Properties

Label 27.7.b.b.26.2
Level $27$
Weight $7$
Character 27.26
Analytic conductor $6.211$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,7,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 27.26
Dual form 27.7.b.b.26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.00000i q^{2} +28.0000 q^{4} -240.000i q^{5} +299.000 q^{7} +552.000i q^{8} +O(q^{10})\) \(q+6.00000i q^{2} +28.0000 q^{4} -240.000i q^{5} +299.000 q^{7} +552.000i q^{8} +1440.00 q^{10} -624.000i q^{11} +2495.00 q^{13} +1794.00i q^{14} -1520.00 q^{16} +1872.00i q^{17} -2509.00 q^{19} -6720.00i q^{20} +3744.00 q^{22} -14352.0i q^{23} -41975.0 q^{25} +14970.0i q^{26} +8372.00 q^{28} +23712.0i q^{29} +5330.00 q^{31} +26208.0i q^{32} -11232.0 q^{34} -71760.0i q^{35} +32591.0 q^{37} -15054.0i q^{38} +132480. q^{40} +66144.0i q^{41} -70630.0 q^{43} -17472.0i q^{44} +86112.0 q^{46} +3984.00i q^{47} -28248.0 q^{49} -251850. i q^{50} +69860.0 q^{52} +190944. i q^{53} -149760. q^{55} +165048. i q^{56} -142272. q^{58} +237360. i q^{59} -61801.0 q^{61} +31980.0i q^{62} -254528. q^{64} -598800. i q^{65} -430261. q^{67} +52416.0i q^{68} +430560. q^{70} -251712. i q^{71} +251615. q^{73} +195546. i q^{74} -70252.0 q^{76} -186576. i q^{77} +660827. q^{79} +364800. i q^{80} -396864. q^{82} +797856. i q^{83} +449280. q^{85} -423780. i q^{86} +344448. q^{88} -270576. i q^{89} +746005. q^{91} -401856. i q^{92} -23904.0 q^{94} +602160. i q^{95} +220727. q^{97} -169488. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} + 598 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{4} + 598 q^{7} + 2880 q^{10} + 4990 q^{13} - 3040 q^{16} - 5018 q^{19} + 7488 q^{22} - 83950 q^{25} + 16744 q^{28} + 10660 q^{31} - 22464 q^{34} + 65182 q^{37} + 264960 q^{40} - 141260 q^{43} + 172224 q^{46} - 56496 q^{49} + 139720 q^{52} - 299520 q^{55} - 284544 q^{58} - 123602 q^{61} - 509056 q^{64} - 860522 q^{67} + 861120 q^{70} + 503230 q^{73} - 140504 q^{76} + 1321654 q^{79} - 793728 q^{82} + 898560 q^{85} + 688896 q^{88} + 1492010 q^{91} - 47808 q^{94} + 441454 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.00000i 0.750000i 0.927025 + 0.375000i \(0.122357\pi\)
−0.927025 + 0.375000i \(0.877643\pi\)
\(3\) 0 0
\(4\) 28.0000 0.437500
\(5\) − 240.000i − 1.92000i −0.280000 0.960000i \(-0.590334\pi\)
0.280000 0.960000i \(-0.409666\pi\)
\(6\) 0 0
\(7\) 299.000 0.871720 0.435860 0.900014i \(-0.356444\pi\)
0.435860 + 0.900014i \(0.356444\pi\)
\(8\) 552.000i 1.07812i
\(9\) 0 0
\(10\) 1440.00 1.44000
\(11\) − 624.000i − 0.468820i −0.972138 0.234410i \(-0.924684\pi\)
0.972138 0.234410i \(-0.0753159\pi\)
\(12\) 0 0
\(13\) 2495.00 1.13564 0.567820 0.823153i \(-0.307787\pi\)
0.567820 + 0.823153i \(0.307787\pi\)
\(14\) 1794.00i 0.653790i
\(15\) 0 0
\(16\) −1520.00 −0.371094
\(17\) 1872.00i 0.381030i 0.981684 + 0.190515i \(0.0610158\pi\)
−0.981684 + 0.190515i \(0.938984\pi\)
\(18\) 0 0
\(19\) −2509.00 −0.365797 −0.182898 0.983132i \(-0.558548\pi\)
−0.182898 + 0.983132i \(0.558548\pi\)
\(20\) − 6720.00i − 0.840000i
\(21\) 0 0
\(22\) 3744.00 0.351615
\(23\) − 14352.0i − 1.17958i −0.807555 0.589792i \(-0.799210\pi\)
0.807555 0.589792i \(-0.200790\pi\)
\(24\) 0 0
\(25\) −41975.0 −2.68640
\(26\) 14970.0i 0.851730i
\(27\) 0 0
\(28\) 8372.00 0.381378
\(29\) 23712.0i 0.972242i 0.873892 + 0.486121i \(0.161588\pi\)
−0.873892 + 0.486121i \(0.838412\pi\)
\(30\) 0 0
\(31\) 5330.00 0.178913 0.0894565 0.995991i \(-0.471487\pi\)
0.0894565 + 0.995991i \(0.471487\pi\)
\(32\) 26208.0i 0.799805i
\(33\) 0 0
\(34\) −11232.0 −0.285772
\(35\) − 71760.0i − 1.67370i
\(36\) 0 0
\(37\) 32591.0 0.643417 0.321708 0.946839i \(-0.395743\pi\)
0.321708 + 0.946839i \(0.395743\pi\)
\(38\) − 15054.0i − 0.274348i
\(39\) 0 0
\(40\) 132480. 2.07000
\(41\) 66144.0i 0.959707i 0.877348 + 0.479854i \(0.159310\pi\)
−0.877348 + 0.479854i \(0.840690\pi\)
\(42\) 0 0
\(43\) −70630.0 −0.888349 −0.444175 0.895940i \(-0.646503\pi\)
−0.444175 + 0.895940i \(0.646503\pi\)
\(44\) − 17472.0i − 0.205109i
\(45\) 0 0
\(46\) 86112.0 0.884688
\(47\) 3984.00i 0.0383730i 0.999816 + 0.0191865i \(0.00610763\pi\)
−0.999816 + 0.0191865i \(0.993892\pi\)
\(48\) 0 0
\(49\) −28248.0 −0.240104
\(50\) − 251850.i − 2.01480i
\(51\) 0 0
\(52\) 69860.0 0.496842
\(53\) 190944.i 1.28256i 0.767306 + 0.641281i \(0.221597\pi\)
−0.767306 + 0.641281i \(0.778403\pi\)
\(54\) 0 0
\(55\) −149760. −0.900135
\(56\) 165048.i 0.939823i
\(57\) 0 0
\(58\) −142272. −0.729181
\(59\) 237360.i 1.15572i 0.816137 + 0.577858i \(0.196111\pi\)
−0.816137 + 0.577858i \(0.803889\pi\)
\(60\) 0 0
\(61\) −61801.0 −0.272274 −0.136137 0.990690i \(-0.543469\pi\)
−0.136137 + 0.990690i \(0.543469\pi\)
\(62\) 31980.0i 0.134185i
\(63\) 0 0
\(64\) −254528. −0.970947
\(65\) − 598800.i − 2.18043i
\(66\) 0 0
\(67\) −430261. −1.43056 −0.715282 0.698835i \(-0.753702\pi\)
