Properties

Label 216.3.r.a.67.2
Level $216$
Weight $3$
Character 216.67
Analytic conductor $5.886$
Analytic rank $0$
Dimension $12$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(43,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.r (of order \(18\), degree \(6\), 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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 67.2
Root \(0.909039 - 1.08335i\) of defining polynomial
Character \(\chi\) \(=\) 216.67
Dual form 216.3.r.a.187.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.347296 - 1.96962i) q^{2} +(1.05203 + 2.80949i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(5.16824 - 3.04783i) q^{6} +(4.00000 + 6.92820i) q^{8} +(-6.78645 + 5.91135i) q^{9} +O(q^{10})\) \(q+(-0.347296 - 1.96962i) q^{2} +(1.05203 + 2.80949i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(5.16824 - 3.04783i) q^{6} +(4.00000 + 6.92820i) q^{8} +(-6.78645 + 5.91135i) q^{9} +(-5.63152 + 4.72540i) q^{11} +(-7.79796 - 9.12096i) q^{12} +(12.2567 - 10.2846i) q^{16} +(-16.8864 + 29.2481i) q^{17} +(14.0000 + 11.3137i) q^{18} +(7.56852 + 13.1091i) q^{19} +(11.2630 + 9.45081i) q^{22} +(-15.2566 + 18.5267i) q^{24} +(4.34120 + 24.6202i) q^{25} +(-23.7474 - 12.8475i) q^{27} +(-24.5134 - 20.5692i) q^{32} +(-19.2005 - 10.8504i) q^{33} +(63.4721 + 23.1019i) q^{34} +(17.4215 - 31.5038i) q^{36} +(23.1913 - 19.4598i) q^{38} +(1.45792 - 8.26827i) q^{41} +(65.8758 - 55.2764i) q^{43} +(14.7028 - 25.4661i) q^{44} +(41.7889 + 23.6153i) q^{48} +(37.5362 + 31.4966i) q^{49} +(46.9846 - 17.1010i) q^{50} +(-99.9372 - 16.6722i) q^{51} +(-17.0573 + 51.2352i) q^{54} +(-28.8674 + 35.0548i) q^{57} +(-89.9014 - 75.4363i) q^{59} +(-32.0000 + 55.4256i) q^{64} +(-14.7028 + 41.5859i) q^{66} +(-3.06159 + 17.3632i) q^{67} +(23.4583 - 133.039i) q^{68} +(-68.1009 - 23.3725i) q^{72} +(-4.38372 - 7.59283i) q^{73} +(-64.6031 + 38.0978i) q^{75} +(-46.3826 - 38.9196i) q^{76} +(11.1118 - 80.2342i) q^{81} -16.7917 q^{82} +(-6.06192 - 34.3788i) q^{83} +(-131.752 - 110.553i) q^{86} +(-55.2646 - 20.1147i) q^{88} +(80.5908 + 139.587i) q^{89} +(32.0000 - 90.5097i) q^{96} +(112.129 - 94.0873i) q^{97} +(49.0000 - 84.8705i) q^{98} +(10.2845 - 65.3586i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67} + 48 q^{68} - 816 q^{76} + 84 q^{86} - 672 q^{88} + 438 q^{89} + 384 q^{96} + 282 q^{97} + 588 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.347296 1.96962i −0.173648 0.984808i
\(3\) 1.05203 + 2.80949i 0.350678 + 0.936496i
\(4\) −3.75877 + 1.36808i −0.939693 + 0.342020i
\(5\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(6\) 5.16824 3.04783i 0.861374 0.507971i
\(7\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(8\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(9\) −6.78645 + 5.91135i −0.754050 + 0.656817i
\(10\) 0 0
\(11\) −5.63152 + 4.72540i −0.511956 + 0.429582i −0.861817 0.507220i \(-0.830673\pi\)
0.349861 + 0.936802i \(0.386229\pi\)
\(12\) −7.79796 9.12096i −0.649830 0.760080i
\(13\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 12.2567 10.2846i 0.766044 0.642788i
\(17\) −16.8864 + 29.2481i −0.993317 + 1.72048i −0.396704 + 0.917947i \(0.629846\pi\)
−0.596613 + 0.802529i \(0.703487\pi\)
\(18\) 14.0000 + 11.3137i 0.777778 + 0.628539i
\(19\) 7.56852 + 13.1091i 0.398343 + 0.689951i 0.993522 0.113643i \(-0.0362520\pi\)
−0.595178 + 0.803594i \(0.702919\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 11.2630 + 9.45081i 0.511956 + 0.429582i
\(23\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(24\) −15.2566 + 18.5267i −0.635691 + 0.771944i
\(25\) 4.34120 + 24.6202i 0.173648 + 0.984808i
\(26\) 0 0
\(27\) −23.7474 12.8475i −0.879535 0.475834i
\(28\) 0 0
\(29\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(30\) 0 0
\(31\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(32\) −24.5134 20.5692i −0.766044 0.642788i
\(33\) −19.2005 10.8504i −0.581834 0.328800i
\(34\) 63.4721 + 23.1019i 1.86683 + 0.679469i
\(35\) 0 0
\(36\) 17.4215 31.5038i 0.483931 0.875106i
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 23.1913 19.4598i 0.610297 0.512100i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.45792 8.26827i 0.0355590 0.201665i −0.961853 0.273568i \(-0.911796\pi\)
0.997412 + 0.0719030i \(0.0229072\pi\)
\(42\) 0 0
