Properties

Label 216.3.b.a.163.12
Level $216$
Weight $3$
Character 216.163
Analytic conductor $5.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,3,Mod(163,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.b (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: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 163.12
Root \(0.941181 + 1.76470i\) of defining polynomial
Character \(\chi\) \(=\) 216.163
Dual form 216.3.b.a.163.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.941181 + 1.76470i) q^{2} +(-2.22836 + 3.32181i) q^{4} +8.60639i q^{5} -4.81948i q^{7} +(-7.95930 - 0.805966i) q^{8} +O(q^{10})\) \(q+(0.941181 + 1.76470i) q^{2} +(-2.22836 + 3.32181i) q^{4} +8.60639i q^{5} -4.81948i q^{7} +(-7.95930 - 0.805966i) q^{8} +(-15.1877 + 8.10017i) q^{10} -9.93139 q^{11} +5.78729i q^{13} +(8.50495 - 4.53600i) q^{14} +(-6.06885 - 14.8044i) q^{16} +30.8667 q^{17} -18.5944 q^{19} +(-28.5888 - 19.1781i) q^{20} +(-9.34723 - 17.5260i) q^{22} +15.6966i q^{23} -49.0699 q^{25} +(-10.2128 + 5.44688i) q^{26} +(16.0094 + 10.7395i) q^{28} +25.9792i q^{29} -12.7847i q^{31} +(20.4134 - 24.6433i) q^{32} +(29.0511 + 54.4705i) q^{34} +41.4783 q^{35} +63.2921i q^{37} +(-17.5007 - 32.8136i) q^{38} +(6.93646 - 68.5008i) q^{40} -14.4473 q^{41} +62.4861 q^{43} +(22.1307 - 32.9902i) q^{44} +(-27.6998 + 14.7733i) q^{46} +26.7557i q^{47} +25.7726 q^{49} +(-46.1837 - 86.5939i) q^{50} +(-19.2243 - 12.8961i) q^{52} -14.2150i q^{53} -85.4734i q^{55} +(-3.88434 + 38.3597i) q^{56} +(-45.8456 + 24.4512i) q^{58} +76.7922 q^{59} -44.5930i q^{61} +(22.5612 - 12.0327i) q^{62} +(62.7008 + 12.8298i) q^{64} -49.8076 q^{65} +88.6047 q^{67} +(-68.7820 + 102.533i) q^{68} +(39.0386 + 73.1969i) q^{70} +114.263i q^{71} +28.3077 q^{73} +(-111.692 + 59.5693i) q^{74} +(41.4350 - 61.7671i) q^{76} +47.8641i q^{77} -115.116i q^{79} +(127.412 - 52.2309i) q^{80} +(-13.5975 - 25.4952i) q^{82} -46.2135 q^{83} +265.651i q^{85} +(58.8107 + 110.269i) q^{86} +(79.0469 + 8.00436i) q^{88} +33.8625 q^{89} +27.8917 q^{91} +(-52.1410 - 34.9775i) q^{92} +(-47.2158 + 25.1819i) q^{94} -160.031i q^{95} -93.6047 q^{97} +(24.2567 + 45.4810i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} - 18 q^{10} + 34 q^{16} + 32 q^{19} - 22 q^{22} - 80 q^{25} + 102 q^{28} + 68 q^{34} - 6 q^{40} + 128 q^{43} + 60 q^{46} - 80 q^{49} - 180 q^{52} - 156 q^{58} - 74 q^{64} + 128 q^{67} - 378 q^{70} - 160 q^{73} + 188 q^{76} - 508 q^{82} + 542 q^{88} - 96 q^{91} + 24 q^{94} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.941181 + 1.76470i 0.470590 + 0.882352i
\(3\) 0 0
\(4\) −2.22836 + 3.32181i −0.557089 + 0.830453i
\(5\) 8.60639i 1.72128i 0.509216 + 0.860639i \(0.329935\pi\)
−0.509216 + 0.860639i \(0.670065\pi\)
\(6\) 0 0
\(7\) 4.81948i 0.688497i −0.938879 0.344249i \(-0.888134\pi\)
0.938879 0.344249i \(-0.111866\pi\)
\(8\) −7.95930 0.805966i −0.994912 0.100746i
\(9\) 0 0
\(10\) −15.1877 + 8.10017i −1.51877 + 0.810017i
\(11\) −9.93139 −0.902854 −0.451427 0.892308i \(-0.649085\pi\)
−0.451427 + 0.892308i \(0.649085\pi\)
\(12\) 0 0
\(13\) 5.78729i 0.445176i 0.974913 + 0.222588i \(0.0714504\pi\)
−0.974913 + 0.222588i \(0.928550\pi\)
\(14\) 8.50495 4.53600i 0.607497 0.324000i
\(15\) 0 0
\(16\) −6.06885 14.8044i −0.379303 0.925273i
\(17\) 30.8667 1.81569 0.907843 0.419309i \(-0.137728\pi\)
0.907843 + 0.419309i \(0.137728\pi\)
\(18\) 0 0
\(19\) −18.5944 −0.978653 −0.489327 0.872101i \(-0.662757\pi\)
−0.489327 + 0.872101i \(0.662757\pi\)
\(20\) −28.5888 19.1781i −1.42944 0.958905i
\(21\) 0 0
\(22\) −9.34723 17.5260i −0.424874 0.796635i
\(23\) 15.6966i 0.682459i 0.939980 + 0.341230i \(0.110843\pi\)
−0.939980 + 0.341230i \(0.889157\pi\)
\(24\) 0 0
\(25\) −49.0699 −1.96280
\(26\) −10.2128 + 5.44688i −0.392802 + 0.209495i
\(27\) 0 0
\(28\) 16.0094 + 10.7395i 0.571764 + 0.383554i
\(29\) 25.9792i 0.895836i 0.894075 + 0.447918i \(0.147834\pi\)
−0.894075 + 0.447918i \(0.852166\pi\)
\(30\) 0 0
\(31\) 12.7847i 0.412409i −0.978509 0.206204i \(-0.933889\pi\)
0.978509 0.206204i \(-0.0661112\pi\)
\(32\) 20.4134 24.6433i 0.637920 0.770103i
\(33\) 0 0
\(34\) 29.0511 + 54.4705i 0.854445 + 1.60207i
\(35\) 41.4783 1.18509
\(36\) 0 0
\(37\) 63.2921i 1.71060i 0.518136 + 0.855298i \(0.326626\pi\)
−0.518136 + 0.855298i \(0.673374\pi\)
\(38\) −17.5007 32.8136i −0.460545 0.863516i
\(39\) 0 0
\(40\) 6.93646 68.5008i 0.173411 1.71252i
\(41\) −14.4473 −0.352373 −0.176187 0.984357i \(-0.556376\pi\)
−0.176187 + 0.984357i \(0.556376\pi\)
\(42\) 0 0
\(43\) 62.4861 1.45317 0.726583 0.687079i \(-0.241107\pi\)
0.726583 + 0.687079i \(0.241107\pi\)
\(44\) 22.1307 32.9902i 0.502970 0.749777i
\(45\) 0 0
\(46\) −27.6998 + 14.7733i −0.602169 + 0.321159i
\(47\) 26.7557i 0.569269i 0.958636 + 0.284635i \(0.0918723\pi\)
−0.958636 + 0.284635i \(0.908128\pi\)
\(48\) 0 0
\(49\) 25.7726 0.525972
\(50\) −46.1837 86.5939i −0.923673 1.73188i
\(51\) 0 0
\(52\) −19.2243 12.8961i −0.369697 0.248003i
\(53\) 14.2150i 0.268207i −0.990967 0.134104i \(-0.957184\pi\)
0.990967 0.134104i \(-0.0428155\pi\)
\(54\) 0 0
\(55\) 85.4734i 1.55406i
\(56\) −3.88434 + 38.3597i −0.0693631 + 0.684994i
\(57\) 0 0
\(58\) −45.8456 + 24.4512i −0.790442 + 0.421572i
\(59\) 76.7922 1.30156 0.650782 0.759265i \(-0.274441\pi\)
0.650782 + 0.759265i \(0.274441\pi\)
\(60\) 0 0
\(61\) 44.5930i 0.731032i −0.930805 0.365516i \(-0.880893\pi\)
0.930805 0.365516i \(-0.119107\pi\)
\(62\) 22.5612 12.0327i 0.363890 0.194076i
\(63\) 0 0
\(64\) 62.7008 + 12.8298i 0.979701 + 0.200466i
\(65\) −49.8076 −0.766271
\(66\) 0 0