−0.715282 + 0.698835i \(0.753702\pi\)
\(68\) 52416.0i 0.166701i
\(69\) 0 0
\(70\) 430560. 1.25528
\(71\) − 251712.i − 0.703281i −0.936135 0.351640i \(-0.885624\pi\)
0.936135 0.351640i \(-0.114376\pi\)
\(72\) 0 0
\(73\) 251615. 0.646797 0.323398 0.946263i \(-0.395175\pi\)
0.323398 + 0.946263i \(0.395175\pi\)
\(74\) 195546.i 0.482563i
\(75\) 0 0
\(76\) −70252.0 −0.160036
\(77\) − 186576.i − 0.408680i
\(78\) 0 0
\(79\) 660827. 1.34031 0.670157 0.742219i \(-0.266227\pi\)
0.670157 + 0.742219i \(0.266227\pi\)
\(80\) 364800.i 0.712500i
\(81\) 0 0
\(82\) −396864. −0.719781
\(83\) 797856.i 1.39537i 0.716403 + 0.697686i \(0.245787\pi\)
−0.716403 + 0.697686i \(0.754213\pi\)
\(84\) 0 0
\(85\) 449280. 0.731577
\(86\) − 423780.i − 0.666262i
\(87\) 0 0
\(88\) 344448. 0.505447
\(89\) − 270576.i − 0.383813i −0.981413 0.191906i \(-0.938533\pi\)
0.981413 0.191906i \(-0.0614670\pi\)
\(90\) 0 0
\(91\) 746005. 0.989960
\(92\) − 401856.i − 0.516068i
\(93\) 0 0
\(94\) −23904.0 −0.0287798
\(95\) 602160.i 0.702330i
\(96\) 0 0
\(97\) 220727. 0.241847 0.120923 0.992662i \(-0.461414\pi\)
0.120923 + 0.992662i \(0.461414\pi\)
\(98\) − 169488.i − 0.180078i
\(99\) 0 0
\(100\) −1.17530e6 −1.17530
\(101\) 289536.i 0.281021i 0.990079 + 0.140510i \(0.0448743\pi\)
−0.990079 + 0.140510i \(0.955126\pi\)
\(102\) 0 0
\(103\) −594205. −0.543782 −0.271891 0.962328i \(-0.587649\pi\)
−0.271891 + 0.962328i \(0.587649\pi\)
\(104\) 1.37724e6i 1.22436i
\(105\) 0 0
\(106\) −1.14566e6 −0.961922
\(107\) − 1.99368e6i − 1.62744i −0.581259 0.813718i \(-0.697440\pi\)
0.581259 0.813718i \(-0.302560\pi\)
\(108\) 0 0
\(109\) 1.51135e6 1.16704 0.583521 0.812098i \(-0.301674\pi\)
0.583521 + 0.812098i \(0.301674\pi\)
\(110\) − 898560.i − 0.675101i
\(111\) 0 0
\(112\) −454480. −0.323490
\(113\) − 1.65048e6i − 1.14387i −0.820301 0.571933i \(-0.806194\pi\)
0.820301 0.571933i \(-0.193806\pi\)
\(114\) 0 0
\(115\) −3.44448e6 −2.26480
\(116\) 663936.i 0.425356i
\(117\) 0 0
\(118\) −1.42416e6 −0.866788
\(119\) 559728.i 0.332151i
\(120\) 0 0
\(121\) 1.38218e6 0.780207
\(122\) − 370806.i − 0.204205i
\(123\) 0 0
\(124\) 149240. 0.0782745
\(125\) 6.32400e6i 3.23789i
\(126\) 0 0
\(127\) 1.41135e6 0.689005 0.344502 0.938785i \(-0.388048\pi\)
0.344502 + 0.938785i \(0.388048\pi\)
\(128\) 150144.i 0.0715942i
\(129\) 0 0
\(130\) 3.59280e6 1.63532
\(131\) − 323232.i − 0.143781i −0.997413 0.0718903i \(-0.977097\pi\)
0.997413 0.0718903i \(-0.0229032\pi\)
\(132\) 0 0
\(133\) −750191. −0.318872
\(134\) − 2.58157e6i − 1.07292i
\(135\) 0 0
\(136\) −1.03334e6 −0.410798
\(137\) − 2.13845e6i − 0.831643i −0.909446 0.415822i \(-0.863494\pi\)
0.909446 0.415822i \(-0.136506\pi\)
\(138\) 0 0
\(139\) −111229. −0.0414165 −0.0207083 0.999786i \(-0.506592\pi\)
−0.0207083 + 0.999786i \(0.506592\pi\)
\(140\) − 2.00928e6i − 0.732245i
\(141\) 0 0
\(142\) 1.51027e6 0.527461
\(143\) − 1.55688e6i − 0.532411i
\(144\) 0 0
\(145\) 5.69088e6 1.86670
\(146\) 1.50969e6i 0.485098i
\(147\) 0 0
\(148\) 912548. 0.281495
\(149\) 3.43661e6i 1.03889i 0.854503 + 0.519447i \(0.173862\pi\)
−0.854503 + 0.519447i \(0.826138\pi\)
\(150\) 0 0
\(151\) −3.93932e6 −1.14417 −0.572086 0.820194i \(-0.693865\pi\)
−0.572086 + 0.820194i \(0.693865\pi\)
\(152\) − 1.38497e6i − 0.394375i
\(153\) 0 0
\(154\) 1.11946e6 0.306510
\(155\) − 1.27920e6i − 0.343513i
\(156\) 0 0
\(157\) 431066. 0.111390 0.0556948 0.998448i \(-0.482263\pi\)
0.0556948 + 0.998448i \(0.482263\pi\)
\(158\) 3.96496e6i 1.00524i
\(159\) 0 0
\(160\) 6.28992e6 1.53563
\(161\) − 4.29125e6i − 1.02827i
\(162\) 0 0
\(163\) −7.94353e6 −1.83422 −0.917109 0.398637i \(-0.869483\pi\)
−0.917109 + 0.398637i \(0.869483\pi\)
\(164\) 1.85203e6i 0.419872i
\(165\) 0 0
\(166\) −4.78714e6 −1.04653
\(167\) − 3.53832e6i − 0.759710i −0.925046 0.379855i \(-0.875974\pi\)
0.925046 0.379855i \(-0.124026\pi\)
\(168\) 0 0
\(169\) 1.39822e6 0.289677
\(170\) 2.69568e6i 0.548683i
\(171\) 0 0
\(172\) −1.97764e6 −0.388653
\(173\) 1.76842e6i 0.341544i 0.985311 + 0.170772i \(0.0546261\pi\)
−0.985311 + 0.170772i \(0.945374\pi\)
\(174\) 0 0
\(175\) −1.25505e7 −2.34179
\(176\) 948480.i 0.173976i
\(177\) 0 0
\(178\) 1.62346e6 0.287859
\(179\) 3.82637e6i 0.667156i 0.942722 + 0.333578i \(0.108256\pi\)
−0.942722 + 0.333578i \(0.891744\pi\)
\(180\) 0 0
\(181\) −2.32721e6 −0.392464 −0.196232 0.980558i \(-0.562871\pi\)
−0.196232 + 0.980558i \(0.562871\pi\)
\(182\) 4.47603e6i 0.742470i
\(183\) 0 0
\(184\) 7.92230e6 1.27174
\(185\) − 7.82184e6i − 1.23536i
\(186\) 0 0
\(187\) 1.16813e6 0.178635
\(188\) 111552.i 0.0167882i
\(189\) 0 0
\(190\) −3.61296e6 −0.526747
\(191\) 5.16235e6i 0.740879i 0.928857 + 0.370440i \(0.120793\pi\)
−0.928857 + 0.370440i \(0.879207\pi\)
\(192\) 0 0
\(193\) −3.47554e6 −0.483448 −0.241724 0.970345i \(-0.577713\pi\)
−0.241724 + 0.970345i \(0.577713\pi\)
\(194\) 1.32436e6i 0.181385i
\(195\) 0 0