\(43\) 65.8758 55.2764i 1.53200 1.28550i 0.758918 0.651186i \(-0.225728\pi\)
0.773078 0.634311i \(-0.218716\pi\)
\(44\) 14.7028 25.4661i 0.334156 0.578774i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(48\) 41.7889 + 23.6153i 0.870603 + 0.491986i
\(49\) 37.5362 + 31.4966i 0.766044 + 0.642788i
\(50\) 46.9846 17.1010i 0.939693 0.342020i
\(51\) −99.9372 16.6722i −1.95955 0.326905i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −17.0573 + 51.2352i −0.315875 + 0.948801i
\(55\) 0 0
\(56\) 0 0
\(57\) −28.8674 + 35.0548i −0.506446 + 0.614997i
\(58\) 0 0
\(59\) −89.9014 75.4363i −1.52375 1.27858i −0.828838 0.559489i \(-0.810997\pi\)
−0.694915 0.719092i \(-0.744558\pi\)
\(60\) 0 0
\(61\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32.0000 + 55.4256i −0.500000 + 0.866025i
\(65\) 0 0
\(66\) −14.7028 + 41.5859i −0.222770 + 0.630090i
\(67\) −3.06159 + 17.3632i −0.0456954 + 0.259152i −0.999094 0.0425626i \(-0.986448\pi\)
0.953398 + 0.301714i \(0.0975589\pi\)
\(68\) 23.4583 133.039i 0.344975 1.95645i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) −68.1009 23.3725i −0.945845 0.324618i
\(73\) −4.38372 7.59283i −0.0600510 0.104011i 0.834437 0.551103i \(-0.185793\pi\)
−0.894488 + 0.447092i \(0.852460\pi\)
\(74\) 0 0
\(75\) −64.6031 + 38.0978i −0.861374 + 0.507971i
\(76\) −46.3826 38.9196i −0.610297 0.512100i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(80\) 0 0
\(81\) 11.1118 80.2342i 0.137183 0.990546i
\(82\) −16.7917 −0.204776
\(83\) −6.06192 34.3788i −0.0730351 0.414203i −0.999303 0.0373364i \(-0.988113\pi\)
0.926268 0.376866i \(-0.122998\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −131.752 110.553i −1.53200 1.28550i
\(87\) 0 0
\(88\) −55.2646 20.1147i −0.628007 0.228576i
\(89\) 80.5908 + 139.587i 0.905515 + 1.56840i 0.820225 + 0.572041i \(0.193848\pi\)
0.0852901 + 0.996356i \(0.472818\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 32.0000 90.5097i 0.333333 0.942809i
\(97\) 112.129 94.0873i 1.15597 0.969972i 0.156126 0.987737i \(-0.450100\pi\)
0.999842 + 0.0177651i \(0.00565510\pi\)
\(98\) 49.0000 84.8705i 0.500000 0.866025i
\(99\) 10.2845 65.3586i 0.103884 0.660188i
\(100\) −50.0000 86.6025i −0.500000 0.866025i
\(101\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(102\) 1.87010 + 202.628i 0.0183343 + 1.98655i
\(103\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −147.899 −1.38223 −0.691115 0.722745i \(-0.742880\pi\)
−0.691115 + 0.722745i \(0.742880\pi\)
\(108\) 106.838 + 15.8024i 0.989237 + 0.146319i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 97.5659 + 81.8675i 0.863415 + 0.724491i 0.962701 0.270568i \(-0.0872114\pi\)
−0.0992858 + 0.995059i \(0.531656\pi\)
\(114\) 79.0701 + 44.6833i 0.693598 + 0.391959i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −117.358 + 203.270i −0.994559 + 1.72263i
\(119\) 0 0
\(120\) 0 0
\(121\) −11.6269 + 65.9394i −0.0960900 + 0.544953i
\(122\) 0 0
\(123\) 24.7634 4.60249i 0.201328 0.0374186i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) 120.281 + 43.7786i 0.939693 + 0.342020i
\(129\) 224.602 + 126.925i 1.74110 + 0.983913i
\(130\) 0 0
\(131\) 236.290 86.0024i 1.80374 0.656507i 0.805808 0.592177i \(-0.201731\pi\)
0.997929 0.0643300i \(-0.0204910\pi\)
\(132\) 87.0145 + 14.5163i 0.659201 + 0.109972i
\(133\) 0 0
\(134\) 35.2620 0.263150
\(135\) 0 0
\(136\) −270.182 −1.98663
\(137\) 30.7727 + 174.520i 0.224618 + 1.27387i 0.863414 + 0.504496i \(0.168322\pi\)
−0.638796 + 0.769376i \(0.720567\pi\)
\(138\) 0 0
\(139\) 241.899 88.0441i 1.74028 0.633411i 0.741007 0.671497i \(-0.234348\pi\)
0.999274 + 0.0380864i \(0.0121262\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −22.3837 + 142.250i −0.155442 + 0.987845i
\(145\) 0 0
\(146\) −13.4325 + 11.2712i −0.0920034 + 0.0772000i
\(147\) −49.0000 + 138.593i −0.333333 + 0.942809i
\(148\) 0 0
\(149\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(150\) 97.4745 + 114.012i 0.649830 + 0.760080i
\(151\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(152\) −60.5482 + 104.873i −0.398343 + 0.689951i
\(153\) −58.2971 298.312i −0.381027 1.94975i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −161.890 + 5.97900i −0.999319 + 0.0369074i
\(163\) 205.091 1.25823 0.629113 0.777314i \(-0.283418\pi\)
0.629113 + 0.777314i \(0.283418\pi\)
\(164\) 5.83168 + 33.0731i 0.0355590 + 0.201665i