\(67\) 88.6047 1.32246 0.661229 0.750184i \(-0.270035\pi\)
0.661229 + 0.750184i \(0.270035\pi\)
\(68\) −68.7820 + 102.533i −1.01150 + 1.50784i
\(69\) 0 0
\(70\) 39.0386 + 73.1969i 0.557694 + 1.04567i
\(71\) 114.263i 1.60933i 0.593728 + 0.804666i \(0.297655\pi\)
−0.593728 + 0.804666i \(0.702345\pi\)
\(72\) 0 0
\(73\) 28.3077 0.387777 0.193889 0.981024i \(-0.437890\pi\)
0.193889 + 0.981024i \(0.437890\pi\)
\(74\) −111.692 + 59.5693i −1.50935 + 0.804990i
\(75\) 0 0
\(76\) 41.4350 61.7671i 0.545197 0.812725i
\(77\) 47.8641i 0.621612i
\(78\) 0 0
\(79\) 115.116i 1.45716i −0.684960 0.728581i \(-0.740180\pi\)
0.684960 0.728581i \(-0.259820\pi\)
\(80\) 127.412 52.2309i 1.59265 0.652886i
\(81\) 0 0
\(82\) −13.5975 25.4952i −0.165823 0.310917i
\(83\) −46.2135 −0.556790 −0.278395 0.960467i \(-0.589802\pi\)
−0.278395 + 0.960467i \(0.589802\pi\)
\(84\) 0 0
\(85\) 265.651i 3.12530i
\(86\) 58.8107 + 110.269i 0.683846 + 1.28220i
\(87\) 0 0
\(88\) 79.0469 + 8.00436i 0.898260 + 0.0909587i
\(89\) 33.8625 0.380478 0.190239 0.981738i \(-0.439074\pi\)
0.190239 + 0.981738i \(0.439074\pi\)
\(90\) 0 0
\(91\) 27.8917 0.306502
\(92\) −52.1410 34.9775i −0.566750 0.380191i
\(93\) 0 0
\(94\) −47.2158 + 25.1819i −0.502296 + 0.267893i
\(95\) 160.031i 1.68453i
\(96\) 0 0
\(97\) −93.6047 −0.964997 −0.482499 0.875897i \(-0.660271\pi\)
−0.482499 + 0.875897i \(0.660271\pi\)
\(98\) 24.2567 + 45.4810i 0.247517 + 0.464092i
\(99\) 0 0
\(100\) 109.345 163.001i 1.09345 1.63001i
\(101\) 28.2006i 0.279214i −0.990207 0.139607i \(-0.955416\pi\)
0.990207 0.139607i \(-0.0445839\pi\)
\(102\) 0 0
\(103\) 55.9057i 0.542774i 0.962470 + 0.271387i \(0.0874823\pi\)
−0.962470 + 0.271387i \(0.912518\pi\)
\(104\) 4.66436 46.0627i 0.0448496 0.442911i
\(105\) 0 0
\(106\) 25.0852 13.3789i 0.236653 0.126216i
\(107\) −169.036 −1.57977 −0.789886 0.613253i \(-0.789860\pi\)
−0.789886 + 0.613253i \(0.789860\pi\)
\(108\) 0 0
\(109\) 84.7359i 0.777393i −0.921366 0.388697i \(-0.872925\pi\)
0.921366 0.388697i \(-0.127075\pi\)
\(110\) 150.835 80.4459i 1.37123 0.731327i
\(111\) 0 0
\(112\) −71.3493 + 29.2487i −0.637047 + 0.261149i
\(113\) 75.2321 0.665771 0.332885 0.942967i \(-0.391978\pi\)
0.332885 + 0.942967i \(0.391978\pi\)
\(114\) 0 0
\(115\) −135.091 −1.17470
\(116\) −86.2981 57.8910i −0.743949 0.499061i
\(117\) 0 0
\(118\) 72.2754 + 135.516i 0.612503 + 1.14844i
\(119\) 148.761i 1.25009i
\(120\) 0 0
\(121\) −22.3675 −0.184855
\(122\) 78.6933 41.9700i 0.645027 0.344017i
\(123\) 0 0
\(124\) 42.4682 + 28.4888i 0.342486 + 0.229748i
\(125\) 207.155i 1.65724i
\(126\) 0 0
\(127\) 58.4530i 0.460260i −0.973160 0.230130i \(-0.926085\pi\)
0.973160 0.230130i \(-0.0739151\pi\)
\(128\) 36.3720 + 122.724i 0.284156 + 0.958778i
\(129\) 0 0
\(130\) −46.8780 87.8957i −0.360600 0.676121i
\(131\) −115.102 −0.878639 −0.439320 0.898331i \(-0.644780\pi\)
−0.439320 + 0.898331i \(0.644780\pi\)
\(132\) 0 0
\(133\) 89.6154i 0.673800i
\(134\) 83.3931 + 156.361i 0.622336 + 1.16687i
\(135\) 0 0
\(136\) −245.677 24.8775i −1.80645 0.182923i
\(137\) −50.2245 −0.366602 −0.183301 0.983057i \(-0.558678\pi\)
−0.183301 + 0.983057i \(0.558678\pi\)
\(138\) 0 0
\(139\) 58.1186 0.418120 0.209060 0.977903i \(-0.432960\pi\)
0.209060 + 0.977903i \(0.432960\pi\)
\(140\) −92.4285 + 137.783i −0.660203 + 0.984165i
\(141\) 0 0
\(142\) −201.639 + 107.542i −1.42000 + 0.757336i
\(143\) 57.4758i 0.401929i
\(144\) 0 0
\(145\) −223.587 −1.54198
\(146\) 26.6427 + 49.9547i 0.182484 + 0.342156i
\(147\) 0 0
\(148\) −210.244 141.037i −1.42057 0.952955i
\(149\) 125.603i 0.842974i 0.906834 + 0.421487i \(0.138492\pi\)
−0.906834 + 0.421487i \(0.861508\pi\)
\(150\) 0 0
\(151\) 162.225i 1.07433i −0.843476 0.537167i \(-0.819494\pi\)
0.843476 0.537167i \(-0.180506\pi\)
\(152\) 147.998 + 14.9865i 0.973674 + 0.0985951i
\(153\) 0 0
\(154\) −84.4660 + 45.0488i −0.548480 + 0.292525i
\(155\) 110.030 0.709870
\(156\) 0 0
\(157\) 157.397i 1.00253i −0.865293 0.501266i \(-0.832868\pi\)
0.865293 0.501266i \(-0.167132\pi\)
\(158\) 203.145 108.345i 1.28573 0.685726i
\(159\) 0 0
\(160\) 212.090 + 175.686i 1.32556 + 1.09804i
\(161\) 75.6492 0.469871
\(162\) 0 0
\(163\) 166.269 1.02006 0.510029 0.860157i \(-0.329635\pi\)
0.510029 + 0.860157i \(0.329635\pi\)
\(164\) 32.1937 47.9912i 0.196303 0.292629i
\(165\) 0 0
\(166\) −43.4953 81.5532i −0.262020 0.491284i
\(167\) 158.709i 0.950355i −0.879890 0.475177i \(-0.842384\pi\)
0.879890 0.475177i \(-0.157616\pi\)
\(168\) 0 0
\(169\) 135.507 0.801818
\(170\) −468.795 + 250.025i −2.75761 + 1.47074i
\(171\) 0 0
\(172\) −139.241 + 207.567i −0.809543 + 1.20678i
\(173\) 45.2251i 0.261417i −0.991421 0.130708i \(-0.958275\pi\)
0.991421 0.130708i \(-0.0417251\pi\)
\(174\) 0 0
\(175\) 236.491i 1.35138i
\(176\) 60.2721 + 147.028i 0.342455 + 0.835386i
\(177\) 0 0
\(178\) 31.8708 + 59.7574i 0.179049 + 0.335715i
\(179\) −34.1175 −0.190601 −0.0953003 0.995449i \(-0.530381\pi\)
−0.0953003 + 0.995449i \(0.530381\pi\)
\(180\) 0 0
\(181\) 21.4438i 0.118474i 0.998244 + 0.0592370i \(0.0188668\pi\)
−0.998244 + 0.0592370i \(0.981133\pi\)
\(182\) 26.2511 + 49.2206i 0.144237 + 0.270443i
\(183\) 0 0
\(184\) 12.6509 124.934i 0.0687548 0.678987i
\(185\) −544.716 −2.94441
\(186\) 0 0
\(187\) −306.549 −1.63930
\(188\) −88.8772 59.6212i −0.472751 0.317134i
\(189\) 0 0
\(190\) 282.407 150.618i 1.48635 0.792725i
\(191\) 172.809i 0.904759i 0.891826 + 0.452380i \(0.149425\pi\)
−0.891826 + 0.452380i \(0.850575\pi\)
\(192\) 0 0
\(193\) 140.210 0.726475 0.363238 0.931697i \(-0.381671\pi\)