\(196\) −790944. −0.105046
\(197\) − 7.36920e6i − 0.963877i −0.876205 0.481939i \(-0.839933\pi\)
0.876205 0.481939i \(-0.160067\pi\)
\(198\) 0 0
\(199\) 8.43100e6 1.06984 0.534921 0.844902i \(-0.320341\pi\)
0.534921 + 0.844902i \(0.320341\pi\)
\(200\) − 2.31702e7i − 2.89628i
\(201\) 0 0
\(202\) −1.73722e6 −0.210766
\(203\) 7.08989e6i 0.847523i
\(204\) 0 0
\(205\) 1.58746e7 1.84264
\(206\) − 3.56523e6i − 0.407836i
\(207\) 0 0
\(208\) −3.79240e6 −0.421429
\(209\) 1.56562e6i 0.171493i
\(210\) 0 0
\(211\) 4.10012e6 0.436465 0.218233 0.975897i \(-0.429971\pi\)
0.218233 + 0.975897i \(0.429971\pi\)
\(212\) 5.34643e6i 0.561121i
\(213\) 0 0
\(214\) 1.19621e7 1.22058
\(215\) 1.69512e7i 1.70563i
\(216\) 0 0
\(217\) 1.59367e6 0.155962
\(218\) 9.06812e6i 0.875282i
\(219\) 0 0
\(220\) −4.19328e6 −0.393809
\(221\) 4.67064e6i 0.432713i
\(222\) 0 0
\(223\) −1.45364e7 −1.31081 −0.655407 0.755276i \(-0.727503\pi\)
−0.655407 + 0.755276i \(0.727503\pi\)
\(224\) 7.83619e6i 0.697206i
\(225\) 0 0
\(226\) 9.90288e6 0.857899
\(227\) − 5.91514e6i − 0.505693i −0.967506 0.252847i \(-0.918633\pi\)
0.967506 0.252847i \(-0.0813668\pi\)
\(228\) 0 0
\(229\) 1.91699e7 1.59630 0.798149 0.602460i \(-0.205813\pi\)
0.798149 + 0.602460i \(0.205813\pi\)
\(230\) − 2.06669e7i − 1.69860i
\(231\) 0 0
\(232\) −1.30890e7 −1.04820
\(233\) − 7.40563e6i − 0.585456i −0.956196 0.292728i \(-0.905437\pi\)
0.956196 0.292728i \(-0.0945631\pi\)
\(234\) 0 0
\(235\) 956160. 0.0736762
\(236\) 6.64608e6i 0.505626i
\(237\) 0 0
\(238\) −3.35837e6 −0.249114
\(239\) − 1.48687e7i − 1.08913i −0.838720 0.544563i \(-0.816695\pi\)
0.838720 0.544563i \(-0.183305\pi\)
\(240\) 0 0
\(241\) −4.22995e6 −0.302193 −0.151097 0.988519i \(-0.548280\pi\)
−0.151097 + 0.988519i \(0.548280\pi\)
\(242\) 8.29311e6i 0.585156i
\(243\) 0 0
\(244\) −1.73043e6 −0.119120
\(245\) 6.77952e6i 0.461000i
\(246\) 0 0
\(247\) −6.25996e6 −0.415413
\(248\) 2.94216e6i 0.192891i
\(249\) 0 0
\(250\) −3.79440e7 −2.42842
\(251\) − 1.32500e7i − 0.837906i −0.908008 0.418953i \(-0.862397\pi\)
0.908008 0.418953i \(-0.137603\pi\)
\(252\) 0 0
\(253\) −8.95565e6 −0.553013
\(254\) 8.46808e6i 0.516754i
\(255\) 0 0
\(256\) −1.71907e7 −1.02464
\(257\) 3.21672e7i 1.89502i 0.319726 + 0.947510i \(0.396409\pi\)
−0.319726 + 0.947510i \(0.603591\pi\)
\(258\) 0 0
\(259\) 9.74471e6 0.560880
\(260\) − 1.67664e7i − 0.953937i
\(261\) 0 0
\(262\) 1.93939e6 0.107835
\(263\) − 2.91608e7i − 1.60299i −0.597999 0.801497i \(-0.704037\pi\)
0.597999 0.801497i \(-0.295963\pi\)
\(264\) 0 0
\(265\) 4.58266e7 2.46252
\(266\) − 4.50115e6i − 0.239154i
\(267\) 0 0
\(268\) −1.20473e7 −0.625872
\(269\) − 1.84710e7i − 0.948930i −0.880274 0.474465i \(-0.842642\pi\)
0.880274 0.474465i \(-0.157358\pi\)
\(270\) 0 0
\(271\) −2.27473e7 −1.14294 −0.571468 0.820624i \(-0.693626\pi\)
−0.571468 + 0.820624i \(0.693626\pi\)
\(272\) − 2.84544e6i − 0.141398i
\(273\) 0 0
\(274\) 1.28307e7 0.623732
\(275\) 2.61924e7i 1.25944i
\(276\) 0 0
\(277\) −2.17765e7 −1.02459 −0.512293 0.858811i \(-0.671204\pi\)
−0.512293 + 0.858811i \(0.671204\pi\)
\(278\) − 667374.i − 0.0310624i
\(279\) 0 0
\(280\) 3.96115e7 1.80446
\(281\) 3.01957e7i 1.36090i 0.732794 + 0.680451i \(0.238216\pi\)
−0.732794 + 0.680451i \(0.761784\pi\)
\(282\) 0 0
\(283\) −1.97788e7 −0.872649 −0.436325 0.899789i \(-0.643720\pi\)
−0.436325 + 0.899789i \(0.643720\pi\)
\(284\) − 7.04794e6i − 0.307685i
\(285\) 0 0
\(286\) 9.34128e6 0.399308
\(287\) 1.97771e7i 0.836596i
\(288\) 0 0
\(289\) 2.06332e7 0.854816
\(290\) 3.41453e7i 1.40003i
\(291\) 0 0
\(292\) 7.04522e6 0.282974
\(293\) − 1.99512e7i − 0.793168i −0.917998 0.396584i \(-0.870196\pi\)
0.917998 0.396584i \(-0.129804\pi\)
\(294\) 0 0
\(295\) 5.69664e7 2.21898
\(296\) 1.79902e7i 0.693684i
\(297\) 0 0
\(298\) −2.06196e7 −0.779170
\(299\) − 3.58082e7i − 1.33958i
\(300\) 0 0
\(301\) −2.11184e7 −0.774392
\(302\) − 2.36360e7i − 0.858128i
\(303\) 0 0
\(304\) 3.81368e6 0.135745
\(305\) 1.48322e7i 0.522766i
\(306\) 0 0
\(307\) 2.43404e7 0.841226 0.420613 0.907240i \(-0.361815\pi\)
0.420613 + 0.907240i \(0.361815\pi\)
\(308\) − 5.22413e6i − 0.178798i
\(309\) 0 0
\(310\) 7.67520e6 0.257635
\(311\) − 5.31455e7i − 1.76679i −0.468629 0.883395i \(-0.655252\pi\)
0.468629 0.883395i \(-0.344748\pi\)
\(312\) 0 0
\(313\) −3.17868e6 −0.103661 −0.0518303 0.998656i \(-0.516506\pi\)
−0.0518303 + 0.998656i \(0.516506\pi\)
\(314\) 2.58640e6i 0.0835422i
\(315\) 0 0
\(316\) 1.85032e7 0.586387
\(317\) − 3.70569e7i − 1.16330i −0.813440 0.581649i \(-0.802408\pi\)
0.813440 0.581649i \(-0.197592\pi\)
\(318\) 0 0
\(319\) 1.47963e7 0.455807
\(320\) 6.10867e7i 1.86422i
\(321\) 0 0
\(322\) 2.57475e7 0.771200
\(323\) − 4.69685e6i − 0.139380i
\(324\) 0 0
\(325\) −1.04728e8 −3.05078
\(326\) − 4.76612e7i − 1.37566i
\(327\) 0 0
\(328\) −3.65115e7 −1.03468
\(329\) 1.19122e6i 0.0334505i