\(165\) 0 0
\(166\) −65.6078 + 23.8793i −0.395228 + 0.143851i
\(167\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(168\) 0 0
\(169\) −158.808 57.8014i −0.939693 0.342020i
\(170\) 0 0
\(171\) −128.856 44.2238i −0.753542 0.258619i
\(172\) −171.990 + 297.895i −0.999939 + 1.73195i
\(173\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.4250 + 115.836i −0.116051 + 0.658158i
\(177\) 117.358 331.938i 0.663039 1.87536i
\(178\) 246.945 207.211i 1.38733 1.16411i
\(179\) −162.818 + 282.009i −0.909597 + 1.57547i −0.0949721 + 0.995480i \(0.530276\pi\)
−0.814625 + 0.579988i \(0.803057\pi\)
\(180\) 0 0
\(181\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −43.1130 244.506i −0.230551 1.30752i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(192\) −189.383 31.5940i −0.986368 0.164552i
\(193\) −296.058 + 107.756i −1.53398 + 0.558323i −0.964592 0.263745i \(-0.915042\pi\)
−0.569388 + 0.822069i \(0.692820\pi\)
\(194\) −224.258 188.175i −1.15597 0.969972i
\(195\) 0 0
\(196\) −184.180 67.0359i −0.939693 0.342020i
\(197\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) −132.303 + 2.44231i −0.668197 + 0.0123349i
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) −153.209 + 128.558i −0.766044 + 0.642788i
\(201\) −52.0025 + 9.66512i −0.258719 + 0.0480852i
\(202\) 0 0
\(203\) 0 0
\(204\) 398.450 74.0554i 1.95319 0.363016i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −104.568 38.0596i −0.500325 0.182103i
\(210\) 0 0
\(211\) 322.992 + 271.022i 1.53077 + 1.28447i 0.792086 + 0.610410i \(0.208995\pi\)
0.738680 + 0.674056i \(0.235449\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 51.3646 + 291.303i 0.240022 + 1.36123i
\(215\) 0 0
\(216\) −5.97959 215.917i −0.0276833 0.999617i
\(217\) 0 0
\(218\) 0 0
\(219\) 16.7201 20.3039i 0.0763477 0.0927120i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(224\) 0 0
\(225\) −175.000 141.421i −0.777778 0.628539i
\(226\) 127.363 220.600i 0.563554 0.976105i
\(227\) 123.341 103.496i 0.543354 0.455928i −0.329329 0.944215i \(-0.606822\pi\)
0.872683 + 0.488287i \(0.162378\pi\)
\(228\) 60.5482 171.256i 0.265562 0.751123i
\(229\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −220.774 + 382.392i −0.947528 + 1.64117i −0.196919 + 0.980420i \(0.563093\pi\)
−0.750609 + 0.660746i \(0.770240\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 441.122 + 160.555i 1.86916 + 0.680319i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(240\) 0 0
\(241\) −57.8616 328.149i −0.240090 1.36162i −0.831626 0.555337i \(-0.812589\pi\)
0.591536 0.806279i \(-0.298522\pi\)
\(242\) 133.913 0.553360
\(243\) 237.107 53.1905i 0.975749 0.218891i
\(244\) 0 0
\(245\) 0 0
\(246\) −17.6654 47.1760i −0.0718105 0.191772i
\(247\) 0 0
\(248\) 0 0
\(249\) 90.2096 53.1986i 0.362288 0.213649i
\(250\) 0 0
\(251\) −51.4601 89.1314i −0.205020 0.355105i 0.745119 0.666932i \(-0.232393\pi\)
−0.950139 + 0.311826i \(0.899059\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 44.4539 252.111i 0.173648 0.984808i
\(257\) −83.1439 + 471.533i −0.323517 + 1.83476i 0.196379 + 0.980528i \(0.437082\pi\)
−0.519896 + 0.854229i \(0.674029\pi\)
\(258\) 171.990 486.460i 0.666626 1.88550i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −251.454 435.531i −0.959748 1.66233i
\(263\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(264\) −1.62828 176.427i −0.00616772 0.668283i
\(265\) 0 0
\(266\) 0 0
\(267\) −307.385 + 373.270i −1.15125 + 1.39801i
\(268\) −12.2464 69.4527i −0.0456954 0.259152i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 93.8333 + 532.155i 0.344975 + 1.95645i
\(273\) 0 0
\(274\) 333.051 121.221i 1.21551 0.442411i
\(275\) −140.788 118.135i −0.511956 0.429582i
\(276\) 0 0
\(277\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(278\) −257.424 445.871i −0.925985 1.60385i
\(279\) 0 0
\(280\) 0 0
\(281\) 428.915 359.902i 1.52639 1.28079i 0.707785 0.706428i \(-0.249695\pi\)
0.818601 0.574362i \(-0.194750\pi\)
\(282\) 0 0
\(283\) −77.0996 + 437.254i −0.272437 + 1.54507i 0.474551 + 0.880228i \(0.342610\pi\)
−0.746988 + 0.664838i \(0.768501\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 287.951 5.31557i 0.999830 0.0184568i
\(289\) −425.800 737.508i −1.47336 2.55193i
\(290\) 0 0
\(291\) 382.301 + 216.042i 1.31375 + 0.742412i
\(292\) 26.8650 + 22.5424i 0.0920034 + 0.0772000i