0.363238 + 0.931697i \(0.381671\pi\)
\(194\) −88.0990 165.185i −0.454118 0.851467i
\(195\) 0 0
\(196\) −57.4306 + 85.6118i −0.293013 + 0.436795i
\(197\) 329.507i 1.67262i −0.548253 0.836312i \(-0.684707\pi\)
0.548253 0.836312i \(-0.315293\pi\)
\(198\) 0 0
\(199\) 249.916i 1.25586i 0.778270 + 0.627930i \(0.216098\pi\)
−0.778270 + 0.627930i \(0.783902\pi\)
\(200\) 390.562 + 39.5487i 1.95281 + 0.197743i
\(201\) 0 0
\(202\) 49.7657 26.5419i 0.246365 0.131395i
\(203\) 125.206 0.616780
\(204\) 0 0
\(205\) 124.339i 0.606532i
\(206\) −98.6570 + 52.6174i −0.478917 + 0.255424i
\(207\) 0 0
\(208\) 85.6771 35.1222i 0.411909 0.168857i
\(209\) 184.668 0.883581
\(210\) 0 0
\(211\) −84.9300 −0.402512 −0.201256 0.979539i \(-0.564502\pi\)
−0.201256 + 0.979539i \(0.564502\pi\)
\(212\) 47.2195 + 31.6761i 0.222733 + 0.149415i
\(213\) 0 0
\(214\) −159.093 298.298i −0.743426 1.39392i
\(215\) 537.780i 2.50130i
\(216\) 0 0
\(217\) −61.6154 −0.283942
\(218\) 149.534 79.7518i 0.685934 0.365834i
\(219\) 0 0
\(220\) 283.926 + 190.465i 1.29057 + 0.865751i
\(221\) 178.634i 0.808300i
\(222\) 0 0
\(223\) 1.94316i 0.00871371i −0.999991 0.00435686i \(-0.998613\pi\)
0.999991 0.00435686i \(-0.00138683\pi\)
\(224\) −118.768 98.3821i −0.530214 0.439206i
\(225\) 0 0
\(226\) 70.8070 + 132.762i 0.313305 + 0.587444i
\(227\) 203.658 0.897171 0.448586 0.893740i \(-0.351928\pi\)
0.448586 + 0.893740i \(0.351928\pi\)
\(228\) 0 0
\(229\) 19.9963i 0.0873202i −0.999046 0.0436601i \(-0.986098\pi\)
0.999046 0.0436601i \(-0.0139019\pi\)
\(230\) −127.145 238.395i −0.552803 1.03650i
\(231\) 0 0
\(232\) 20.9384 206.776i 0.0902516 0.891278i
\(233\) −91.1197 −0.391072 −0.195536 0.980697i \(-0.562645\pi\)
−0.195536 + 0.980697i \(0.562645\pi\)
\(234\) 0 0
\(235\) −230.270 −0.979870
\(236\) −171.121 + 255.089i −0.725087 + 1.08089i
\(237\) 0 0
\(238\) 262.520 140.011i 1.10302 0.588283i
\(239\) 36.8504i 0.154186i −0.997024 0.0770929i \(-0.975436\pi\)
0.997024 0.0770929i \(-0.0245638\pi\)
\(240\) 0 0
\(241\) 308.182 1.27876 0.639381 0.768890i \(-0.279191\pi\)
0.639381 + 0.768890i \(0.279191\pi\)
\(242\) −21.0519 39.4720i −0.0869911 0.163107i
\(243\) 0 0
\(244\) 148.129 + 99.3690i 0.607087 + 0.407250i
\(245\) 221.809i 0.905344i
\(246\) 0 0
\(247\) 107.611i 0.435673i
\(248\) −10.3040 + 101.757i −0.0415484 + 0.410310i
\(249\) 0 0
\(250\) 365.567 194.970i 1.46227 0.779881i
\(251\) −228.334 −0.909696 −0.454848 0.890569i \(-0.650306\pi\)
−0.454848 + 0.890569i \(0.650306\pi\)
\(252\) 0 0
\(253\) 155.889i 0.616161i
\(254\) 103.152 55.0149i 0.406111 0.216594i
\(255\) 0 0
\(256\) −182.338 + 179.691i −0.712259 + 0.701917i
\(257\) 292.265 1.13722 0.568609 0.822608i \(-0.307482\pi\)
0.568609 + 0.822608i \(0.307482\pi\)
\(258\) 0 0
\(259\) 305.035 1.17774
\(260\) 110.989 165.451i 0.426881 0.636352i
\(261\) 0 0
\(262\) −108.332 203.120i −0.413479 0.775269i
\(263\) 488.233i 1.85640i −0.372083 0.928199i \(-0.621356\pi\)
0.372083 0.928199i \(-0.378644\pi\)
\(264\) 0 0
\(265\) 122.340 0.461659
\(266\) −158.145 + 84.3443i −0.594528 + 0.317084i
\(267\) 0 0
\(268\) −197.443 + 294.328i −0.736728 + 1.09824i
\(269\) 38.8093i 0.144273i 0.997395 + 0.0721363i \(0.0229816\pi\)
−0.997395 + 0.0721363i \(0.977018\pi\)
\(270\) 0 0
\(271\) 327.987i 1.21028i 0.796118 + 0.605142i \(0.206884\pi\)
−0.796118 + 0.605142i \(0.793116\pi\)
\(272\) −187.325 456.961i −0.688695 1.68001i
\(273\) 0 0
\(274\) −47.2703 88.6313i −0.172519 0.323472i
\(275\) 487.332 1.77212
\(276\) 0 0
\(277\) 418.966i 1.51251i −0.654275 0.756257i \(-0.727026\pi\)
0.654275 0.756257i \(-0.272974\pi\)
\(278\) 54.7001 + 102.562i 0.196763 + 0.368929i
\(279\) 0 0
\(280\) −330.138 33.4301i −1.17906 0.119393i
\(281\) 201.185 0.715960 0.357980 0.933729i \(-0.383466\pi\)
0.357980 + 0.933729i \(0.383466\pi\)
\(282\) 0 0
\(283\) 242.161 0.855692 0.427846 0.903852i \(-0.359273\pi\)
0.427846 + 0.903852i \(0.359273\pi\)
\(284\) −379.558 254.618i −1.33647 0.896541i
\(285\) 0 0
\(286\) 101.428 54.0951i 0.354642 0.189144i
\(287\) 69.6285i 0.242608i
\(288\) 0 0
\(289\) 663.752 2.29672
\(290\) −210.436 394.565i −0.725642 1.36057i
\(291\) 0 0
\(292\) −63.0797 + 94.0329i −0.216026 + 0.322030i
\(293\) 268.900i 0.917749i 0.888501 + 0.458874i \(0.151747\pi\)
−0.888501 + 0.458874i \(0.848253\pi\)
\(294\) 0 0
\(295\) 660.904i 2.24035i
\(296\) 51.0112 503.760i 0.172335 1.70189i
\(297\) 0 0
\(298\) −221.652 + 118.215i −0.743800 + 0.396696i
\(299\) −90.8405 −0.303814
\(300\) 0 0
\(301\) 301.151i 1.00050i
\(302\) 286.278 152.683i 0.947941 0.505572i
\(303\) 0 0
\(304\) 112.847 + 275.278i 0.371206 + 0.905521i
\(305\) 383.784 1.25831
\(306\) 0 0
\(307\) −88.1714 −0.287203 −0.143602 0.989636i \(-0.545868\pi\)
−0.143602 + 0.989636i \(0.545868\pi\)
\(308\) −158.996 106.658i −0.516219 0.346293i
\(309\) 0 0
\(310\) 103.558 + 194.170i 0.334058 + 0.626355i
\(311\) 175.687i 0.564909i −0.959281 0.282455i \(-0.908851\pi\)
0.959281 0.282455i \(-0.0911486\pi\)
\(312\) 0 0
\(313\) −298.025 −0.952157 −0.476078 0.879403i \(-0.657942\pi\)
−0.476078 + 0.879403i \(0.657942\pi\)
\(314\) 277.760 148.140i 0.884586 0.471782i
\(315\) 0 0
\(316\) 382.393 + 256.519i 1.21010 + 0.811769i
\(317\) 157.658i 0.497343i 0.968588 + 0.248671i \(0.0799939\pi\)
−0.968588 + 0.248671i \(0.920006\pi\)
\(318\) 0 0
\(319\) 258.010i 0.808809i
\(320\) −110.419 + 539.628i −0.345058 + 1.68634i
\(321\) 0 0
\(322\) 71.1996 + 133.498i 0.221117 + 0.414592i
\(323\) −573.948 −1.77693
\(324\) 0 0
\(325\) 283.982i 0.873790i
\(326\) 156.490 + 293.416i 0.480029 + 0.900049i
\(327\) 0 0