\(330\) 0 0
\(331\) 5.46748e6 0.150766 0.0753829 0.997155i \(-0.475982\pi\)
0.0753829 + 0.997155i \(0.475982\pi\)
\(332\) 2.23400e7i 0.610476i
\(333\) 0 0
\(334\) 2.12299e7 0.569782
\(335\) 1.03263e8i 2.74668i
\(336\) 0 0
\(337\) −4.09219e7 −1.06922 −0.534609 0.845099i \(-0.679541\pi\)
−0.534609 + 0.845099i \(0.679541\pi\)
\(338\) 8.38930e6i 0.217258i
\(339\) 0 0
\(340\) 1.25798e7 0.320065
\(341\) − 3.32592e6i − 0.0838781i
\(342\) 0 0
\(343\) −4.36232e7 −1.08102
\(344\) − 3.89878e7i − 0.957752i
\(345\) 0 0
\(346\) −1.06105e7 −0.256158
\(347\) 5.36465e7i 1.28397i 0.766719 + 0.641983i \(0.221888\pi\)
−0.766719 + 0.641983i \(0.778112\pi\)
\(348\) 0 0
\(349\) 1.60887e7 0.378482 0.189241 0.981931i \(-0.439397\pi\)
0.189241 + 0.981931i \(0.439397\pi\)
\(350\) − 7.53032e7i − 1.75634i
\(351\) 0 0
\(352\) 1.63538e7 0.374965
\(353\) 4.11130e6i 0.0934662i 0.998907 + 0.0467331i \(0.0148810\pi\)
−0.998907 + 0.0467331i \(0.985119\pi\)
\(354\) 0 0
\(355\) −6.04109e7 −1.35030
\(356\) − 7.57613e6i − 0.167918i
\(357\) 0 0
\(358\) −2.29582e7 −0.500367
\(359\) 1.41496e7i 0.305816i 0.988240 + 0.152908i \(0.0488638\pi\)
−0.988240 + 0.152908i \(0.951136\pi\)
\(360\) 0 0
\(361\) −4.07508e7 −0.866193
\(362\) − 1.39633e7i − 0.294348i
\(363\) 0 0
\(364\) 2.08881e7 0.433107
\(365\) − 6.03876e7i − 1.24185i
\(366\) 0 0
\(367\) 4.13390e7 0.836300 0.418150 0.908378i \(-0.362679\pi\)
0.418150 + 0.908378i \(0.362679\pi\)
\(368\) 2.18150e7i 0.437736i
\(369\) 0 0
\(370\) 4.69310e7 0.926520
\(371\) 5.70923e7i 1.11804i
\(372\) 0 0
\(373\) 5.83380e7 1.12415 0.562076 0.827085i \(-0.310003\pi\)
0.562076 + 0.827085i \(0.310003\pi\)
\(374\) 7.00877e6i 0.133976i
\(375\) 0 0
\(376\) −2.19917e6 −0.0413709
\(377\) 5.91614e7i 1.10412i
\(378\) 0 0
\(379\) 7.49065e7 1.37595 0.687974 0.725735i \(-0.258500\pi\)
0.687974 + 0.725735i \(0.258500\pi\)
\(380\) 1.68605e7i 0.307269i
\(381\) 0 0
\(382\) −3.09741e7 −0.555660
\(383\) − 2.84444e7i − 0.506292i −0.967428 0.253146i \(-0.918535\pi\)
0.967428 0.253146i \(-0.0814652\pi\)
\(384\) 0 0
\(385\) −4.47782e7 −0.784666
\(386\) − 2.08532e7i − 0.362586i
\(387\) 0 0
\(388\) 6.18036e6 0.105808
\(389\) − 6.11364e7i − 1.03861i −0.854590 0.519303i \(-0.826192\pi\)
0.854590 0.519303i \(-0.173808\pi\)
\(390\) 0 0
\(391\) 2.68669e7 0.449457
\(392\) − 1.55929e7i − 0.258862i
\(393\) 0 0
\(394\) 4.42152e7 0.722908
\(395\) − 1.58598e8i − 2.57340i
\(396\) 0 0
\(397\) −3.70144e7 −0.591561 −0.295780 0.955256i \(-0.595580\pi\)
−0.295780 + 0.955256i \(0.595580\pi\)
\(398\) 5.05860e7i 0.802381i
\(399\) 0 0
\(400\) 6.38020e7 0.996906
\(401\) − 2.08769e7i − 0.323768i −0.986810 0.161884i \(-0.948243\pi\)
0.986810 0.161884i \(-0.0517570\pi\)
\(402\) 0 0
\(403\) 1.32984e7 0.203181
\(404\) 8.10701e6i 0.122947i
\(405\) 0 0
\(406\) −4.25393e7 −0.635642
\(407\) − 2.03368e7i − 0.301647i
\(408\) 0 0
\(409\) 9.10530e7 1.33084 0.665418 0.746471i \(-0.268254\pi\)
0.665418 + 0.746471i \(0.268254\pi\)
\(410\) 9.52474e7i 1.38198i
\(411\) 0 0
\(412\) −1.66377e7 −0.237905
\(413\) 7.09706e7i 1.00746i
\(414\) 0 0
\(415\) 1.91485e8 2.67912
\(416\) 6.53890e7i 0.908290i
\(417\) 0 0
\(418\) −9.39370e6 −0.128620
\(419\) 1.15211e8i 1.56622i 0.621885 + 0.783108i \(0.286367\pi\)
−0.621885 + 0.783108i \(0.713633\pi\)
\(420\) 0 0
\(421\) −1.31590e8 −1.76350 −0.881750 0.471718i \(-0.843634\pi\)
−0.881750 + 0.471718i \(0.843634\pi\)
\(422\) 2.46007e7i 0.327349i
\(423\) 0 0
\(424\) −1.05401e8 −1.38276
\(425\) − 7.85772e7i − 1.02360i
\(426\) 0 0
\(427\) −1.84785e7 −0.237347
\(428\) − 5.58230e7i − 0.712004i
\(429\) 0 0
\(430\) −1.01707e8 −1.27922
\(431\) 7.15543e7i 0.893725i 0.894603 + 0.446863i \(0.147459\pi\)
−0.894603 + 0.446863i \(0.852541\pi\)
\(432\) 0 0
\(433\) 1.83937e7 0.226572 0.113286 0.993562i \(-0.463862\pi\)
0.113286 + 0.993562i \(0.463862\pi\)
\(434\) 9.56202e6i 0.116972i
\(435\) 0 0
\(436\) 4.23179e7 0.510581
\(437\) 3.60092e7i 0.431488i
\(438\) 0 0
\(439\) −9.90611e7 −1.17087 −0.585436 0.810719i \(-0.699077\pi\)
−0.585436 + 0.810719i \(0.699077\pi\)
\(440\) − 8.26675e7i − 0.970458i
\(441\) 0 0
\(442\) −2.80238e7 −0.324534
\(443\) − 1.35397e8i − 1.55739i −0.627403 0.778695i \(-0.715882\pi\)
0.627403 0.778695i \(-0.284118\pi\)
\(444\) 0 0
\(445\) −6.49382e7 −0.736920
\(446\) − 8.72182e7i − 0.983111i
\(447\) 0 0
\(448\) −7.61039e7 −0.846394
\(449\) 6.85948e7i 0.757796i 0.925438 + 0.378898i \(0.123697\pi\)
−0.925438 + 0.378898i \(0.876303\pi\)
\(450\) 0 0
\(451\) 4.12739e7 0.449930
\(452\) − 4.62134e7i − 0.500441i
\(453\) 0 0
\(454\) 3.54908e7 0.379270
\(455\) − 1.79041e8i − 1.90072i
\(456\) 0 0
\(457\) 7.12649e7 0.746667 0.373334 0.927697i \(-0.378215\pi\)
0.373334 + 0.927697i \(0.378215\pi\)
\(458\) 1.15020e8i 1.19722i
\(459\) 0 0
\(460\) −9.64454e7 −0.990851
\(461\) 1.56559e8i 1.59800i 0.601332 + 0.798999i \(0.294637\pi\)
−0.601332 + 0.798999i \(0.705363\pi\)
