479521i
\(836\) 0 0
\(837\) 11.4115i 0.394441i
\(838\) 0 0
\(839\) 10.9019 + 18.8827i 0.376376 + 0.651903i 0.990532 0.137282i \(-0.0438367\pi\)
−0.614156 + 0.789185i \(0.710503\pi\)
\(840\) 0 0
\(841\) 12.0885 + 20.9378i 0.416843 + 0.721994i
\(842\) 0 0
\(843\) 29.7846 1.02584
\(844\) 0 0
\(845\) 30.7583 + 17.7583i 1.05812 + 0.610905i
\(846\) 0 0
\(847\) 11.3205i 0.388977i
\(848\) 0 0
\(849\) 27.6962 + 15.9904i 0.950529 + 0.548788i
\(850\) 0 0
\(851\) 17.1962 + 9.92820i 0.589477 + 0.340334i
\(852\) 0 0
\(853\) 13.1769 7.60770i 0.451169 0.260483i −0.257155 0.966370i \(-0.582785\pi\)
0.708324 + 0.705888i \(0.249452\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.50000 + 2.59808i −0.153717 + 0.0887486i −0.574886 0.818234i \(-0.694953\pi\)
0.421168 + 0.906982i \(0.361620\pi\)
\(858\) 0 0
\(859\) 7.69615 13.3301i 0.262589 0.454818i −0.704340 0.709863i \(-0.748757\pi\)
0.966929 + 0.255045i \(0.0820902\pi\)
\(860\) 0 0
\(861\) 14.7058 + 8.49038i 0.501171 + 0.289351i
\(862\) 0 0
\(863\) −8.44486 −0.287467 −0.143733 0.989616i \(-0.545911\pi\)
−0.143733 + 0.989616i \(0.545911\pi\)
\(864\) 0 0
\(865\) −61.1769 35.3205i −2.08008 1.20093i
\(866\) 0 0
\(867\) 22.2679i 0.756259i
\(868\) 0 0
\(869\) −36.5885 63.3731i −1.24118 2.14978i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.78461 0.296974
\(876\) 0 0
\(877\) 16.5622 + 28.6865i 0.559265 + 0.968675i 0.997558 + 0.0698432i \(0.0222499\pi\)
−0.438293 + 0.898832i \(0.644417\pi\)
\(878\) 0 0
\(879\) 7.09808 4.09808i 0.239412 0.138225i
\(880\) 0 0
\(881\) −19.5359 −0.658181 −0.329091 0.944298i \(-0.606742\pi\)
−0.329091 + 0.944298i \(0.606742\pi\)
\(882\) 0 0
\(883\) −3.50000 + 6.06218i −0.117784 + 0.204009i −0.918889 0.394515i \(-0.870912\pi\)
0.801105 + 0.598524i \(0.204246\pi\)
\(884\) 0 0
\(885\) 3.80385i 0.127865i
\(886\) 0 0
\(887\) 24.2487 42.0000i 0.814192 1.41022i −0.0957146 0.995409i \(-0.530514\pi\)
0.909907 0.414813i \(-0.136153\pi\)
\(888\) 0 0
\(889\) −2.78461 1.60770i −0.0933928 0.0539204i
\(890\) 0 0
\(891\) −20.0885 34.7942i −0.672989 1.16565i
\(892\) 0 0
\(893\) −25.0981 + 33.8109i −0.839875 + 1.13144i
\(894\) 0 0
\(895\) 7.09808 + 12.2942i 0.237263 + 0.410951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.41154 + 4.17691i −0.0804295 + 0.139308i
\(900\) 0 0
\(901\) −51.7128 −1.72280
\(902\) 0 0
\(903\) −3.80385 + 6.58846i −0.126584 + 0.219250i
\(904\) 0 0
\(905\) 12.9282i 0.429748i
\(906\) 0 0
\(907\) 46.5000 26.8468i 1.54401 0.891433i 0.545427 0.838158i \(-0.316367\pi\)
0.998580 0.0532748i \(-0.0169659\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49.5167 1.64056 0.820280 0.571962i \(-0.193818\pi\)
0.820280 + 0.571962i \(0.193818\pi\)
\(912\) 0 0
\(913\) −15.1436 −0.501180
\(914\) 0 0
\(915\) 49.9808 28.8564i 1.65231 0.953963i
\(916\) 0 0
\(917\) 2.11731 1.22243i 0.0699199 0.0403683i
\(918\) 0 0
\(919\) 8.78461i 0.289778i −0.989448 0.144889i \(-0.953718\pi\)
0.989448 0.144889i \(-0.0462824\pi\)
\(920\) 0 0
\(921\) −11.0885 + 19.2058i −0.365377 + 0.632852i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.83013 10.0981i 0.191693 0.332023i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.0885 33.0622i −0.626272 1.08473i −0.988293 0.152565i \(-0.951247\pi\)