\(328\) 114.990 + 11.6440i 0.350580 + 0.0355001i
\(329\) 128.948 0.391940
\(330\) 0 0
\(331\) 171.458 0.518000 0.259000 0.965877i \(-0.416607\pi\)
0.259000 + 0.965877i \(0.416607\pi\)
\(332\) 102.980 153.513i 0.310182 0.462387i
\(333\) 0 0
\(334\) 280.075 149.374i 0.838547 0.447228i
\(335\) 762.567i 2.27632i
\(336\) 0 0
\(337\) −358.818 −1.06474 −0.532371 0.846511i \(-0.678699\pi\)
−0.532371 + 0.846511i \(0.678699\pi\)
\(338\) 127.537 + 239.130i 0.377328 + 0.707486i
\(339\) 0 0
\(340\) −882.441 591.964i −2.59541 1.74107i
\(341\) 126.970i 0.372345i
\(342\) 0 0
\(343\) 360.365i 1.05063i
\(344\) −497.346 50.3617i −1.44577 0.146400i
\(345\) 0 0
\(346\) 79.8088 42.5650i 0.230661 0.123020i
\(347\) −167.974 −0.484075 −0.242037 0.970267i \(-0.577816\pi\)
−0.242037 + 0.970267i \(0.577816\pi\)
\(348\) 0 0
\(349\) 496.148i 1.42163i 0.703380 + 0.710814i \(0.251673\pi\)
−0.703380 + 0.710814i \(0.748327\pi\)
\(350\) −417.337 + 222.581i −1.19239 + 0.635946i
\(351\) 0 0
\(352\) −202.734 + 244.742i −0.575948 + 0.695290i
\(353\) 132.906 0.376505 0.188253 0.982121i \(-0.439718\pi\)
0.188253 + 0.982121i \(0.439718\pi\)
\(354\) 0 0
\(355\) −983.388 −2.77011
\(356\) −75.4578 + 112.485i −0.211960 + 0.315969i
\(357\) 0 0
\(358\) −32.1107 60.2073i −0.0896948 0.168177i
\(359\) 565.184i 1.57433i 0.616743 + 0.787164i \(0.288452\pi\)
−0.616743 + 0.787164i \(0.711548\pi\)
\(360\) 0 0
\(361\) −15.2479 −0.0422381
\(362\) −37.8420 + 20.1825i −0.104536 + 0.0557528i
\(363\) 0 0
\(364\) −62.1527 + 92.6509i −0.170749 + 0.254536i
\(365\) 243.627i 0.667472i
\(366\) 0 0
\(367\) 615.679i 1.67760i 0.544441 + 0.838799i \(0.316742\pi\)
−0.544441 + 0.838799i \(0.683258\pi\)
\(368\) 232.377 95.2600i 0.631461 0.258859i
\(369\) 0 0
\(370\) −512.676 961.262i −1.38561 2.59801i
\(371\) −68.5088 −0.184660
\(372\) 0 0
\(373\) 455.597i 1.22144i −0.791847 0.610720i \(-0.790880\pi\)
0.791847 0.610720i \(-0.209120\pi\)
\(374\) −288.518 540.968i −0.771439 1.44644i
\(375\) 0 0
\(376\) 21.5641 212.956i 0.0573514 0.566373i
\(377\) −150.349 −0.398804
\(378\) 0 0
\(379\) −337.902 −0.891562 −0.445781 0.895142i \(-0.647074\pi\)
−0.445781 + 0.895142i \(0.647074\pi\)
\(380\) 531.592 + 356.606i 1.39893 + 0.938436i
\(381\) 0 0
\(382\) −304.957 + 162.644i −0.798316 + 0.425771i
\(383\) 218.893i 0.571522i −0.958301 0.285761i \(-0.907754\pi\)
0.958301 0.285761i \(-0.0922464\pi\)
\(384\) 0 0
\(385\) −411.937 −1.06997
\(386\) 131.963 + 247.429i 0.341872 + 0.641007i
\(387\) 0 0
\(388\) 208.585 310.937i 0.537590 0.801384i
\(389\) 142.868i 0.367270i −0.982994 0.183635i \(-0.941214\pi\)
0.982994 0.183635i \(-0.0587865\pi\)
\(390\) 0 0
\(391\) 484.501i 1.23913i
\(392\) −205.132 20.7719i −0.523296 0.0529894i
\(393\) 0 0
\(394\) 581.482 310.126i 1.47584 0.787121i
\(395\) 990.731 2.50818
\(396\) 0 0
\(397\) 709.501i 1.78716i 0.448907 + 0.893579i \(0.351814\pi\)
−0.448907 + 0.893579i \(0.648186\pi\)
\(398\) −441.028 + 235.216i −1.10811 + 0.590996i
\(399\) 0 0
\(400\) 297.798 + 726.449i 0.744495 + 1.81612i
\(401\) −157.900 −0.393766 −0.196883 0.980427i \(-0.563082\pi\)
−0.196883 + 0.980427i \(0.563082\pi\)
\(402\) 0 0
\(403\) 73.9885 0.183594
\(404\) 93.6771 + 62.8410i 0.231874 + 0.155547i
\(405\) 0 0
\(406\) 117.842 + 220.952i 0.290251 + 0.544217i
\(407\) 628.578i 1.54442i
\(408\) 0 0
\(409\) 121.735 0.297641 0.148820 0.988864i \(-0.452452\pi\)
0.148820 + 0.988864i \(0.452452\pi\)
\(410\) 219.422 117.026i 0.535175 0.285428i
\(411\) 0 0
\(412\) −185.708 124.578i −0.450748 0.302373i
\(413\) 370.099i 0.896123i
\(414\) 0 0
\(415\) 397.732i 0.958390i
\(416\) 142.618 + 118.138i 0.342831 + 0.283986i
\(417\) 0 0
\(418\) 173.806 + 325.885i 0.415805 + 0.779629i
\(419\) 24.2370 0.0578449 0.0289224 0.999582i \(-0.490792\pi\)
0.0289224 + 0.999582i \(0.490792\pi\)
\(420\) 0 0
\(421\) 280.461i 0.666177i −0.942895 0.333089i \(-0.891909\pi\)
0.942895 0.333089i \(-0.108091\pi\)
\(422\) −79.9345 149.876i −0.189418 0.355157i
\(423\) 0 0
\(424\) −11.4568 + 113.141i −0.0270208 + 0.266843i
\(425\) −1514.63 −3.56382
\(426\) 0 0
\(427\) −214.915 −0.503313
\(428\) 376.672 561.504i 0.880074 1.31193i
\(429\) 0 0
\(430\) −949.022 + 506.148i −2.20703 + 1.17709i
\(431\) 537.398i 1.24686i −0.781878 0.623432i \(-0.785738\pi\)
0.781878 0.623432i \(-0.214262\pi\)
\(432\) 0 0
\(433\) −103.584 −0.239224 −0.119612 0.992821i \(-0.538165\pi\)
−0.119612 + 0.992821i \(0.538165\pi\)
\(434\) −57.9913 108.733i −0.133620 0.250537i
\(435\) 0 0
\(436\) 281.476 + 188.822i 0.645588 + 0.433078i
\(437\) 291.868i 0.667891i
\(438\) 0 0
\(439\) 557.858i 1.27075i −0.772205 0.635373i \(-0.780846\pi\)
0.772205 0.635373i \(-0.219154\pi\)
\(440\) −68.8886 + 680.308i −0.156565 + 1.54615i
\(441\) 0 0
\(442\) −315.237 + 168.127i −0.713205 + 0.380378i
\(443\) 4.79025 0.0108132 0.00540660 0.999985i \(-0.498279\pi\)
0.00540660 + 0.999985i \(0.498279\pi\)
\(444\) 0 0
\(445\) 291.434i 0.654908i
\(446\) 3.42910 1.82886i 0.00768856 0.00410059i
\(447\) 0 0
\(448\) 61.8332 302.185i 0.138020 0.674521i
\(449\) 838.004 1.86638 0.933190 0.359384i \(-0.117013\pi\)
0.933190 + 0.359384i \(0.117013\pi\)
\(450\) 0 0
\(451\) 143.482 0.318141
\(452\) −167.644 + 249.907i −0.370894 + 0.552891i
\(453\) 0 0
\(454\) 191.679 + 359.396i 0.422200 + 0.791620i
\(455\) 240.047i 0.527575i
\(456\) 0 0
\(457\) 537.817 1.17684 0.588422 0.808554i \(-0.299750\pi\)
0.588422 + 0.808554i \(0.299750\pi\)
\(458\) 35.2876 18.8202i 0.0770471 0.0410920i
\(459\) 0 0
\(460\) 301.030 448.746i 0.654414 0.975534i