\(462\) 0 0
\(463\) −553813. −0.00557982 −0.00278991 0.999996i \(-0.500888\pi\)
−0.00278991 + 0.999996i \(0.500888\pi\)
\(464\) − 3.60422e7i − 0.360793i
\(465\) 0 0
\(466\) 4.44338e7 0.439092
\(467\) − 3.14739e7i − 0.309030i −0.987990 0.154515i \(-0.950619\pi\)
0.987990 0.154515i \(-0.0493815\pi\)
\(468\) 0 0
\(469\) −1.28648e8 −1.24705
\(470\) 5.73696e6i 0.0552571i
\(471\) 0 0
\(472\) −1.31023e8 −1.24601
\(473\) 4.40731e7i 0.416476i
\(474\) 0 0
\(475\) 1.05315e8 0.982676
\(476\) 1.56724e7i 0.145316i
\(477\) 0 0
\(478\) 8.92120e7 0.816845
\(479\) 6.67838e7i 0.607666i 0.952725 + 0.303833i \(0.0982665\pi\)
−0.952725 + 0.303833i \(0.901734\pi\)
\(480\) 0 0
\(481\) 8.13145e7 0.730690
\(482\) − 2.53797e7i − 0.226645i
\(483\) 0 0
\(484\) 3.87012e7 0.341341
\(485\) − 5.29745e7i − 0.464346i
\(486\) 0 0
\(487\) 2.00508e8 1.73598 0.867992 0.496579i \(-0.165411\pi\)
0.867992 + 0.496579i \(0.165411\pi\)
\(488\) − 3.41142e7i − 0.293545i
\(489\) 0 0
\(490\) −4.06771e7 −0.345750
\(491\) 8.61838e7i 0.728083i 0.931383 + 0.364042i \(0.118603\pi\)
−0.931383 + 0.364042i \(0.881397\pi\)
\(492\) 0 0
\(493\) −4.43889e7 −0.370453
\(494\) − 3.75597e7i − 0.311560i
\(495\) 0 0
\(496\) −8.10160e6 −0.0663935
\(497\) − 7.52619e7i − 0.613064i
\(498\) 0 0
\(499\) 1.06673e7 0.0858525 0.0429263 0.999078i \(-0.486332\pi\)
0.0429263 + 0.999078i \(0.486332\pi\)
\(500\) 1.77072e8i 1.41658i
\(501\) 0 0
\(502\) 7.95001e7 0.628429
\(503\) − 1.07893e8i − 0.847790i −0.905711 0.423895i \(-0.860663\pi\)
0.905711 0.423895i \(-0.139337\pi\)
\(504\) 0 0
\(505\) 6.94886e7 0.539560
\(506\) − 5.37339e7i − 0.414760i
\(507\) 0 0
\(508\) 3.95177e7 0.301440
\(509\) 1.54033e8i 1.16805i 0.811736 + 0.584025i \(0.198523\pi\)
−0.811736 + 0.584025i \(0.801477\pi\)
\(510\) 0 0
\(511\) 7.52329e7 0.563826
\(512\) − 9.35347e7i − 0.696888i
\(513\) 0 0
\(514\) −1.93003e8 −1.42127
\(515\) 1.42609e8i 1.04406i
\(516\) 0 0
\(517\) 2.48602e6 0.0179900
\(518\) 5.84683e7i 0.420660i
\(519\) 0 0
\(520\) 3.30538e8 2.35077
\(521\) − 1.85442e8i − 1.31128i −0.755074 0.655640i \(-0.772399\pi\)
0.755074 0.655640i \(-0.227601\pi\)
\(522\) 0 0
\(523\) −1.46950e7 −0.102723 −0.0513613 0.998680i \(-0.516356\pi\)
−0.0513613 + 0.998680i \(0.516356\pi\)
\(524\) − 9.05050e6i − 0.0629040i
\(525\) 0 0
\(526\) 1.74965e8 1.20224
\(527\) 9.97776e6i 0.0681712i
\(528\) 0 0
\(529\) −5.79440e7 −0.391419
\(530\) 2.74959e8i 1.84689i
\(531\) 0 0
\(532\) −2.10053e7 −0.139507
\(533\) 1.65029e8i 1.08988i
\(534\) 0 0
\(535\) −4.78483e8 −3.12468
\(536\) − 2.37504e8i − 1.54233i
\(537\) 0 0
\(538\) 1.10826e8 0.711697
\(539\) 1.76268e7i 0.112566i
\(540\) 0 0
\(541\) −1.03570e7 −0.0654098 −0.0327049 0.999465i \(-0.510412\pi\)
−0.0327049 + 0.999465i \(0.510412\pi\)
\(542\) − 1.36484e8i − 0.857202i
\(543\) 0 0
\(544\) −4.90614e7 −0.304750
\(545\) − 3.62725e8i − 2.24072i
\(546\) 0 0
\(547\) 8.03197e7 0.490750 0.245375 0.969428i \(-0.421089\pi\)
0.245375 + 0.969428i \(0.421089\pi\)
\(548\) − 5.98765e7i − 0.363844i
\(549\) 0 0
\(550\) −1.57154e8 −0.944579
\(551\) − 5.94934e7i − 0.355643i
\(552\) 0 0
\(553\) 1.97587e8 1.16838
\(554\) − 1.30659e8i − 0.768439i
\(555\) 0 0
\(556\) −3.11441e6 −0.0181197
\(557\) − 3.87935e7i − 0.224488i −0.993681 0.112244i \(-0.964196\pi\)
0.993681 0.112244i \(-0.0358038\pi\)
\(558\) 0 0
\(559\) −1.76222e8 −1.00884
\(560\) 1.09075e8i 0.621101i
\(561\) 0 0
\(562\) −1.81174e8 −1.02068
\(563\) 2.45635e8i 1.37647i 0.725490 + 0.688233i \(0.241613\pi\)
−0.725490 + 0.688233i \(0.758387\pi\)
\(564\) 0 0
\(565\) −3.96115e8 −2.19622
\(566\) − 1.18673e8i − 0.654487i
\(567\) 0 0
\(568\) 1.38945e8 0.758225
\(569\) − 3.37004e7i − 0.182935i −0.995808 0.0914677i \(-0.970844\pi\)
0.995808 0.0914677i \(-0.0291558\pi\)
\(570\) 0 0
\(571\) −2.36969e8 −1.27287 −0.636435 0.771330i \(-0.719592\pi\)
−0.636435 + 0.771330i \(0.719592\pi\)
\(572\) − 4.35926e7i − 0.232930i
\(573\) 0 0
\(574\) −1.18662e8 −0.627447
\(575\) 6.02425e8i 3.16883i
\(576\) 0 0
\(577\) 4.09015e6 0.0212918 0.0106459 0.999943i \(-0.496611\pi\)
0.0106459 + 0.999943i \(0.496611\pi\)
\(578\) 1.23799e8i 0.641112i
\(579\) 0 0
\(580\) 1.59345e8 0.816683
\(581\) 2.38559e8i 1.21637i
\(582\) 0 0
\(583\) 1.19149e8 0.601291
\(584\) 1.38891e8i 0.697328i
\(585\) 0 0
\(586\) 1.19707e8 0.594876
\(587\) − 3.64756e8i − 1.80338i −0.432379 0.901692i \(-0.642326\pi\)
0.432379 0.901692i \(-0.357674\pi\)
\(588\) 0 0
\(589\) −1.33730e7 −0.0654458
\(590\) 3.41798e8i 1.66423i
\(591\) 0 0
\(592\) −4.95383e7 −0.238768
\(593\) − 3.38297e8i − 1.62231i −0.584832 0.811154i \(-0.698840\pi\)
0.584832 0.811154i \(-0.301160\pi\)
\(594\) 0 0
\(595\) 1.34335e8 0.637731
\(596\) 9.62250e7i 0.454516i
\(597\) 0 0
\(598\) 2.14849e8 1.00469
\(599\) − 2.52534e8i − 1.17500i −0.809223 0.587502i \(-0.800111\pi\)
0.809223 0.587502i \(-0.199889\pi\)
\(600\) 0 0
