Properties

Label 338.6.b.a.337.1
Level $338$
Weight $6$
Character 338.337
Analytic conductor $54.210$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,6,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), not 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: \(54.2097310968\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.6.b.a.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} -16.0000 q^{4} -14.0000i q^{5} +170.000i q^{7} +64.0000i q^{8} -243.000 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} -16.0000 q^{4} -14.0000i q^{5} +170.000i q^{7} +64.0000i q^{8} -243.000 q^{9} -56.0000 q^{10} +250.000i q^{11} +680.000 q^{14} +256.000 q^{16} -1062.00 q^{17} +972.000i q^{18} -78.0000i q^{19} +224.000i q^{20} +1000.00 q^{22} -1576.00 q^{23} +2929.00 q^{25} -2720.00i q^{28} +2578.00 q^{29} -8654.00i q^{31} -1024.00i q^{32} +4248.00i q^{34} +2380.00 q^{35} +3888.00 q^{36} -10986.0i q^{37} -312.000 q^{38} +896.000 q^{40} +1050.00i q^{41} +5900.00 q^{43} -4000.00i q^{44} +3402.00i q^{45} +6304.00i q^{46} +5962.00i q^{47} -12093.0 q^{49} -11716.0i q^{50} +29046.0 q^{53} +3500.00 q^{55} -10880.0 q^{56} -10312.0i q^{58} +13922.0i q^{59} -32882.0 q^{61} -34616.0 q^{62} -41310.0i q^{63} -4096.00 q^{64} -69566.0i q^{67} +16992.0 q^{68} -9520.00i q^{70} -50542.0i q^{71} -15552.0i q^{72} +46750.0i q^{73} -43944.0 q^{74} +1248.00i q^{76} -42500.0 q^{77} -19348.0 q^{79} -3584.00i q^{80} +59049.0 q^{81} +4200.00 q^{82} -87438.0i q^{83} +14868.0i q^{85} -23600.0i q^{86} -16000.0 q^{88} -94170.0i q^{89} +13608.0 q^{90} +25216.0 q^{92} +23848.0 q^{94} -1092.00 q^{95} +182786. i q^{97} +48372.0i q^{98} -60750.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 486 q^{9} - 112 q^{10} + 1360 q^{14} + 512 q^{16} - 2124 q^{17} + 2000 q^{22} - 3152 q^{23} + 5858 q^{25} + 5156 q^{29} + 4760 q^{35} + 7776 q^{36} - 624 q^{38} + 1792 q^{40} + 11800 q^{43} - 24186 q^{49} + 58092 q^{53} + 7000 q^{55} - 21760 q^{56} - 65764 q^{61} - 69232 q^{62} - 8192 q^{64} + 33984 q^{68} - 87888 q^{74} - 85000 q^{77} - 38696 q^{79} + 118098 q^{81} + 8400 q^{82} - 32000 q^{88} + 27216 q^{90} + 50432 q^{92} + 47696 q^{94} - 2184 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −16.0000 −0.500000
\(5\) − 14.0000i − 0.250440i −0.992129 0.125220i \(-0.960036\pi\)
0.992129 0.125220i \(-0.0399636\pi\)
\(6\) 0 0
\(7\) 170.000i 1.31131i 0.755063 + 0.655653i \(0.227606\pi\)
−0.755063 + 0.655653i \(0.772394\pi\)
\(8\) 64.0000i 0.353553i
\(9\) −243.000 −1.00000
\(10\) −56.0000 −0.177088
\(11\) 250.000i 0.622957i 0.950253 + 0.311479i \(0.100824\pi\)
−0.950253 + 0.311479i \(0.899176\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 680.000 0.927233