\(461\) 161.391i 0.350088i −0.984561 0.175044i \(-0.943993\pi\)
0.984561 0.175044i \(-0.0560068\pi\)
\(462\) 0 0
\(463\) 485.576i 1.04876i −0.851484 0.524381i \(-0.824297\pi\)
0.851484 0.524381i \(-0.175703\pi\)
\(464\) 384.606 157.664i 0.828892 0.339793i
\(465\) 0 0
\(466\) −85.7601 160.799i −0.184035 0.345063i
\(467\) 239.041 0.511864 0.255932 0.966695i \(-0.417618\pi\)
0.255932 + 0.966695i \(0.417618\pi\)
\(468\) 0 0
\(469\) 427.029i 0.910509i
\(470\) −216.725 406.357i −0.461118 0.864590i
\(471\) 0 0
\(472\) −611.212 61.8919i −1.29494 0.131127i
\(473\) −620.574 −1.31200
\(474\) 0 0
\(475\) 912.426 1.92090
\(476\) 494.157 + 331.493i 1.03814 + 0.696415i
\(477\) 0 0
\(478\) 65.0300 34.6829i 0.136046 0.0725583i
\(479\) 416.528i 0.869579i 0.900532 + 0.434789i \(0.143177\pi\)
−0.900532 + 0.434789i \(0.856823\pi\)
\(480\) 0 0
\(481\) −366.289 −0.761516
\(482\) 290.055 + 543.849i 0.601773 + 1.12832i
\(483\) 0 0
\(484\) 49.8428 74.3006i 0.102981 0.153514i
\(485\) 805.599i 1.66103i
\(486\) 0 0
\(487\) 404.821i 0.831254i 0.909535 + 0.415627i \(0.136438\pi\)
−0.909535 + 0.415627i \(0.863562\pi\)
\(488\) −35.9404 + 354.929i −0.0736484 + 0.727313i
\(489\) 0 0
\(490\) −391.427 + 208.763i −0.798832 + 0.426046i
\(491\) −508.547 −1.03574 −0.517869 0.855460i \(-0.673275\pi\)
−0.517869 + 0.855460i \(0.673275\pi\)
\(492\) 0 0
\(493\) 801.893i 1.62656i
\(494\) 189.902 101.282i 0.384417 0.205023i
\(495\) 0 0
\(496\) −189.269 + 77.5882i −0.381590 + 0.156428i
\(497\) 550.686 1.10802
\(498\) 0 0
\(499\) 745.549 1.49409 0.747043 0.664776i \(-0.231473\pi\)
0.747043 + 0.664776i \(0.231473\pi\)
\(500\) 688.130 + 461.615i 1.37626 + 0.923231i
\(501\) 0 0
\(502\) −214.903 402.941i −0.428094 0.802672i
\(503\) 290.768i 0.578068i −0.957319 0.289034i \(-0.906666\pi\)
0.957319 0.289034i \(-0.0933341\pi\)
\(504\) 0 0
\(505\) 242.705 0.480605
\(506\) 275.097 146.719i 0.543670 0.289959i
\(507\) 0 0
\(508\) 194.170 + 130.254i 0.382224 + 0.256406i
\(509\) 215.775i 0.423920i −0.977278 0.211960i \(-0.932015\pi\)
0.977278 0.211960i \(-0.0679846\pi\)
\(510\) 0 0
\(511\) 136.428i 0.266983i
\(512\) −488.714 152.651i −0.954520 0.298147i
\(513\) 0 0
\(514\) 275.074 + 515.761i 0.535164 + 1.00343i
\(515\) −481.146 −0.934264
\(516\) 0 0
\(517\) 265.721i 0.513967i
\(518\) 287.093 + 538.296i 0.554233 + 1.03918i
\(519\) 0 0
\(520\) 396.434 + 40.1433i 0.762373 + 0.0771986i
\(521\) −747.187 −1.43414 −0.717070 0.697001i \(-0.754517\pi\)
−0.717070 + 0.697001i \(0.754517\pi\)
\(522\) 0 0
\(523\) 289.683 0.553886 0.276943 0.960886i \(-0.410679\pi\)
0.276943 + 0.960886i \(0.410679\pi\)
\(524\) 256.488 382.346i 0.489481 0.729668i
\(525\) 0 0
\(526\) 861.586 459.515i 1.63800 0.873604i
\(527\) 394.620i 0.748805i
\(528\) 0 0
\(529\) 282.618 0.534250
\(530\) 115.144 + 215.893i 0.217252 + 0.407346i
\(531\) 0 0
\(532\) −297.685 199.695i −0.559559 0.375367i
\(533\) 83.6107i 0.156868i
\(534\) 0 0
\(535\) 1454.79i 2.71923i
\(536\) −705.231 71.4124i −1.31573 0.133232i
\(537\) 0 0
\(538\) −68.4869 + 36.5266i −0.127299 + 0.0678933i
\(539\) −255.958 −0.474876
\(540\) 0 0
\(541\) 348.783i 0.644700i 0.946621 + 0.322350i \(0.104473\pi\)
−0.946621 + 0.322350i \(0.895527\pi\)
\(542\) −578.800 + 308.695i −1.06790 + 0.569548i
\(543\) 0 0
\(544\) 630.095 760.657i 1.15826 1.39827i
\(545\) 729.270 1.33811
\(546\) 0 0
\(547\) −730.849 −1.33610 −0.668052 0.744115i \(-0.732872\pi\)
−0.668052 + 0.744115i \(0.732872\pi\)
\(548\) 111.918 166.836i 0.204230 0.304446i
\(549\) 0 0
\(550\) 458.668 + 859.997i 0.833942 + 1.56363i
\(551\) 483.069i 0.876712i
\(552\) 0 0
\(553\) −554.798 −1.00325
\(554\) 739.351 394.323i 1.33457 0.711774i
\(555\) 0 0
\(556\) −129.509 + 193.059i −0.232930 + 0.347228i
\(557\) 462.185i 0.829776i 0.909872 + 0.414888i \(0.136179\pi\)
−0.909872 + 0.414888i \(0.863821\pi\)
\(558\) 0 0
\(559\) 361.625i 0.646914i
\(560\) −251.726 614.060i −0.449510 1.09654i
\(561\) 0 0
\(562\) 189.351 + 355.032i 0.336924 + 0.631729i
\(563\) −161.225 −0.286367 −0.143184 0.989696i \(-0.545734\pi\)
−0.143184 + 0.989696i \(0.545734\pi\)
\(564\) 0 0
\(565\) 647.476i 1.14598i
\(566\) 227.917 + 427.342i 0.402680 + 0.755021i
\(567\) 0 0
\(568\) 92.0917 909.449i 0.162133 1.60114i
\(569\) −63.2844 −0.111220 −0.0556102 0.998453i \(-0.517710\pi\)
−0.0556102 + 0.998453i \(0.517710\pi\)
\(570\) 0 0
\(571\) 268.584 0.470375 0.235188 0.971950i \(-0.424430\pi\)
0.235188 + 0.971950i \(0.424430\pi\)
\(572\) 190.924 + 128.077i 0.333783 + 0.223910i
\(573\) 0 0
\(574\) −122.874 + 65.5330i −0.214066 + 0.114169i
\(575\) 770.229i 1.33953i
\(576\) 0 0
\(577\) −989.322 −1.71460 −0.857298 0.514821i \(-0.827858\pi\)
−0.857298 + 0.514821i \(0.827858\pi\)
\(578\) 624.710 + 1171.33i 1.08081 + 2.02651i
\(579\) 0 0
\(580\) 498.233 742.715i 0.859022 1.28054i
\(581\) 222.725i 0.383348i
\(582\) 0 0
\(583\) 141.175i 0.242152i
\(584\) −225.310 22.8151i −0.385804 0.0390669i
\(585\) 0 0
\(586\) −474.529 + 253.084i −0.809777 + 0.431884i
\(587\) −856.358 −1.45887 −0.729436 0.684049i \(-0.760217\pi\)
−0.729436 + 0.684049i \(0.760217\pi\)
\(588\) 0 0
\(589\) 237.723i 0.403605i
\(590\) −1166.30 + 622.030i −1.97678 + 1.05429i
\(591\) 0 0
\(592\) 936.998 384.110i 1.58277 0.648834i
\(593\) 108.067 0.182238 0.0911189 0.995840i \(-0.470956\pi\)
0.0911189 + 0.995840i \(0.470956\pi\)
\(594\) 0 0
\(595\) 1280.30 2.15176
\(596\) −417.230 279.889i −0.700050 0.469612i
\(597\) 0 0
\(598\) −85.4973 160.306i −0.142972 0.268071i
\(599\) 556.750i 0.929466i 0.885451 + 0.464733i \(0.153850\pi\)
−0.885451 + 0.464733i \(0.846150\pi\)