0.362021 0.932170i \(-0.382087\pi\)
\(930\) 0 0
\(931\) −2.69615 + 23.3494i −0.0883628 + 0.765245i
\(932\) 0 0
\(933\) −16.2224 28.0981i −0.531099 0.919890i
\(934\) 0 0
\(935\) −57.7128 33.3205i −1.88741 1.08970i
\(936\) 0 0
\(937\) −24.8923 + 43.1147i −0.813196 + 1.40850i 0.0974198 + 0.995243i \(0.468941\pi\)
−0.910616 + 0.413254i \(0.864392\pi\)
\(938\) 0 0
\(939\) 33.5885i 1.09612i
\(940\) 0 0
\(941\) 19.5167 33.8038i 0.636225 1.10197i −0.350029 0.936739i \(-0.613828\pi\)
0.986254 0.165235i \(-0.0528383\pi\)
\(942\) 0 0
\(943\) 32.4449 1.05655
\(944\) 0 0
\(945\) −15.5885 + 9.00000i −0.507093 + 0.292770i
\(946\) 0 0
\(947\) 2.80385 + 4.85641i 0.0911128 + 0.157812i 0.907980 0.419014i \(-0.137624\pi\)
−0.816867 + 0.576826i \(0.804291\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 52.3923i 1.69894i
\(952\) 0 0
\(953\) 14.3038 8.25833i 0.463347 0.267514i −0.250104 0.968219i \(-0.580465\pi\)
0.713451 + 0.700706i \(0.247131\pi\)
\(954\) 0 0
\(955\) −24.3923 42.2487i −0.789316 1.36714i
\(956\) 0 0
\(957\) 16.9808i 0.548910i
\(958\) 0 0
\(959\) 19.6865 + 11.3660i 0.635711 + 0.367028i
\(960\) 0 0
\(961\) −26.1769 −0.844417
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −34.0526 + 58.9808i −1.09619 + 1.89866i
\(966\) 0 0
\(967\) 10.3923 6.00000i 0.334194 0.192947i −0.323508 0.946226i \(-0.604862\pi\)
0.657702 + 0.753279i \(0.271529\pi\)
\(968\) 0 0
\(969\) 24.5885 33.1244i 0.789895 1.06411i
\(970\) 0 0
\(971\) 9.10770 5.25833i 0.292280 0.168748i −0.346690 0.937980i \(-0.612694\pi\)
0.638970 + 0.769232i \(0.279361\pi\)
\(972\) 0 0
\(973\) −5.92116 3.41858i −0.189824 0.109595i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.2679i 0.520458i 0.965547 + 0.260229i \(0.0837980\pi\)
−0.965547 + 0.260229i \(0.916202\pi\)
\(978\) 0 0
\(979\) 49.9808 + 28.8564i 1.59739 + 0.922255i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.5885 + 21.8038i 0.401509 + 0.695435i 0.993908 0.110210i \(-0.0351524\pi\)
−0.592399 + 0.805645i \(0.701819\pi\)
\(984\) 0 0
\(985\) 10.6603 + 18.4641i 0.339664 + 0.588315i
\(986\) 0 0
\(987\) 21.2154i 0.675293i
\(988\) 0 0
\(989\) 14.5359i 0.462215i
\(990\) 0 0
\(991\) 16.0526 + 27.8038i 0.509926 + 0.883218i 0.999934 + 0.0115001i \(0.00366069\pi\)
−0.490008 + 0.871718i \(0.663006\pi\)
\(992\) 0 0
\(993\) −27.4808 47.5981i −0.872076 1.51048i
\(994\) 0 0
\(995\) 33.8564 1.07332
\(996\) 0 0
\(997\) −10.5622 6.09808i −0.334508 0.193128i 0.323333 0.946285i \(-0.395197\pi\)
−0.657841 + 0.753157i \(0.728530\pi\)
\(998\) 0 0
\(999\) 24.5885i 0.777944i
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 1216.2.s.b.31.2 yes 4
4.3 odd 2 1216.2.s.f.31.2 yes 4
8.3 odd 2 1216.2.s.a.31.1 4
8.5 even 2 1216.2.s.e.31.1 yes 4
19.8 odd 6 1216.2.s.a.863.1 yes 4
76.27 even 6 1216.2.s.e.863.1 yes 4
152.27 even 6 inner 1216.2.s.b.863.2 yes 4
152.141 odd 6 1216.2.s.f.863.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.a.31.1 4 8.3 odd 2
1216.2.s.a.863.1 yes 4 19.8 odd 6
1216.2.s.b.31.2 yes 4 1.1 even 1 trivial
1216.2.s.b.863.2 yes 4 152.27 even 6 inner
1216.2.s.e.31.1 yes 4 8.5 even 2
1216.2.s.e.863.1 yes 4 76.27 even 6
1216.2.s.f.31.2 yes 4 4.3 odd 2
1216.2.s.f.863.2 yes 4 152.141 odd 6