\(293\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(294\) 289.992 + 48.3783i 0.986368 + 0.164552i
\(295\) 0 0
\(296\) 0 0
\(297\) 194.444 39.8653i 0.654693 0.134227i
\(298\) 0 0
\(299\) 0 0
\(300\) 190.707 231.583i 0.635691 0.771944i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 227.587 + 82.8348i 0.748640 + 0.272483i
\(305\) 0 0
\(306\) −567.314 + 218.426i −1.85397 + 0.713809i
\(307\) −114.876 + 198.971i −0.374189 + 0.648115i −0.990205 0.139619i \(-0.955412\pi\)
0.616016 + 0.787734i \(0.288746\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(312\) 0 0
\(313\) −475.797 + 399.241i −1.52012 + 1.27553i −0.679862 + 0.733340i \(0.737960\pi\)
−0.840256 + 0.542191i \(0.817595\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −155.594 415.519i −0.484717 1.29445i
\(322\) 0 0
\(323\) −511.220 −1.58272
\(324\) 68.0000 + 316.784i 0.209877 + 0.977728i
\(325\) 0 0
\(326\) −71.2273 403.950i −0.218489 1.23911i
\(327\) 0 0
\(328\) 63.1160 22.9723i 0.192427 0.0700376i
\(329\) 0 0
\(330\) 0 0
\(331\) −532.036 193.645i −1.60736 0.585031i −0.626444 0.779467i \(-0.715490\pi\)
−0.980915 + 0.194436i \(0.937712\pi\)
\(332\) 69.8184 + 120.929i 0.210296 + 0.364244i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −95.6733 + 542.590i −0.283897 + 1.61006i 0.425302 + 0.905052i \(0.360168\pi\)
−0.709199 + 0.705009i \(0.750943\pi\)
\(338\) −58.6931 + 332.865i −0.173648 + 0.984808i
\(339\) −127.363 + 360.238i −0.375703 + 1.06265i
\(340\) 0 0
\(341\) 0 0
\(342\) −42.3528 + 269.155i −0.123839 + 0.787003i
\(343\) 0 0
\(344\) 646.469 + 235.296i 1.87927 + 0.683999i
\(345\) 0 0
\(346\) 0 0
\(347\) 340.401 123.896i 0.980983 0.357048i 0.198761 0.980048i \(-0.436308\pi\)
0.782222 + 0.623000i \(0.214086\pi\)
\(348\) 0 0
\(349\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 235.246 0.668311
\(353\) −110.236 625.178i −0.312282 1.77104i −0.587070 0.809536i \(-0.699719\pi\)
0.274788 0.961505i \(-0.411392\pi\)
\(354\) −694.549 115.869i −1.96200 0.327314i
\(355\) 0 0
\(356\) −493.889 414.422i −1.38733 1.16411i
\(357\) 0 0
\(358\) 611.995 + 222.748i 1.70948 + 0.622201i
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 65.9350 114.203i 0.182645 0.316351i
\(362\) 0 0
\(363\) −197.488 + 36.7048i −0.544043 + 0.101115i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(368\) 0 0
\(369\) 38.9826 + 64.7305i 0.105644 + 0.175421i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(374\) −466.610 + 169.832i −1.24762 + 0.454097i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 732.767 1.93342 0.966712 0.255869i \(-0.0823614\pi\)
0.966712 + 0.255869i \(0.0823614\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(384\) 3.54386 + 383.984i 0.00922881 + 0.999957i
\(385\) 0 0
\(386\) 315.059 + 545.698i 0.816214 + 1.41372i
\(387\) −120.305 + 764.546i −0.310865 + 1.97557i
\(388\) −292.748 + 507.054i −0.754504 + 1.30684i
\(389\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −68.0701 + 386.045i −0.173648 + 0.984808i
\(393\) 490.207 + 573.375i 1.24735 + 1.45897i
\(394\) 0 0
\(395\) 0 0
\(396\) 50.7588 + 259.738i 0.128179 + 0.655904i
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 306.418 + 257.115i 0.766044 + 0.642788i
\(401\) 694.911 252.927i 1.73294 0.630740i 0.734111 0.679029i \(-0.237599\pi\)
0.998833 + 0.0482891i \(0.0153769\pi\)
\(402\) 37.0969 + 99.0683i 0.0922807 + 0.246439i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −284.241 759.074i −0.696669 1.86048i
\(409\) −636.512 + 231.671i −1.55626 + 0.566434i −0.969877 0.243596i \(-0.921673\pi\)
−0.586388 + 0.810030i \(0.699451\pi\)
\(410\) 0 0
\(411\) −457.939 + 270.057i −1.11421 + 0.657072i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 501.845 + 586.988i 1.20346 + 1.40764i
\(418\) −38.6467 + 219.176i −0.0924563 + 0.524346i
\(419\) 144.159 817.568i 0.344056 1.95124i 0.0378083 0.999285i \(-0.487962\pi\)
0.306247 0.951952i \(-0.400927\pi\)
\(420\) 0 0
\(421\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(422\) 421.636 730.294i 0.999137 1.73056i
\(423\) 0 0
\(424\) 0 0
\(425\) −793.401 288.774i −1.86683 0.679469i
\(426\) 0 0
\(427\) 0 0
\(428\) 555.917 202.337i 1.29887 0.472750i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −423.197 + 86.7648i −0.979623 + 0.200844i
\(433\) 735.453 1.69851 0.849253 0.527987i \(-0.177053\pi\)