\(600\) 0 0
\(601\) 260.078 0.432742 0.216371 0.976311i \(-0.430578\pi\)
0.216371 + 0.976311i \(0.430578\pi\)
\(602\) 531.441 283.437i 0.882793 0.470826i
\(603\) 0 0
\(604\) 538.879 + 361.494i 0.892184 + 0.598500i
\(605\) 192.503i 0.318187i
\(606\) 0 0
\(607\) 8.91926i 0.0146940i −0.999973 0.00734701i \(-0.997661\pi\)
0.999973 0.00734701i \(-0.00233865\pi\)
\(608\) −379.576 + 458.228i −0.624302 + 0.753664i
\(609\) 0 0
\(610\) 361.210 + 677.265i 0.592148 + 1.11027i
\(611\) −154.843 −0.253425
\(612\) 0 0
\(613\) 738.776i 1.20518i 0.798051 + 0.602590i \(0.205865\pi\)
−0.798051 + 0.602590i \(0.794135\pi\)
\(614\) −82.9852 155.596i −0.135155 0.253414i
\(615\) 0 0
\(616\) 38.5769 380.965i 0.0626248 0.618449i
\(617\) 357.548 0.579495 0.289747 0.957103i \(-0.406429\pi\)
0.289747 + 0.957103i \(0.406429\pi\)
\(618\) 0 0
\(619\) −408.256 −0.659541 −0.329770 0.944061i \(-0.606971\pi\)
−0.329770 + 0.944061i \(0.606971\pi\)
\(620\) −245.186 + 365.498i −0.395461 + 0.589513i
\(621\) 0 0
\(622\) 310.035 165.353i 0.498449 0.265841i
\(623\) 163.200i 0.261958i
\(624\) 0 0
\(625\) 556.109 0.889774
\(626\) −280.495 525.926i −0.448076 0.840137i
\(627\) 0 0
\(628\) 522.845 + 350.738i 0.832555 + 0.558500i
\(629\) 1953.62i 3.10591i
\(630\) 0 0
\(631\) 230.148i 0.364735i −0.983230 0.182367i \(-0.941624\pi\)
0.983230 0.182367i \(-0.0583760\pi\)
\(632\) −92.7794 + 916.241i −0.146803 + 1.44975i
\(633\) 0 0
\(634\) −278.219 + 148.384i −0.438831 + 0.234045i
\(635\) 503.069 0.792235
\(636\) 0 0
\(637\) 149.154i 0.234150i
\(638\) 455.311 242.834i 0.713654 0.380618i
\(639\) 0 0
\(640\) −1056.21 + 313.031i −1.65032 + 0.489111i
\(641\) 736.093 1.14835 0.574176 0.818732i \(-0.305323\pi\)
0.574176 + 0.818732i \(0.305323\pi\)
\(642\) 0 0
\(643\) 21.9654 0.0341609 0.0170804 0.999854i \(-0.494563\pi\)
0.0170804 + 0.999854i \(0.494563\pi\)
\(644\) −168.574 + 251.292i −0.261760 + 0.390206i
\(645\) 0 0
\(646\) −540.188 1012.85i −0.836205 1.56788i
\(647\) 493.227i 0.762329i 0.924507 + 0.381164i \(0.124477\pi\)
−0.924507 + 0.381164i \(0.875523\pi\)
\(648\) 0 0
\(649\) −762.654 −1.17512
\(650\) 501.143 267.278i 0.770990 0.411197i
\(651\) 0 0
\(652\) −370.508 + 552.315i −0.568263 + 0.847109i
\(653\) 1024.37i 1.56871i −0.620310 0.784356i \(-0.712993\pi\)
0.620310 0.784356i \(-0.287007\pi\)
\(654\) 0 0
\(655\) 990.610i 1.51238i
\(656\) 87.6785 + 213.883i 0.133656 + 0.326041i
\(657\) 0 0
\(658\) 121.364 + 227.556i 0.184443 + 0.345829i
\(659\) 759.973 1.15322 0.576611 0.817019i \(-0.304375\pi\)
0.576611 + 0.817019i \(0.304375\pi\)
\(660\) 0 0
\(661\) 215.108i 0.325427i 0.986673 + 0.162714i \(0.0520247\pi\)
−0.986673 + 0.162714i \(0.947975\pi\)
\(662\) 161.373 + 302.573i 0.243766 + 0.457058i
\(663\) 0 0
\(664\) 367.827 + 37.2465i 0.553957 + 0.0560942i
\(665\) −771.265 −1.15980
\(666\) 0 0
\(667\) −407.785 −0.611371
\(668\) 527.202 + 353.661i 0.789224 + 0.529432i
\(669\) 0 0
\(670\) −1345.70 + 717.713i −2.00851 + 1.07121i
\(671\) 442.870i 0.660015i
\(672\) 0 0
\(673\) 475.353 0.706320 0.353160 0.935563i \(-0.385107\pi\)
0.353160 + 0.935563i \(0.385107\pi\)
\(674\) −337.713 633.207i −0.501057 0.939477i
\(675\) 0 0
\(676\) −301.959 + 450.130i −0.446685 + 0.665872i
\(677\) 47.6348i 0.0703615i 0.999381 + 0.0351808i \(0.0112007\pi\)
−0.999381 + 0.0351808i \(0.988799\pi\)
\(678\) 0 0
\(679\) 451.126i 0.664398i
\(680\) 214.105 2114.39i 0.314861 3.10940i
\(681\) 0 0
\(682\) −224.064 + 119.501i −0.328539 + 0.175222i
\(683\) −857.260 −1.25514 −0.627570 0.778561i \(-0.715950\pi\)
−0.627570 + 0.778561i \(0.715950\pi\)
\(684\) 0 0
\(685\) 432.251i 0.631024i
\(686\) 635.938 339.169i 0.927023 0.494415i
\(687\) 0 0
\(688\) −379.219 925.067i −0.551190 1.34457i
\(689\) 82.2662 0.119399
\(690\) 0 0
\(691\) −437.965 −0.633813 −0.316906 0.948457i \(-0.602644\pi\)
−0.316906 + 0.948457i \(0.602644\pi\)
\(692\) 150.229 + 100.778i 0.217094 + 0.145632i
\(693\) 0 0
\(694\) −158.094 296.424i −0.227801 0.427124i
\(695\) 500.191i 0.719700i
\(696\) 0 0
\(697\) −445.940 −0.639799
\(698\) −875.554 + 466.965i −1.25438 + 0.669004i
\(699\) 0 0
\(700\) −785.580 526.987i −1.12226 0.752839i
\(701\) 4.79962i 0.00684682i 0.999994 + 0.00342341i \(0.00108971\pi\)
−0.999994 + 0.00342341i \(0.998910\pi\)
\(702\) 0 0
\(703\) 1176.88i 1.67408i
\(704\) −622.706 127.418i −0.884526 0.180992i
\(705\) 0 0
\(706\) 125.089 + 234.540i 0.177180 + 0.332210i
\(707\) −135.912 −0.192238
\(708\) 0 0
\(709\) 117.092i 0.165151i −0.996585 0.0825753i \(-0.973685\pi\)
0.996585 0.0825753i \(-0.0263145\pi\)
\(710\) −925.546 1735.39i −1.30359 2.44421i
\(711\) 0 0
\(712\) −269.522 27.2921i −0.378542 0.0383315i
\(713\) 200.675 0.281452
\(714\) 0 0
\(715\) 494.659 0.691831
\(716\) 76.0260 113.332i 0.106182 0.158285i
\(717\) 0 0
\(718\) −997.382 + 531.940i −1.38911 + 0.740864i
\(719\) 430.448i 0.598676i −0.954147 0.299338i \(-0.903234\pi\)
0.954147 0.299338i \(-0.0967658\pi\)
\(720\) 0 0
\(721\) 269.436 0.373698
\(722\) −14.3511 26.9081i −0.0198768 0.0372688i
\(723\) 0 0
\(724\) −71.2323 47.7845i −0.0983871 0.0660007i
\(725\) 1274.80i 1.75834i
\(726\) 0 0
\(727\) 1080.06i 1.48564i −0.669493 0.742818i \(-0.733489\pi\)
0.669493 0.742818i \(-0.266511\pi\)
\(728\) −221.998 22.4798i −0.304943 0.0308788i
\(729\) 0 0
\(730\) −429.930 + 229.297i −0.588945 + 0.314106i
\(731\) 1928.74 2.63849
\(732\) 0 0
\(733\) 756.874i 1.03257i 0.856417 + 0.516285i \(0.172685\pi\)
−0.856417 + 0.516285i \(0.827315\pi\)
\(734\) −1086.49 + 579.465i −1.48023 + 0.789462i
\(735\) 0 0
\(736\) 386.815 + 320.420i 0.525564 + 0.435354i