\(601\) 2.67857e8 1.23390 0.616949 0.787003i \(-0.288368\pi\)
0.616949 + 0.787003i \(0.288368\pi\)
\(602\) − 1.26710e8i − 0.580794i
\(603\) 0 0
\(604\) −1.10301e8 −0.500575
\(605\) − 3.31724e8i − 1.49800i
\(606\) 0 0
\(607\) −2.72153e8 −1.21688 −0.608438 0.793601i \(-0.708204\pi\)
−0.608438 + 0.793601i \(0.708204\pi\)
\(608\) − 6.57559e7i − 0.292566i
\(609\) 0 0
\(610\) −8.89934e7 −0.392074
\(611\) 9.94008e6i 0.0435779i
\(612\) 0 0
\(613\) −4.91383e7 −0.213323 −0.106662 0.994295i \(-0.534016\pi\)
−0.106662 + 0.994295i \(0.534016\pi\)
\(614\) 1.46042e8i 0.630919i
\(615\) 0 0
\(616\) 1.02990e8 0.440608
\(617\) 1.24682e8i 0.530823i 0.964135 + 0.265412i \(0.0855078\pi\)
−0.964135 + 0.265412i \(0.914492\pi\)
\(618\) 0 0
\(619\) −4.53573e8 −1.91239 −0.956193 0.292738i \(-0.905434\pi\)
−0.956193 + 0.292738i \(0.905434\pi\)
\(620\) − 3.58176e7i − 0.150287i
\(621\) 0 0
\(622\) 3.18873e8 1.32509
\(623\) − 8.09022e7i − 0.334577i
\(624\) 0 0
\(625\) 8.61901e8 3.53034
\(626\) − 1.90721e7i − 0.0777455i
\(627\) 0 0
\(628\) 1.20698e7 0.0487330
\(629\) 6.10104e7i 0.245161i
\(630\) 0 0
\(631\) −5.31523e7 −0.211560 −0.105780 0.994390i \(-0.533734\pi\)
−0.105780 + 0.994390i \(0.533734\pi\)
\(632\) 3.64777e8i 1.44503i
\(633\) 0 0
\(634\) 2.22341e8 0.872473
\(635\) − 3.38723e8i − 1.32289i
\(636\) 0 0
\(637\) −7.04788e7 −0.272672
\(638\) 8.87777e7i 0.341855i
\(639\) 0 0
\(640\) 3.60346e7 0.137461
\(641\) 3.65442e8i 1.38754i 0.720199 + 0.693768i \(0.244051\pi\)
−0.720199 + 0.693768i \(0.755949\pi\)
\(642\) 0 0
\(643\) 1.52561e8 0.573867 0.286933 0.957951i \(-0.407364\pi\)
0.286933 + 0.957951i \(0.407364\pi\)
\(644\) − 1.20155e8i − 0.449867i
\(645\) 0 0
\(646\) 2.81811e7 0.104535
\(647\) − 1.22990e7i − 0.0454107i −0.999742 0.0227054i \(-0.992772\pi\)
0.999742 0.0227054i \(-0.00722796\pi\)
\(648\) 0 0
\(649\) 1.48113e8 0.541824
\(650\) − 6.28366e8i − 2.28809i
\(651\) 0 0
\(652\) −2.22419e8 −0.802470
\(653\) − 5.72096e7i − 0.205461i −0.994709 0.102730i \(-0.967242\pi\)
0.994709 0.102730i \(-0.0327579\pi\)
\(654\) 0 0
\(655\) −7.75757e7 −0.276059
\(656\) − 1.00539e8i − 0.356141i
\(657\) 0 0
\(658\) −7.14730e6 −0.0250879
\(659\) 2.30093e8i 0.803982i 0.915644 + 0.401991i \(0.131682\pi\)
−0.915644 + 0.401991i \(0.868318\pi\)
\(660\) 0 0
\(661\) 4.81046e7 0.166564 0.0832822 0.996526i \(-0.473460\pi\)
0.0832822 + 0.996526i \(0.473460\pi\)
\(662\) 3.28048e7i 0.113074i
\(663\) 0 0
\(664\) −4.40417e8 −1.50439
\(665\) 1.80046e8i 0.612235i
\(666\) 0 0
\(667\) 3.40315e8 1.14684
\(668\) − 9.90730e7i − 0.332373i
\(669\) 0 0
\(670\) −6.19576e8 −2.06001
\(671\) 3.85638e7i 0.127648i
\(672\) 0 0
\(673\) 2.30653e8 0.756681 0.378341 0.925666i \(-0.376495\pi\)
0.378341 + 0.925666i \(0.376495\pi\)
\(674\) − 2.45532e8i − 0.801914i
\(675\) 0 0
\(676\) 3.91500e7 0.126734
\(677\) 5.64246e8i 1.81846i 0.416299 + 0.909228i \(0.363327\pi\)
−0.416299 + 0.909228i \(0.636673\pi\)
\(678\) 0 0
\(679\) 6.59974e7 0.210823
\(680\) 2.48003e8i 0.788732i
\(681\) 0 0
\(682\) 1.99555e7 0.0629086
\(683\) − 2.05363e8i − 0.644555i −0.946645 0.322278i \(-0.895552\pi\)
0.946645 0.322278i \(-0.104448\pi\)
\(684\) 0 0
\(685\) −5.13228e8 −1.59675
\(686\) − 2.61739e8i − 0.810768i
\(687\) 0 0
\(688\) 1.07358e8 0.329661
\(689\) 4.76405e8i 1.45653i
\(690\) 0 0
\(691\) −1.08669e8 −0.329360 −0.164680 0.986347i \(-0.552659\pi\)
−0.164680 + 0.986347i \(0.552659\pi\)
\(692\) 4.95156e7i 0.149425i
\(693\) 0 0
\(694\) −3.21879e8 −0.962974
\(695\) 2.66950e7i 0.0795197i
\(696\) 0 0
\(697\) −1.23822e8 −0.365677
\(698\) 9.65324e7i 0.283862i
\(699\) 0 0
\(700\) −3.51415e8 −1.02453
\(701\) 1.08271e8i 0.314310i 0.987574 + 0.157155i \(0.0502321\pi\)
−0.987574 + 0.157155i \(0.949768\pi\)
\(702\) 0 0
\(703\) −8.17708e7 −0.235360
\(704\) 1.58825e8i 0.455200i
\(705\) 0 0
\(706\) −2.46678e7 −0.0700997
\(707\) 8.65713e7i 0.244971i
\(708\) 0 0
\(709\) 1.68535e8 0.472881 0.236440 0.971646i \(-0.424019\pi\)
0.236440 + 0.971646i \(0.424019\pi\)
\(710\) − 3.62465e8i − 1.01272i
\(711\) 0 0
\(712\) 1.49358e8 0.413798
\(713\) − 7.64962e7i − 0.211043i
\(714\) 0 0
\(715\) −3.73651e8 −1.02223
\(716\) 1.07138e8i 0.291881i
\(717\) 0 0
\(718\) −8.48975e7 −0.229362
\(719\) − 6.10328e8i − 1.64201i −0.570918 0.821007i \(-0.693413\pi\)
0.570918 0.821007i \(-0.306587\pi\)
\(720\) 0 0
\(721\) −1.77667e8 −0.474025
\(722\) − 2.44505e8i − 0.649645i
\(723\) 0 0
\(724\) −6.51619e7 −0.171703
\(725\) − 9.95311e8i − 2.61183i
\(726\) 0 0
\(727\) 6.15925e8 1.60297 0.801483 0.598017i \(-0.204045\pi\)
0.801483 + 0.598017i \(0.204045\pi\)
\(728\) 4.11795e8i 1.06730i
\(729\) 0 0
\(730\) 3.62326e8 0.931388
\(731\) − 1.32219e8i − 0.338488i
\(732\) 0 0
\(733\) 3.65043e8 0.926897 0.463449 0.886124i \(-0.346612\pi\)
0.463449 + 0.886124i \(0.346612\pi\)
\(734\) 2.48034e8i 0.627225i
\(735\) 0 0
\(736\) 3.76137e8 0.943437
\(737\) 2.68483e8i 0.670678i