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −1062.00 −0.891255 −0.445628 0.895218i \(-0.647019\pi\)
−0.445628 + 0.895218i \(0.647019\pi\)
\(18\) 972.000i 0.707107i
\(19\) − 78.0000i − 0.0495691i −0.999693 0.0247845i \(-0.992110\pi\)
0.999693 0.0247845i \(-0.00788997\pi\)
\(20\) 224.000i 0.125220i
\(21\) 0 0
\(22\) 1000.00 0.440497
\(23\) −1576.00 −0.621207 −0.310604 0.950539i \(-0.600531\pi\)
−0.310604 + 0.950539i \(0.600531\pi\)
\(24\) 0 0
\(25\) 2929.00 0.937280
\(26\) 0 0
\(27\) 0 0
\(28\) − 2720.00i − 0.655653i
\(29\) 2578.00 0.569230 0.284615 0.958642i \(-0.408134\pi\)
0.284615 + 0.958642i \(0.408134\pi\)
\(30\) 0 0
\(31\) − 8654.00i − 1.61738i −0.588234 0.808691i \(-0.700176\pi\)
0.588234 0.808691i \(-0.299824\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 0 0
\(34\) 4248.00i 0.630213i
\(35\) 2380.00 0.328403
\(36\) 3888.00 0.500000
\(37\) − 10986.0i − 1.31927i −0.751584 0.659637i \(-0.770710\pi\)
0.751584 0.659637i \(-0.229290\pi\)
\(38\) −312.000 −0.0350506
\(39\) 0 0
\(40\) 896.000 0.0885438
\(41\) 1050.00i 0.0975505i 0.998810 + 0.0487753i \(0.0155318\pi\)
−0.998810 + 0.0487753i \(0.984468\pi\)
\(42\) 0 0
\(43\) 5900.00 0.486610 0.243305 0.969950i \(-0.421768\pi\)
0.243305 + 0.969950i \(0.421768\pi\)
\(44\) − 4000.00i − 0.311479i
\(45\) 3402.00i 0.250440i
\(46\) 6304.00i 0.439260i
\(47\) 5962.00i 0.393684i 0.980435 + 0.196842i \(0.0630685\pi\)
−0.980435 + 0.196842i \(0.936931\pi\)
\(48\) 0 0
\(49\) −12093.0 −0.719522
\(50\) − 11716.0i − 0.662757i
\(51\) 0 0
\(52\) 0 0
\(53\) 29046.0 1.42035 0.710177 0.704023i \(-0.248615\pi\)
0.710177 + 0.704023i \(0.248615\pi\)
\(54\) 0 0
\(55\) 3500.00 0.156013
\(56\) −10880.0 −0.463616
\(57\) 0 0
\(58\) − 10312.0i − 0.402507i
\(59\) 13922.0i 0.520681i 0.965517 + 0.260340i \(0.0838348\pi\)
−0.965517 + 0.260340i \(0.916165\pi\)
\(60\) 0 0
\(61\) −32882.0 −1.13145 −0.565723 0.824596i \(-0.691403\pi\)
−0.565723 + 0.824596i \(0.691403\pi\)
\(62\) −34616.0 −1.14366
\(63\) − 41310.0i − 1.31131i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 69566.0i − 1.89326i −0.322324 0.946629i \(-0.604464\pi\)
0.322324 0.946629i \(-0.395536\pi\)
\(68\) 16992.0 0.445628
\(69\) 0 0
\(70\) − 9520.00i − 0.232216i
\(71\) − 50542.0i − 1.18989i −0.803767 0.594945i \(-0.797174\pi\)
0.803767 0.594945i \(-0.202826\pi\)
\(72\) − 15552.0i − 0.353553i
\(73\) 46750.0i 1.02677i 0.858157 + 0.513387i \(0.171609\pi\)
−0.858157 + 0.513387i \(0.828391\pi\)
\(74\) −43944.0 −0.932868
\(75\) 0 0
\(76\) 1248.00i 0.0247845i
\(77\) −42500.0 −0.816887
\(78\) 0 0
\(79\) −19348.0 −0.348793 −0.174397 0.984675i \(-0.555798\pi\)
−0.174397 + 0.984675i \(0.555798\pi\)
\(80\) − 3584.00i − 0.0626099i
\(81\) 59049.0 1.00000