0.849253 + 0.527987i \(0.177053\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −45.7978 25.8808i −0.104561 0.0590885i
\(439\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(440\) 0 0
\(441\) −440.925 + 8.13946i −0.999830 + 0.0184568i
\(442\) 0 0
\(443\) 573.726 481.413i 1.29509 1.08671i 0.304122 0.952633i \(-0.401637\pi\)
0.990971 0.134079i \(-0.0428076\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 447.517 775.122i 0.996697 1.72633i 0.428015 0.903772i \(-0.359213\pi\)
0.568681 0.822558i \(-0.307454\pi\)
\(450\) −217.769 + 393.798i −0.483931 + 0.875106i
\(451\) 30.8606 + 53.4522i 0.0684271 + 0.118519i
\(452\) −478.729 174.243i −1.05914 0.385494i
\(453\) 0 0
\(454\) −246.683 206.991i −0.543354 0.455928i
\(455\) 0 0
\(456\) −358.337 59.7800i −0.785826 0.131097i
\(457\) 66.8410 + 379.074i 0.146260 + 0.829483i 0.966347 + 0.257244i \(0.0828143\pi\)
−0.820086 + 0.572240i \(0.806075\pi\)
\(458\) 0 0
\(459\) 776.774 477.619i 1.69232 1.04056i
\(460\) 0 0
\(461\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(462\) 0 0
\(463\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 829.839 + 302.037i 1.78077 + 0.648147i
\(467\) −286.960 497.029i −0.614476 1.06430i −0.990476 0.137684i \(-0.956034\pi\)
0.376001 0.926619i \(-0.377299\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 163.032 924.600i 0.345407 1.95890i
\(473\) −109.778 + 622.580i −0.232088 + 1.31624i
\(474\) 0 0
\(475\) −289.891 + 243.248i −0.610297 + 0.512100i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −626.233 + 227.930i −1.29924 + 0.472884i
\(483\) 0 0
\(484\) −46.5076 263.757i −0.0960900 0.544953i
\(485\) 0 0
\(486\) −187.111 448.537i −0.385003 0.922916i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 215.762 + 576.200i 0.441232 + 1.17832i
\(490\) 0 0
\(491\) 552.118 + 463.282i 1.12448 + 0.943548i 0.998822 0.0485262i \(-0.0154524\pi\)
0.125655 + 0.992074i \(0.459897\pi\)
\(492\) −86.7834 + 51.1780i −0.176389 + 0.104020i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −136.110 159.203i −0.273314 0.319684i
\(499\) 77.3915 438.909i 0.155093 0.879578i −0.803607 0.595160i \(-0.797089\pi\)
0.958700 0.284418i \(-0.0918002\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −157.683 + 132.312i −0.314109 + 0.263569i
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4.67901 506.978i −0.00922881 0.999957i
\(508\) 0 0
\(509\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −1.00000
\(513\) −11.3142 408.544i −0.0220549 0.796381i
\(514\) 957.614 1.86306
\(515\) 0 0
\(516\) −1017.87 169.808i −1.97262 0.329084i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 221.047 + 382.865i 0.424274 + 0.734865i 0.996352 0.0853344i \(-0.0271959\pi\)
−0.572078 + 0.820199i \(0.693863\pi\)
\(522\) 0 0
\(523\) −319.363 + 553.153i −0.610636 + 1.05765i 0.380497 + 0.924782i \(0.375753\pi\)
−0.991133 + 0.132871i \(0.957580\pi\)
\(524\) −770.500 + 646.526i −1.47042 + 1.23383i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −346.927 + 64.4794i −0.657059 + 0.122120i
\(529\) 405.238 340.035i 0.766044 0.642788i
\(530\) 0 0
\(531\) 1056.04 19.4945i 1.98878 0.0367128i
\(532\) 0 0
\(533\) 0 0
\(534\) 841.951 + 475.795i 1.57669 + 0.891002i
\(535\) 0 0
\(536\) −132.542 + 48.2413i −0.247280 + 0.0900025i
\(537\) −963.590 160.752i −1.79440 0.299352i
\(538\) 0 0
\(539\) −360.220 −0.668311
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1015.55 369.631i 1.86683 0.679469i
\(545\) 0 0
\(546\) 0 0
\(547\) −527.977 192.168i −0.965223 0.351313i −0.189145 0.981949i \(-0.560572\pi\)
−0.776078 + 0.630637i \(0.782794\pi\)
\(548\) −354.425 613.883i −0.646761 1.12022i
\(549\) 0 0
\(550\) −183.786 + 318.326i −0.334156 + 0.578774i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −788.792 + 661.875i −1.41869 + 1.19042i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 641.581 378.354i 1.14364 0.674428i
\(562\) −857.829 719.804i −1.52639 1.28079i
\(563\) 154.962 56.4016i 0.275244 0.100180i −0.200710 0.979651i \(-0.564325\pi\)
0.475954 + 0.879470i \(0.342103\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 887.998 1.56890
\(567\) 0 0
\(568\) 0 0
\(569\) −22.8053 129.335i −0.0400796 0.227303i 0.958188 0.286139i \(-0.0923720\pi\)
−0.998268 + 0.0588367i \(0.981261\pi\)
\(570\) 0 0
\(571\) 971.136 353.465i 1.70076 0.619027i 0.704851 0.709355i \(-0.251014\pi\)