\(738\) 0 0
\(739\) −7.10760e8 −1.76112 −0.880562 0.473931i \(-0.842835\pi\)
−0.880562 + 0.473931i \(0.842835\pi\)
\(740\) − 2.19012e8i − 0.540470i
\(741\) 0 0
\(742\) −3.42554e8 −0.838526
\(743\) − 2.11394e8i − 0.515378i −0.966228 0.257689i \(-0.917039\pi\)
0.966228 0.257689i \(-0.0829611\pi\)
\(744\) 0 0
\(745\) 8.24786e8 1.99468
\(746\) 3.50028e8i 0.843115i
\(747\) 0 0
\(748\) 3.27076e7 0.0781526
\(749\) − 5.96110e8i − 1.41867i
\(750\) 0 0
\(751\) −1.90625e8 −0.450050 −0.225025 0.974353i \(-0.572246\pi\)
−0.225025 + 0.974353i \(0.572246\pi\)
\(752\) − 6.05568e6i − 0.0142400i
\(753\) 0 0
\(754\) −3.54969e8 −0.828087
\(755\) 9.45438e8i 2.19681i
\(756\) 0 0
\(757\) 6.27405e8 1.44631 0.723153 0.690688i \(-0.242692\pi\)
0.723153 + 0.690688i \(0.242692\pi\)
\(758\) 4.49439e8i 1.03196i
\(759\) 0 0
\(760\) −3.32392e8 −0.757199
\(761\) 7.99042e7i 0.181307i 0.995882 + 0.0906537i \(0.0288956\pi\)
−0.995882 + 0.0906537i \(0.971104\pi\)
\(762\) 0 0
\(763\) 4.51895e8 1.01733
\(764\) 1.44546e8i 0.324135i
\(765\) 0 0
\(766\) 1.70666e8 0.379719
\(767\) 5.92213e8i 1.31248i
\(768\) 0 0
\(769\) 6.64475e8 1.46117 0.730583 0.682824i \(-0.239249\pi\)
0.730583 + 0.682824i \(0.239249\pi\)
\(770\) − 2.68669e8i − 0.588499i
\(771\) 0 0
\(772\) −9.73150e7 −0.211509
\(773\) 6.85987e6i 0.0148517i 0.999972 + 0.00742587i \(0.00236375\pi\)
−0.999972 + 0.00742587i \(0.997636\pi\)
\(774\) 0 0
\(775\) −2.23727e8 −0.480632
\(776\) 1.21841e8i 0.260741i
\(777\) 0 0
\(778\) 3.66818e8 0.778955
\(779\) − 1.65955e8i − 0.351058i
\(780\) 0 0
\(781\) −1.57068e8 −0.329712
\(782\) 1.61202e8i 0.337093i
\(783\) 0 0
\(784\) 4.29370e7 0.0891011
\(785\) − 1.03456e8i − 0.213868i
\(786\) 0 0
\(787\) −3.05590e8 −0.626923 −0.313462 0.949601i \(-0.601489\pi\)
−0.313462 + 0.949601i \(0.601489\pi\)
\(788\) − 2.06338e8i − 0.421696i
\(789\) 0 0
\(790\) 9.51591e8 1.93005
\(791\) − 4.93494e8i − 0.997131i
\(792\) 0 0
\(793\) −1.54193e8 −0.309205
\(794\) − 2.22086e8i − 0.443670i
\(795\) 0 0
\(796\) 2.36068e8 0.468056
\(797\) 2.01445e8i 0.397906i 0.980009 + 0.198953i \(0.0637542\pi\)
−0.980009 + 0.198953i \(0.936246\pi\)
\(798\) 0 0
\(799\) −7.45805e6 −0.0146213
\(800\) − 1.10008e9i − 2.14860i
\(801\) 0 0
\(802\) 1.25262e8 0.242826
\(803\) − 1.57008e8i − 0.303232i
\(804\) 0 0
\(805\) −1.02990e9 −1.97427
\(806\) 7.97901e7i 0.152386i
\(807\) 0 0
\(808\) −1.59824e8 −0.302976
\(809\) 1.11534e8i 0.210650i 0.994438 + 0.105325i \(0.0335882\pi\)
−0.994438 + 0.105325i \(0.966412\pi\)
\(810\) 0 0
\(811\) −1.72124e8 −0.322685 −0.161343 0.986898i \(-0.551582\pi\)
−0.161343 + 0.986898i \(0.551582\pi\)
\(812\) 1.98517e8i 0.370791i
\(813\) 0 0
\(814\) 1.22021e8 0.226235
\(815\) 1.90645e9i 3.52170i
\(816\) 0 0
\(817\) 1.77211e8 0.324955
\(818\) 5.46318e8i 0.998126i
\(819\) 0 0
\(820\) 4.44488e8 0.806154
\(821\) − 8.62483e8i − 1.55855i −0.626681 0.779276i \(-0.715587\pi\)
0.626681 0.779276i \(-0.284413\pi\)
\(822\) 0 0
\(823\) −5.89009e6 −0.0105663 −0.00528315 0.999986i \(-0.501682\pi\)
−0.00528315 + 0.999986i \(0.501682\pi\)
\(824\) − 3.28001e8i − 0.586265i
\(825\) 0 0
\(826\) −4.25824e8 −0.755596
\(827\) − 7.60618e8i − 1.34478i −0.740199 0.672388i \(-0.765269\pi\)
0.740199 0.672388i \(-0.234731\pi\)
\(828\) 0 0
\(829\) −1.06302e9 −1.86585 −0.932926 0.360067i \(-0.882754\pi\)
−0.932926 + 0.360067i \(0.882754\pi\)
\(830\) 1.14891e9i 2.00934i
\(831\) 0 0
\(832\) −6.35047e8 −1.10265
\(833\) − 5.28803e7i − 0.0914868i
\(834\) 0 0
\(835\) −8.49197e8 −1.45864
\(836\) 4.38372e7i 0.0750282i
\(837\) 0 0
\(838\) −6.91266e8 −1.17466
\(839\) − 4.66992e8i − 0.790722i −0.918526 0.395361i \(-0.870619\pi\)
0.918526 0.395361i \(-0.129381\pi\)
\(840\) 0 0
\(841\) 3.25644e7 0.0547463
\(842\) − 7.89538e8i − 1.32262i
\(843\) 0 0
\(844\) 1.14803e8 0.190953
\(845\) − 3.35572e8i − 0.556180i
\(846\) 0 0
\(847\) 4.13273e8 0.680122
\(848\) − 2.90235e8i − 0.475951i
\(849\) 0 0
\(850\) 4.71463e8 0.767699
\(851\) − 4.67746e8i − 0.758964i
\(852\) 0 0
\(853\) −1.02054e9 −1.64430 −0.822150 0.569272i \(-0.807225\pi\)
−0.822150 + 0.569272i \(0.807225\pi\)
\(854\) − 1.10871e8i − 0.178010i
\(855\) 0 0
\(856\) 1.10051e9 1.75458
\(857\) − 6.31488e7i − 0.100328i −0.998741 0.0501641i \(-0.984026\pi\)
0.998741 0.0501641i \(-0.0159744\pi\)
\(858\) 0 0
\(859\) 1.03556e9 1.63378 0.816892 0.576791i \(-0.195695\pi\)
0.816892 + 0.576791i \(0.195695\pi\)
\(860\) 4.74634e8i 0.746214i
\(861\) 0 0
\(862\) −4.29326e8 −0.670294
\(863\) − 4.60575e8i − 0.716585i −0.933609 0.358292i \(-0.883359\pi\)
0.933609 0.358292i \(-0.116641\pi\)
\(864\) 0 0
\(865\) 4.24420e8 0.655764
\(866\) 1.10362e8i 0.169929i
\(867\) 0 0
\(868\) 4.46228e7 0.0682334
\(869\) − 4.12356e8i − 0.628367i
\(870\) 0 0
\(871\) −1.07350e9 −1.62461
\(872\) 8.34267e8i 1.25822i
\(873\) 0 0
\(874\) −2.16055e8 −0.323616
\(875\) 1.89088e9i 2.82253i
\(876\) 0 0