\(82\) 4200.00 0.0689786
\(83\) − 87438.0i − 1.39317i −0.717473 0.696586i \(-0.754701\pi\)
0.717473 0.696586i \(-0.245299\pi\)
\(84\) 0 0
\(85\) 14868.0i 0.223206i
\(86\) − 23600.0i − 0.344085i
\(87\) 0 0
\(88\) −16000.0 −0.220249
\(89\) − 94170.0i − 1.26019i −0.776516 0.630097i \(-0.783015\pi\)
0.776516 0.630097i \(-0.216985\pi\)
\(90\) 13608.0 0.177088
\(91\) 0 0
\(92\) 25216.0 0.310604
\(93\) 0 0
\(94\) 23848.0 0.278376
\(95\) −1092.00 −0.0124141
\(96\) 0 0
\(97\) 182786.i 1.97248i 0.165307 + 0.986242i \(0.447139\pi\)
−0.165307 + 0.986242i \(0.552861\pi\)
\(98\) 48372.0i 0.508779i
\(99\) − 60750.0i − 0.622957i
\(100\) −46864.0 −0.468640
\(101\) 18514.0 0.180591 0.0902957 0.995915i \(-0.471219\pi\)
0.0902957 + 0.995915i \(0.471219\pi\)
\(102\) 0 0
\(103\) −116056. −1.07789 −0.538945 0.842341i \(-0.681177\pi\)
−0.538945 + 0.842341i \(0.681177\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 116184.i − 1.00434i
\(107\) 153520. 1.29630 0.648150 0.761513i \(-0.275543\pi\)
0.648150 + 0.761513i \(0.275543\pi\)
\(108\) 0 0
\(109\) − 178622.i − 1.44002i −0.693963 0.720010i \(-0.744137\pi\)
0.693963 0.720010i \(-0.255863\pi\)
\(110\) − 14000.0i − 0.110318i
\(111\) 0 0
\(112\) 43520.0i 0.327826i
\(113\) −244754. −1.80316 −0.901579 0.432615i \(-0.857591\pi\)
−0.901579 + 0.432615i \(0.857591\pi\)
\(114\) 0 0
\(115\) 22064.0i 0.155575i
\(116\) −41248.0 −0.284615
\(117\) 0 0
\(118\) 55688.0 0.368177
\(119\) − 180540.i − 1.16871i
\(120\) 0 0
\(121\) 98551.0 0.611924
\(122\) 131528.i 0.800053i
\(123\) 0 0
\(124\) 138464.i 0.808691i
\(125\) − 84756.0i − 0.485172i
\(126\) −165240. −0.927233
\(127\) −256600. −1.41172 −0.705858 0.708353i \(-0.749438\pi\)
−0.705858 + 0.708353i \(0.749438\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −262736. −1.33765 −0.668823 0.743421i \(-0.733202\pi\)
−0.668823 + 0.743421i \(0.733202\pi\)
\(132\) 0 0
\(133\) 13260.0 0.0650002
\(134\) −278264. −1.33874
\(135\) 0 0
\(136\) − 67968.0i − 0.315106i
\(137\) 38286.0i 0.174276i 0.996196 + 0.0871382i \(0.0277722\pi\)
−0.996196 + 0.0871382i \(0.972228\pi\)
\(138\) 0 0
\(139\) −57776.0 −0.253636 −0.126818 0.991926i \(-0.540476\pi\)
−0.126818 + 0.991926i \(0.540476\pi\)
\(140\) −38080.0 −0.164201
\(141\) 0 0
\(142\) −202168. −0.841379
\(143\) 0 0
\(144\) −62208.0 −0.250000
\(145\) − 36092.0i − 0.142558i
\(146\) 187000. 0.726038
\(147\) 0 0
\(148\) 175776.i 0.659637i
\(149\) 28866.0i 0.106517i 0.998581 + 0.0532587i \(0.0169608\pi\)
−0.998581 + 0.0532587i \(0.983039\pi\)
\(150\) 0 0
\(151\) − 39870.0i − 0.142300i −0.997466 0.0711498i \(-0.977333\pi\)
0.997466 0.0711498i \(-0.0226668\pi\)
\(152\) 4992.00 0.0175253
\(153\) 258066. 0.891255
\(154\) 170000.i 0.577627i