\(737\) −879.968 −1.19399
\(738\) 0 0
\(739\) 269.971 0.365319 0.182659 0.983176i \(-0.441529\pi\)
0.182659 + 0.983176i \(0.441529\pi\)
\(740\) 1213.82 1809.44i 1.64030 2.44519i
\(741\) 0 0
\(742\) −64.4792 120.898i −0.0868992 0.162935i
\(743\) 412.457i 0.555123i 0.960708 + 0.277562i \(0.0895263\pi\)
−0.960708 + 0.277562i \(0.910474\pi\)
\(744\) 0 0
\(745\) −1080.99 −1.45099
\(746\) 803.994 428.799i 1.07774 0.574798i
\(747\) 0 0
\(748\) 683.101 1018.30i 0.913236 1.36136i
\(749\) 814.664i 1.08767i
\(750\) 0 0
\(751\) 982.058i 1.30767i 0.756639 + 0.653833i \(0.226840\pi\)
−0.756639 + 0.653833i \(0.773160\pi\)
\(752\) 396.100 162.376i 0.526729 0.215925i
\(753\) 0 0
\(754\) −141.506 265.322i −0.187674 0.351886i
\(755\) 1396.17 1.84923
\(756\) 0 0
\(757\) 96.0126i 0.126833i 0.997987 + 0.0634165i \(0.0201997\pi\)
−0.997987 + 0.0634165i \(0.979800\pi\)
\(758\) −318.027 596.297i −0.419561 0.786671i
\(759\) 0 0
\(760\) −128.979 + 1273.73i −0.169710 + 1.67596i
\(761\) −483.966 −0.635960 −0.317980 0.948097i \(-0.603005\pi\)
−0.317980 + 0.948097i \(0.603005\pi\)
\(762\) 0 0
\(763\) −408.383 −0.535233
\(764\) −574.039 385.080i −0.751359 0.504032i
\(765\) 0 0
\(766\) 386.281 206.018i 0.504284 0.268953i
\(767\) 444.419i 0.579425i
\(768\) 0 0
\(769\) 395.882 0.514801 0.257400 0.966305i \(-0.417134\pi\)
0.257400 + 0.966305i \(0.417134\pi\)
\(770\) −387.707 726.947i −0.503516 0.944087i
\(771\) 0 0
\(772\) −312.437 + 465.750i −0.404712 + 0.603303i
\(773\) 51.0498i 0.0660411i −0.999455 0.0330206i \(-0.989487\pi\)
0.999455 0.0330206i \(-0.0105127\pi\)
\(774\) 0 0
\(775\) 627.343i 0.809474i
\(776\) 745.028 + 75.4422i 0.960087 + 0.0972194i
\(777\) 0 0
\(778\) 252.120 134.465i 0.324062 0.172834i
\(779\) 268.639 0.344851
\(780\) 0 0
\(781\) 1134.79i 1.45299i
\(782\) −855.000 + 456.003i −1.09335 + 0.583124i
\(783\) 0 0
\(784\) −156.410 381.547i −0.199503 0.486667i
\(785\) 1354.62 1.72564
\(786\) 0 0
\(787\) −973.748 −1.23729 −0.618646 0.785670i \(-0.712318\pi\)
−0.618646 + 0.785670i \(0.712318\pi\)
\(788\) 1094.56 + 734.260i 1.38904 + 0.931801i
\(789\) 0 0
\(790\) 932.457 + 1748.35i 1.18033 + 2.21310i
\(791\) 362.579i 0.458381i
\(792\) 0 0
\(793\) 258.072 0.325438
\(794\) −1252.06 + 667.769i −1.57690 + 0.841019i
\(795\) 0 0
\(796\) −830.174 556.903i −1.04293 0.699627i
\(797\) 849.870i 1.06634i 0.846009 + 0.533168i \(0.178999\pi\)
−0.846009 + 0.533168i \(0.821001\pi\)
\(798\) 0 0
\(799\) 825.858i 1.03361i
\(800\) −1001.69 + 1209.24i −1.25211 + 1.51156i
\(801\) 0 0
\(802\) −148.613 278.647i −0.185303 0.347441i
\(803\) −281.135 −0.350106
\(804\) 0 0
\(805\) 651.067i 0.808778i
\(806\) 69.6366 + 130.568i 0.0863977 + 0.161995i
\(807\) 0 0
\(808\) −22.7287 + 224.457i −0.0281296 + 0.277793i
\(809\) −396.410 −0.490000 −0.245000 0.969523i \(-0.578788\pi\)
−0.245000 + 0.969523i \(0.578788\pi\)
\(810\) 0 0
\(811\) −1531.93 −1.88894 −0.944470 0.328598i \(-0.893424\pi\)
−0.944470 + 0.328598i \(0.893424\pi\)
\(812\) −279.005 + 415.912i −0.343602 + 0.512207i
\(813\) 0 0
\(814\) 1109.25 591.606i 1.36272 0.726788i
\(815\) 1430.98i 1.75580i
\(816\) 0 0
\(817\) −1161.89 −1.42214
\(818\) 114.575 + 214.826i 0.140067 + 0.262624i
\(819\) 0 0
\(820\) 413.031 + 277.072i 0.503696 + 0.337893i
\(821\) 790.420i 0.962752i −0.876514 0.481376i \(-0.840137\pi\)
0.876514 0.481376i \(-0.159863\pi\)
\(822\) 0 0
\(823\) 1061.92i 1.29031i 0.764054 + 0.645153i \(0.223206\pi\)
−0.764054 + 0.645153i \(0.776794\pi\)
\(824\) 45.0581 444.970i 0.0546821 0.540012i
\(825\) 0 0
\(826\) 653.114 348.330i 0.790695 0.421707i
\(827\) 575.872 0.696339 0.348170 0.937432i \(-0.386803\pi\)
0.348170 + 0.937432i \(0.386803\pi\)
\(828\) 0 0
\(829\) 849.875i 1.02518i −0.858633 0.512590i \(-0.828686\pi\)
0.858633 0.512590i \(-0.171314\pi\)
\(830\) 701.879 374.337i 0.845637 0.451009i
\(831\) 0 0
\(832\) −74.2500 + 362.868i −0.0892428 + 0.436139i
\(833\) 795.515 0.955000
\(834\) 0 0
\(835\) 1365.91 1.63582
\(836\) −411.507 + 613.433i −0.492233 + 0.733772i
\(837\) 0 0
\(838\) 22.8114 + 42.7711i 0.0272212 + 0.0510395i
\(839\) 1187.72i 1.41564i 0.706394 + 0.707819i \(0.250321\pi\)
−0.706394 + 0.707819i \(0.749679\pi\)
\(840\) 0 0
\(841\) 166.079 0.197478
\(842\) 494.930 263.964i 0.587803 0.313497i
\(843\) 0 0
\(844\) 189.254 282.121i 0.224235 0.334267i
\(845\) 1166.23i 1.38015i
\(846\) 0 0
\(847\) 107.800i 0.127272i
\(848\) −210.444 + 86.2686i −0.248165 + 0.101732i
\(849\) 0 0
\(850\) −1425.54 2672.86i −1.67710 3.14455i
\(851\) −993.467 −1.16741
\(852\) 0 0
\(853\) 941.579i 1.10384i 0.833896 + 0.551922i \(0.186105\pi\)
−0.833896 + 0.551922i \(0.813895\pi\)
\(854\) −202.274 379.261i −0.236854 0.444099i
\(855\) 0 0
\(856\) 1345.41 + 136.237i 1.57173 + 0.159155i
\(857\) −1094.02 −1.27657 −0.638284 0.769801i \(-0.720355\pi\)
−0.638284 + 0.769801i \(0.720355\pi\)
\(858\) 0 0
\(859\) 811.069 0.944201 0.472101 0.881545i \(-0.343496\pi\)
0.472101 + 0.881545i \(0.343496\pi\)
\(860\) −1786.40 1198.37i −2.07721 1.39345i
\(861\) 0 0
\(862\) 948.348 505.789i 1.10017 0.586762i
\(863\) 64.9665i 0.0752799i −0.999291 0.0376399i \(-0.988016\pi\)
0.999291 0.0376399i \(-0.0119840\pi\)
\(864\) 0 0
\(865\) 389.224 0.449970
\(866\) −97.4912 182.795i −0.112576 0.211079i
\(867\) 0 0
\(868\) 137.301 204.675i 0.158181 0.235800i
\(869\) 1143.26i 1.31560i
\(870\) 0 0
\(871\) 512.781i 0.588727i
\(872\) −68.2942 + 674.438i −0.0783191 + 0.773438i
\(873\) 0 0
\(874\) 515.061 274.701i 0.589314 0.314303i
\(875\) −998.379 −1.14100
\(876\) 0 0