0.995912 + 0.0903277i \(0.0287914\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −110.474 565.307i −0.191795 0.981435i
\(577\) 198.285 343.440i 0.343648 0.595216i −0.641459 0.767157i \(-0.721671\pi\)
0.985107 + 0.171941i \(0.0550039\pi\)
\(578\) −1304.73 + 1094.80i −2.25731 + 1.89411i
\(579\) −614.204 718.409i −1.06080 1.24078i
\(580\) 0 0
\(581\) 0 0
\(582\) 292.748 828.016i 0.503003 1.42271i
\(583\) 0 0
\(584\) 35.0698 60.7426i 0.0600510 0.104011i
\(585\) 0 0
\(586\) 0 0
\(587\) −912.157 331.998i −1.55393 0.565585i −0.584595 0.811325i \(-0.698747\pi\)
−0.969336 + 0.245741i \(0.920969\pi\)
\(588\) −5.42654 587.975i −0.00922881 0.999957i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1136.45 1.91645 0.958223 0.286021i \(-0.0923326\pi\)
0.958223 + 0.286021i \(0.0923326\pi\)
\(594\) −146.049 369.135i −0.245874 0.621439i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(600\) −522.362 295.192i −0.870603 0.491986i
\(601\) 556.188 + 202.436i 0.925437 + 0.336831i 0.760399 0.649456i \(-0.225003\pi\)
0.165038 + 0.986287i \(0.447225\pi\)
\(602\) 0 0
\(603\) −81.8624 135.932i −0.135759 0.225427i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(608\) 84.1126 477.026i 0.138343 0.784583i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 627.240 + 1041.53i 1.02490 + 1.70185i
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 431.793 + 157.160i 0.703246 + 0.255960i
\(615\) 0 0
\(616\) 0 0
\(617\) −676.726 + 246.308i −1.09680 + 0.399203i −0.826136 0.563471i \(-0.809466\pi\)
−0.270665 + 0.962674i \(0.587243\pi\)
\(618\) 0 0
\(619\) −197.884 1122.25i −0.319683 1.81301i −0.544670 0.838651i \(-0.683345\pi\)
0.224987 0.974362i \(-0.427766\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −587.308 + 213.763i −0.939693 + 0.342020i
\(626\) 951.594 + 798.482i 1.52012 + 1.27553i
\(627\) −3.08091 333.822i −0.00491374 0.532412i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0 0
\(633\) −421.636 + 1192.57i −0.666091 + 1.88399i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1064.51 387.449i −1.66070 0.604445i −0.670227 0.742156i \(-0.733803\pi\)
−0.990472 + 0.137711i \(0.956025\pi\)
\(642\) −764.376 + 450.769i −1.19062 + 0.702133i
\(643\) −762.735 640.011i −1.18621 0.995351i −0.999917 0.0128574i \(-0.995907\pi\)
−0.186296 0.982494i \(-0.559648\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 177.545 + 1006.91i 0.274837 + 1.55868i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 600.326 243.952i 0.926429 0.376469i
\(649\) 862.748 1.32935
\(650\) 0 0
\(651\) 0 0
\(652\) −770.889 + 280.581i −1.18235 + 0.430339i
\(653\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −67.1666 116.336i −0.102388 0.177341i
\(657\) 74.6338 + 25.6146i 0.113598 + 0.0389873i
\(658\) 0 0
\(659\) −193.460 + 162.332i −0.293566 + 0.246331i −0.777660 0.628685i \(-0.783594\pi\)
0.484094 + 0.875016i \(0.339149\pi\)
\(660\) 0 0
\(661\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(662\) −196.632 + 1115.16i −0.297028 + 1.68453i
\(663\) 0 0
\(664\) 213.936 179.514i 0.322193 0.270352i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 184.746 + 1047.74i 0.274511 + 1.55683i 0.740512 + 0.672043i \(0.234583\pi\)
−0.466001 + 0.884784i \(0.654306\pi\)
\(674\) 1101.92 1.63490
\(675\) 213.216 640.440i 0.315875 0.948801i
\(676\) 676.000 1.00000
\(677\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(678\) 753.763 + 125.747i 1.11174 + 0.185468i
\(679\) 0 0
\(680\) 0 0
\(681\) 420.529 + 237.645i 0.617517 + 0.348965i
\(682\) 0 0
\(683\) 410.463 + 710.943i 0.600971 + 1.04091i 0.992674 + 0.120821i \(0.0385527\pi\)
−0.391703 + 0.920092i \(0.628114\pi\)
\(684\) 544.841 10.0577i 0.796551 0.0147043i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 238.925 1355.01i 0.347275 1.96950i
\(689\) 0 0
\(690\) 0 0
\(691\) −1057.98 + 887.747i −1.53108 + 1.28473i −0.742201 + 0.670178i \(0.766218\pi\)
−0.788878 + 0.614550i \(0.789338\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −362.247 627.430i −0.521970 0.904079i
\(695\) 0 0
\(696\) 0 0
\(697\) 217.212 + 182.263i 0.311639 + 0.261496i
\(698\) 0 0
\(699\) −1306.59 217.973i −1.86922 0.311836i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −81.6999 463.343i −0.116051 0.658158i
\(705\) 0 0
\(706\) −1193.07 + 434.244i −1.68991 + 0.615076i