\(877\) −9.04580e8 −1.34106 −0.670530 0.741883i \(-0.733933\pi\)
−0.670530 + 0.741883i \(0.733933\pi\)
\(878\) − 5.94366e8i − 0.878154i
\(879\) 0 0
\(880\) 2.27635e8 0.334035
\(881\) 1.06708e9i 1.56051i 0.625460 + 0.780257i \(0.284912\pi\)
−0.625460 + 0.780257i \(0.715088\pi\)
\(882\) 0 0
\(883\) −8.52780e7 −0.123867 −0.0619334 0.998080i \(-0.519727\pi\)
−0.0619334 + 0.998080i \(0.519727\pi\)
\(884\) 1.30778e8i 0.189312i
\(885\) 0 0
\(886\) 8.12381e8 1.16804
\(887\) 2.43209e8i 0.348505i 0.984701 + 0.174252i \(0.0557509\pi\)
−0.984701 + 0.174252i \(0.944249\pi\)
\(888\) 0 0
\(889\) 4.21992e8 0.600619
\(890\) − 3.89629e8i − 0.552690i
\(891\) 0 0
\(892\) −4.07018e8 −0.573481
\(893\) − 9.99586e6i − 0.0140367i
\(894\) 0 0
\(895\) 9.18328e8 1.28094
\(896\) 4.48931e7i 0.0624101i
\(897\) 0 0
\(898\) −4.11569e8 −0.568347
\(899\) 1.26385e8i 0.173947i
\(900\) 0 0
\(901\) −3.57447e8 −0.488695
\(902\) 2.47643e8i 0.337448i
\(903\) 0 0
\(904\) 9.11065e8 1.23323
\(905\) 5.58530e8i 0.753531i
\(906\) 0 0
\(907\) 8.62425e8 1.15584 0.577922 0.816092i \(-0.303864\pi\)
0.577922 + 0.816092i \(0.303864\pi\)
\(908\) − 1.65624e8i − 0.221241i
\(909\) 0 0
\(910\) 1.07425e9 1.42554
\(911\) − 1.14635e9i − 1.51622i −0.652126 0.758111i \(-0.726123\pi\)
0.652126 0.758111i \(-0.273877\pi\)
\(912\) 0 0
\(913\) 4.97862e8 0.654179
\(914\) 4.27589e8i 0.560000i
\(915\) 0 0
\(916\) 5.36758e8 0.698381
\(917\) − 9.66464e7i − 0.125336i
\(918\) 0 0
\(919\) 8.23240e8 1.06067 0.530335 0.847788i \(-0.322066\pi\)
0.530335 + 0.847788i \(0.322066\pi\)
\(920\) − 1.90135e9i − 2.44174i
\(921\) 0 0
\(922\) −9.39356e8 −1.19850
\(923\) − 6.28021e8i − 0.798674i
\(924\) 0 0
\(925\) −1.36801e9 −1.72848
\(926\) − 3.32288e6i − 0.00418486i
\(927\) 0 0
\(928\) −6.21444e8 −0.777603
\(929\) 7.28737e8i 0.908916i 0.890768 + 0.454458i \(0.150167\pi\)
−0.890768 + 0.454458i \(0.849833\pi\)
\(930\) 0 0
\(931\) 7.08742e7 0.0878293
\(932\) − 2.07358e8i − 0.256137i
\(933\) 0 0
\(934\) 1.88844e8 0.231772
\(935\) − 2.80351e8i − 0.342978i
\(936\) 0 0
\(937\) 5.88557e8 0.715434 0.357717 0.933830i \(-0.383555\pi\)
0.357717 + 0.933830i \(0.383555\pi\)
\(938\) − 7.71888e8i − 0.935289i
\(939\) 0 0
\(940\) 2.67725e7 0.0322333
\(941\) 9.73643e8i 1.16851i 0.811572 + 0.584253i \(0.198612\pi\)
−0.811572 + 0.584253i \(0.801388\pi\)
\(942\) 0 0
\(943\) 9.49299e8 1.13206
\(944\) − 3.60787e8i − 0.428879i
\(945\) 0 0
\(946\) −2.64439e8 −0.312357
\(947\) − 3.90596e8i − 0.459916i −0.973201 0.229958i \(-0.926141\pi\)
0.973201 0.229958i \(-0.0738588\pi\)
\(948\) 0 0
\(949\) 6.27779e8 0.734528
\(950\) 6.31892e8i 0.737007i
\(951\) 0 0
\(952\) −3.08970e8 −0.358101
\(953\) 4.41062e7i 0.0509590i 0.999675 + 0.0254795i \(0.00811125\pi\)
−0.999675 + 0.0254795i \(0.991889\pi\)
\(954\) 0 0
\(955\) 1.23896e9 1.42249
\(956\) − 4.16323e8i − 0.476493i
\(957\) 0 0
\(958\) −4.00703e8 −0.455749
\(959\) − 6.39396e8i − 0.724960i
\(960\) 0 0
\(961\) −8.59095e8 −0.967990
\(962\) 4.87887e8i 0.548017i
\(963\) 0 0
\(964\) −1.18439e8 −0.132209
\(965\) 8.34129e8i 0.928221i
\(966\) 0 0
\(967\) −9.63378e8 −1.06541 −0.532705 0.846301i \(-0.678825\pi\)
−0.532705 + 0.846301i \(0.678825\pi\)
\(968\) 7.62966e8i 0.841161i
\(969\) 0 0
\(970\) 3.17847e8 0.348259
\(971\) 7.36207e8i 0.804160i 0.915605 + 0.402080i \(0.131713\pi\)
−0.915605 + 0.402080i \(0.868287\pi\)
\(972\) 0 0
\(973\) −3.32575e7 −0.0361036
\(974\) 1.20305e9i 1.30199i
\(975\) 0 0
\(976\) 9.39375e7 0.101039
\(977\) − 1.60897e9i − 1.72530i −0.505805 0.862648i \(-0.668804\pi\)
0.505805 0.862648i \(-0.331196\pi\)
\(978\) 0 0
\(979\) −1.68839e8 −0.179939
\(980\) 1.89827e8i 0.201687i
\(981\) 0 0
\(982\) −5.17103e8 −0.546062
\(983\) 4.46977e8i 0.470570i 0.971926 + 0.235285i \(0.0756024\pi\)
−0.971926 + 0.235285i \(0.924398\pi\)
\(984\) 0 0
\(985\) −1.76861e9 −1.85064
\(986\) − 2.66333e8i − 0.277840i
\(987\) 0 0
\(988\) −1.75279e8 −0.181743
\(989\) 1.01368e9i 1.04788i
\(990\) 0 0
\(991\) −1.40384e9 −1.44244 −0.721220 0.692706i \(-0.756418\pi\)
−0.721220 + 0.692706i \(0.756418\pi\)
\(992\) 1.39689e8i 0.143096i
\(993\) 0 0
\(994\) 4.51571e8 0.459798
\(995\) − 2.02344e9i − 2.05410i
\(996\) 0 0
\(997\) 1.33142e9 1.34348 0.671738 0.740789i \(-0.265548\pi\)
0.671738 + 0.740789i \(0.265548\pi\)
\(998\) 6.40038e7i 0.0643894i
\(999\) 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 27.7.b.b.26.2 yes 2
3.2 odd 2 inner 27.7.b.b.26.1 2
4.3 odd 2 432.7.e.d.161.1 2
9.2 odd 6 81.7.d.b.53.1 4
9.4 even 3 81.7.d.b.26.1 4
9.5 odd 6 81.7.d.b.26.2 4
9.7 even 3 81.7.d.b.53.2 4
12.11 even 2 432.7.e.d.161.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.7.b.b.26.1 2 3.2 odd 2 inner
27.7.b.b.26.2 yes 2 1.1 even 1 trivial
81.7.d.b.26.1 4 9.4 even 3
81.7.d.b.26.2 4 9.5 odd 6
81.7.d.b.53.1 4 9.2 odd 6
81.7.d.b.53.2 4 9.7 even 3
432.7.e.d.161.1 2 4.3 odd 2
432.7.e.d.161.2 2 12.11 even 2