\(877\) 180.847i 0.206211i −0.994670 0.103106i \(-0.967122\pi\)
0.994670 0.103106i \(-0.0328779\pi\)
\(878\) 984.454 525.045i 1.12125 0.598001i
\(879\) 0 0
\(880\) −1265.38 + 518.725i −1.43793 + 0.589460i
\(881\) 896.400 1.01748 0.508740 0.860920i \(-0.330112\pi\)
0.508740 + 0.860920i \(0.330112\pi\)
\(882\) 0 0
\(883\) 490.293 0.555258 0.277629 0.960688i \(-0.410451\pi\)
0.277629 + 0.960688i \(0.410451\pi\)
\(884\) −593.389 398.061i −0.671255 0.450295i
\(885\) 0 0
\(886\) 4.50849 + 8.45336i 0.00508859 + 0.00954104i
\(887\) 450.485i 0.507875i 0.967221 + 0.253938i \(0.0817258\pi\)
−0.967221 + 0.253938i \(0.918274\pi\)
\(888\) 0 0
\(889\) −281.713 −0.316888
\(890\) −514.295 + 274.292i −0.577860 + 0.308194i
\(891\) 0 0
\(892\) 6.45480 + 4.33005i 0.00723632 + 0.00485432i
\(893\) 497.506i 0.557117i
\(894\) 0 0
\(895\) 293.629i 0.328077i
\(896\) 591.464 175.294i 0.660116 0.195640i
\(897\) 0 0
\(898\) 788.714 + 1478.83i 0.878300 + 1.64680i
\(899\) 332.136 0.369450
\(900\) 0 0
\(901\) 438.769i 0.486981i
\(902\) 135.042 + 253.203i 0.149714 + 0.280713i
\(903\) 0 0
\(904\) −598.794 60.6345i −0.662383 0.0670735i
\(905\) −184.554 −0.203927
\(906\) 0 0
\(907\) 113.210 0.124818 0.0624091 0.998051i \(-0.480122\pi\)
0.0624091 + 0.998051i \(0.480122\pi\)
\(908\) −453.822 + 676.513i −0.499804 + 0.745058i
\(909\) 0 0
\(910\) −423.611 + 225.927i −0.465507 + 0.248272i
\(911\) 1580.60i 1.73502i −0.497423 0.867508i \(-0.665720\pi\)
0.497423 0.867508i \(-0.334280\pi\)
\(912\) 0 0
\(913\) 458.965 0.502700
\(914\) 506.183 + 949.088i 0.553811 + 1.03839i
\(915\) 0 0
\(916\) 66.4240 + 44.5589i 0.0725153 + 0.0486451i
\(917\) 554.730i 0.604941i
\(918\) 0 0
\(919\) 846.401i 0.921002i −0.887659 0.460501i \(-0.847670\pi\)
0.887659 0.460501i \(-0.152330\pi\)
\(920\) 1075.23 + 108.878i 1.16872 + 0.118346i
\(921\) 0 0
\(922\) 284.807 151.898i 0.308901 0.164748i
\(923\) −661.270 −0.716435
\(924\) 0 0
\(925\) 3105.74i 3.35755i
\(926\) 856.898 457.015i 0.925376 0.493537i
\(927\) 0 0
\(928\) 640.214 + 530.325i 0.689886 + 0.571471i
\(929\) 579.755 0.624063 0.312032 0.950072i \(-0.398991\pi\)
0.312032 + 0.950072i \(0.398991\pi\)
\(930\) 0 0
\(931\) −479.227 −0.514744
\(932\) 203.047 302.682i 0.217862 0.324767i
\(933\) 0 0
\(934\) 224.980 + 421.836i 0.240878 + 0.451644i
\(935\) 2638.28i 2.82169i
\(936\) 0 0
\(937\) 1810.65 1.93239 0.966196 0.257808i \(-0.0830002\pi\)
0.966196 + 0.257808i \(0.0830002\pi\)
\(938\) 753.579 401.911i 0.803389 0.428477i
\(939\) 0 0
\(940\) 513.123 764.912i 0.545875 0.813736i
\(941\) 1154.89i 1.22730i −0.789579 0.613649i \(-0.789701\pi\)
0.789579 0.613649i \(-0.210299\pi\)
\(942\) 0 0
\(943\) 226.773i 0.240480i
\(944\) −466.040 1136.86i −0.493687 1.20430i
\(945\) 0 0
\(946\) −584.072 1095.13i −0.617413 1.15764i
\(947\) 1.41679 0.00149608 0.000748039 1.00000i \(-0.499762\pi\)
0.000748039 1.00000i \(0.499762\pi\)
\(948\) 0 0
\(949\) 163.825i 0.172629i
\(950\) 858.758 + 1610.16i 0.903956 + 1.69491i
\(951\) 0 0
\(952\) −119.897 + 1184.04i −0.125942 + 1.24373i
\(953\) −662.875 −0.695566 −0.347783 0.937575i \(-0.613065\pi\)
−0.347783 + 0.937575i \(0.613065\pi\)
\(954\) 0 0
\(955\) −1487.26 −1.55734
\(956\) 122.410 + 82.1158i 0.128044 + 0.0858952i
\(957\) 0 0
\(958\) −735.049 + 392.028i −0.767274 + 0.409215i
\(959\) 242.056i 0.252404i
\(960\) 0 0
\(961\) 797.552 0.829919
\(962\) −344.744 646.392i −0.358362 0.671925i
\(963\) 0 0
\(964\) −686.739 + 1023.72i −0.712385 + 1.06195i
\(965\) 1206.70i 1.25047i
\(966\) 0 0
\(967\) 843.335i 0.872114i 0.899919 + 0.436057i \(0.143625\pi\)
−0.899919 + 0.436057i \(0.856375\pi\)
\(968\) 178.030 + 18.0274i 0.183915 + 0.0186234i
\(969\) 0 0
\(970\) 1421.64 758.214i 1.46561 0.781664i
\(971\) −289.643 −0.298293 −0.149147 0.988815i \(-0.547653\pi\)
−0.149147 + 0.988815i \(0.547653\pi\)
\(972\) 0 0
\(973\) 280.101i 0.287874i
\(974\) −714.388 + 381.009i −0.733458 + 0.391180i
\(975\) 0 0
\(976\) −660.170 + 270.628i −0.676404 + 0.277283i
\(977\) 1403.92 1.43697 0.718484 0.695544i \(-0.244836\pi\)
0.718484 + 0.695544i \(0.244836\pi\)
\(978\) 0 0
\(979\) −336.302 −0.343516
\(980\) −736.808 494.270i −0.751845 0.504357i
\(981\) 0 0
\(982\) −478.635 897.435i −0.487408 0.913885i
\(983\) 531.529i 0.540722i 0.962759 + 0.270361i \(0.0871430\pi\)
−0.962759 + 0.270361i \(0.912857\pi\)
\(984\) 0 0
\(985\) 2835.87 2.87905
\(986\) −1415.10 + 754.726i −1.43520 + 0.765442i
\(987\) 0 0
\(988\) 357.464 + 239.796i 0.361806 + 0.242709i
\(989\) 980.817i 0.991726i
\(990\) 0 0
\(991\) 1549.03i 1.56310i 0.623846 + 0.781548i \(0.285569\pi\)
−0.623846 + 0.781548i \(0.714431\pi\)
\(992\) −315.056 260.979i −0.317597 0.263084i
\(993\) 0 0
\(994\) 518.295 + 971.797i 0.521423 + 0.977663i
\(995\) −2150.88 −2.16168
\(996\) 0 0
\(997\) 411.253i 0.412491i −0.978500 0.206245i \(-0.933876\pi\)
0.978500 0.206245i \(-0.0661245\pi\)
\(998\) 701.696 + 1315.67i 0.703102 + 1.31831i
\(999\) 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.b.a.163.12 yes 16
3.2 odd 2 inner 216.3.b.a.163.5 16
4.3 odd 2 864.3.b.a.271.16 16
8.3 odd 2 inner 216.3.b.a.163.11 yes 16
8.5 even 2 864.3.b.a.271.1 16
12.11 even 2 864.3.b.a.271.2 16
24.5 odd 2 864.3.b.a.271.15 16
24.11 even 2 inner 216.3.b.a.163.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.b.a.163.5 16 3.2 odd 2 inner
216.3.b.a.163.6 yes 16 24.11 even 2 inner
216.3.b.a.163.11 yes 16 8.3 odd 2 inner
216.3.b.a.163.12 yes 16 1.1 even 1 trivial
864.3.b.a.271.1 16 8.5 even 2
864.3.b.a.271.2 16 12.11 even 2
864.3.b.a.271.15 16 24.5 odd 2
864.3.b.a.271.16 16 4.3 odd 2