\(707\) 0 0
\(708\) 12.9969 + 1408.24i 0.0183572 + 1.98903i
\(709\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −644.727 + 1116.70i −0.905515 + 1.56840i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 226.184 1282.75i 0.315900 1.79156i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −247.834 90.2043i −0.343261 0.124937i
\(723\) 861.059 507.785i 1.19095 0.702331i
\(724\) 0 0
\(725\) 0 0
\(726\) 140.881 + 376.228i 0.194051 + 0.518220i
\(727\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(728\) 0 0
\(729\) 398.883 + 610.191i 0.547164 + 0.837025i
\(730\) 0 0
\(731\) 504.323 + 2860.16i 0.689909 + 3.91267i
\(732\) 0 0
\(733\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −64.8065 112.248i −0.0879329 0.152304i
\(738\) 113.956 99.2614i 0.154412 0.134500i
\(739\) −397.972 + 689.308i −0.538528 + 0.932758i 0.460456 + 0.887683i \(0.347686\pi\)
−0.998984 + 0.0450751i \(0.985647\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 244.364 + 197.476i 0.327128 + 0.264359i
\(748\) 496.556 + 860.060i 0.663845 + 1.14981i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(752\) 0 0
\(753\) 196.276 238.346i 0.260659 0.316528i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −254.487 1443.27i −0.335735 1.90405i
\(759\) 0 0
\(760\) 0 0
\(761\) 1067.87 + 896.046i 1.40324 + 1.17746i 0.959639 + 0.281234i \(0.0907438\pi\)
0.443601 + 0.896224i \(0.353701\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 755.069 140.336i 0.983163 0.182729i
\(769\) 259.951 1474.26i 0.338038 1.91711i −0.0568609 0.998382i \(-0.518109\pi\)
0.394899 0.918725i \(-0.370780\pi\)
\(770\) 0 0
\(771\) −1412.24 + 262.476i −1.83169 + 0.340436i
\(772\) 965.396 810.063i 1.25051 1.04930i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 1547.64 28.5694i 1.99954 0.0369114i
\(775\) 0 0
\(776\) 1100.37 + 400.502i 1.41800 + 0.516111i
\(777\) 0 0
\(778\) 0 0
\(779\) 119.424 43.4666i 0.153304 0.0557980i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 959.082 1164.65i 1.22021 1.48174i
\(787\) −600.718 + 218.643i −0.763301 + 0.277819i −0.694191 0.719790i \(-0.744238\pi\)
−0.0691092 + 0.997609i \(0.522016\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 493.956 190.181i 0.623681 0.240128i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 400.000 692.820i 0.500000 0.866025i
\(801\) −1372.08 470.902i −1.71295 0.587893i
\(802\) −739.509 1280.87i −0.922081 1.59709i
\(803\) 60.5662 + 22.0443i 0.0754249 + 0.0274524i
\(804\) 182.243 107.473i 0.226670 0.133672i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 59.5434 0.0736013 0.0368006 0.999323i \(-0.488283\pi\)
0.0368006 + 0.999323i \(0.488283\pi\)
\(810\) 0 0
\(811\) −384.141 −0.473663 −0.236832 0.971551i \(-0.576109\pi\)
−0.236832 + 0.971551i \(0.576109\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −1396.37 + 823.469i −1.71124 + 1.00915i
\(817\) 1223.20 + 445.210i 1.49719 + 0.544933i
\(818\) 677.362 + 1173.23i 0.828071 + 1.43426i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(822\) 690.948 + 808.174i 0.840570 + 0.983181i
\(823\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(824\) 0 0
\(825\) 183.786 519.824i 0.222770 0.630090i
\(826\) 0 0
\(827\) −631.000 + 1092.92i −0.762999 + 1.32155i 0.178299 + 0.983976i \(0.442940\pi\)
−0.941298 + 0.337576i \(0.890393\pi\)
\(828\) 0 0
\(829\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1555.07 + 565.998i −1.86683 + 0.679469i
\(834\) 981.851 1192.30i 1.17728 1.42962i
\(835\) 0 0
\(836\) 445.115 0.532434
\(837\) 0 0
\(838\) −1660.36 −1.98134
\(839\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(840\) 0 0
\(841\) −790.281 + 287.639i −0.939693 + 0.342020i
\(842\) 0 0
\(843\) 1462.37 + 826.401i 1.73472 + 0.980310i
\(844\) −1584.83 576.832i −1.87776 0.683450i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1309.57 + 243.395i −1.54249 + 0.286684i
\(850\) −293.229 + 1662.98i −0.344975 + 1.95645i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −591.594 1024.67i −0.691115 1.19705i
\(857\) −1129.51 411.108i −1.31798 0.479706i −0.415171 0.909743i \(-0.636278\pi\)
−0.902811 + 0.430037i \(0.858500\pi\)
\(858\) 0 0
\(859\) −1313.81 1102.42i −1.52947 1.28337i −0.802334 0.596875i \(-0.796409\pi\)
−0.727131 0.686499i \(-0.759147\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 317.868 + 803.403i 0.367903 + 0.929864i
\(865\) 0 0
\(866\) −255.420 1448.56i −0.294942 1.67270i
\(867\) 1624.06 1972.16i 1.87320 2.27470i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −204.774 + 1301.35i −0.234564 + 1.49067i
\(874\) 0 0
\(875\) 0 0
\(876\) −35.0698 + 99.1923i −0.0400340 + 0.113233i
\(877\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 81.4082 141.003i 0.0924043 0.160049i −0.816118 0.577885i \(-0.803878\pi\)
0.908522 + 0.417836i \(0.137211\pi\)
\(882\) 169.163 + 865.626i 0.191795 + 0.981435i
\(883\) 211.479 + 366.292i 0.239500 + 0.414826i 0.960571 0.278035i \(-0.0896831\pi\)
−0.721071 + 0.692861i \(0.756350\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1147.45 962.827i −1.29509 1.08671i
\(887\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 316.562 + 504.348i 0.355289 + 0.566047i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1682.11 612.239i −1.87318 0.681781i
\(899\) 0 0
\(900\) 851.261 + 292.156i 0.945845 + 0.324618i
\(901\) 0 0
\(902\) 94.5625 79.3473i 0.104836 0.0879682i
\(903\) 0 0
\(904\) −176.931 + 1003.43i −0.195720 + 1.10999i
\(905\) 0 0
\(906\) 0 0
\(907\) 1078.19 904.713i 1.18875 0.997478i 0.188868 0.982002i \(-0.439518\pi\)
0.999880 0.0154761i \(-0.00492638\pi\)
\(908\) −322.022 + 557.758i −0.354649 + 0.614271i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(912\) 6.70545 + 726.547i 0.00735247 + 0.796653i
\(913\) 196.592 + 164.960i 0.215325 + 0.180679i
\(914\) 723.416 263.302i 0.791484 0.288077i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −1210.50 1364.07i −1.31862 1.48592i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −679.861 113.419i −0.738177 0.123147i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 810.475 680.069i 0.872417 0.732044i −0.0921889 0.995742i \(-0.529386\pi\)
0.964606 + 0.263697i \(0.0849419\pi\)
\(930\) 0 0
\(931\) −128.797 + 730.447i −0.138343 + 0.784583i
\(932\) 306.696 1739.36i 0.329073 1.86627i
\(933\) 0 0
\(934\) −879.297 + 737.818i −0.941431 + 0.789955i
\(935\) 0 0
\(936\) 0 0
\(937\) −929.044 1609.15i −0.991509 1.71734i −0.608371 0.793653i \(-0.708177\pi\)
−0.383138 0.923691i \(-0.625157\pi\)
\(938\) 0 0
\(939\) −1622.22 916.731i −1.72760 0.976284i
\(940\) 0 0
\(941\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1877.73 −1.98912
\(945\) 0 0
\(946\) 1264.37 1.33654
\(947\) 245.730 + 1393.61i 0.259483 + 1.47160i 0.784298 + 0.620384i \(0.213023\pi\)
−0.524815 + 0.851216i \(0.675866\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 579.782 + 486.495i 0.610297 + 0.512100i
\(951\) 0 0
\(952\) 0 0
\(953\) −391.758 678.544i −0.411078 0.712008i 0.583930 0.811804i \(-0.301514\pi\)
−0.995008 + 0.0997959i \(0.968181\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 736.169 617.719i 0.766044 0.642788i
\(962\) 0 0
\(963\) 1003.71 874.280i 1.04227 0.907872i
\(964\) 666.423 + 1154.28i 0.691310 + 1.19738i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(968\) −503.349 + 183.204i −0.519989 + 0.189260i
\(969\) −537.821 1436.27i −0.555026 1.48222i
\(970\) 0 0
\(971\) 974.000 1.00309 0.501545 0.865132i \(-0.332765\pi\)
0.501545 + 0.865132i \(0.332765\pi\)
\(972\) −818.462 + 524.312i −0.842039 + 0.539416i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −941.688 790.170i −0.963857 0.808772i 0.0177191 0.999843i \(-0.494360\pi\)
−0.981576 + 0.191071i \(0.938804\pi\)
\(978\) 1059.96 625.081i 1.08380 0.639142i
\(979\) −1113.46 405.265i −1.13734 0.413958i
\(980\) 0 0
\(981\) 0 0
\(982\) 720.739 1248.36i 0.733950 1.27124i
\(983\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(984\) 130.941 + 153.156i 0.133070 + 0.155646i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(992\) 0 0
\(993\) −15.6755 1698.47i −0.0157860 1.71044i
\(994\) 0 0
\(995\) 0 0
\(996\) −266.297 + 323.375i −0.267367 + 0.324674i
\(997\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(998\) −891.360 −0.893147
\(999\) 0 0
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 216.3.r.a.67.2 12
8.3 odd 2 CM 216.3.r.a.67.2 12
27.25 even 9 inner 216.3.r.a.187.2 yes 12
216.187 odd 18 inner 216.3.r.a.187.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.r.a.67.2 12 1.1 even 1 trivial
216.3.r.a.67.2 12 8.3 odd 2 CM
216.3.r.a.187.2 yes 12 27.25 even 9 inner
216.3.r.a.187.2 yes 12 216.187 odd 18 inner