\(155\) −121156. −0.405057
\(156\) 0 0
\(157\) 161042. 0.521423 0.260711 0.965417i \(-0.416043\pi\)
0.260711 + 0.965417i \(0.416043\pi\)
\(158\) 77392.0i 0.246634i
\(159\) 0 0
\(160\) −14336.0 −0.0442719
\(161\) − 267920.i − 0.814593i
\(162\) − 236196.i − 0.707107i
\(163\) − 312830.i − 0.922230i −0.887340 0.461115i \(-0.847450\pi\)
0.887340 0.461115i \(-0.152550\pi\)
\(164\) − 16800.0i − 0.0487753i
\(165\) 0 0
\(166\) −349752. −0.985122
\(167\) − 532926.i − 1.47869i −0.673329 0.739343i \(-0.735136\pi\)
0.673329 0.739343i \(-0.264864\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 59472.0 0.157830
\(171\) 18954.0i 0.0495691i
\(172\) −94400.0 −0.243305
\(173\) 630458. 1.60155 0.800776 0.598964i \(-0.204421\pi\)
0.800776 + 0.598964i \(0.204421\pi\)
\(174\) 0 0
\(175\) 497930.i 1.22906i
\(176\) 64000.0i 0.155739i
\(177\) 0 0
\(178\) −376680. −0.891092
\(179\) 674916. 1.57441 0.787204 0.616693i \(-0.211528\pi\)
0.787204 + 0.616693i \(0.211528\pi\)
\(180\) − 54432.0i − 0.125220i
\(181\) −186282. −0.422644 −0.211322 0.977417i \(-0.567777\pi\)
−0.211322 + 0.977417i \(0.567777\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 100864.i − 0.219630i
\(185\) −153804. −0.330399
\(186\) 0 0
\(187\) − 265500.i − 0.555214i
\(188\) − 95392.0i − 0.196842i
\(189\) 0 0
\(190\) 4368.00i 0.00877806i
\(191\) 812180. 1.61090 0.805451 0.592663i \(-0.201923\pi\)
0.805451 + 0.592663i \(0.201923\pi\)
\(192\) 0 0
\(193\) 150142.i 0.290141i 0.989421 + 0.145070i \(0.0463409\pi\)
−0.989421 + 0.145070i \(0.953659\pi\)
\(194\) 731144. 1.39476
\(195\) 0 0
\(196\) 193488. 0.359761
\(197\) 236394.i 0.433981i 0.976174 + 0.216991i \(0.0696241\pi\)
−0.976174 + 0.216991i \(0.930376\pi\)
\(198\) −243000. −0.440497
\(199\) 39376.0 0.0704854 0.0352427 0.999379i \(-0.488780\pi\)
0.0352427 + 0.999379i \(0.488780\pi\)
\(200\) 187456.i 0.331379i
\(201\) 0 0
\(202\) − 74056.0i − 0.127697i
\(203\) 438260.i 0.746435i
\(204\) 0 0
\(205\) 14700.0 0.0244305
\(206\) 464224.i 0.762183i
\(207\) 382968. 0.621207
\(208\) 0 0
\(209\) 19500.0 0.0308794
\(210\) 0 0
\(211\) −410776. −0.635183 −0.317592 0.948228i \(-0.602874\pi\)
−0.317592 + 0.948228i \(0.602874\pi\)
\(212\) −464736. −0.710177
\(213\) 0 0
\(214\) − 614080.i − 0.916623i
\(215\) − 82600.0i − 0.121866i
\(216\) 0 0
\(217\) 1.47118e6 2.12088
\(218\) −714488. −1.01825
\(219\) 0 0
\(220\) −56000.0 −0.0780066
\(221\) 0 0
\(222\) 0 0
\(223\) 1.08688e6i 1.46359i 0.681523 + 0.731796i \(0.261318\pi\)
−0.681523 + 0.731796i \(0.738682\pi\)
\(224\) 174080. 0.231808
\(225\) −711747. −0.937280
\(226\) 979016.i 1.27502i
\(227\) − 256470.i − 0.330348i −0.986264 0.165174i \(-0.947181\pi\)
0.986264 0.165174i \(-0.0528186\pi\)
\(228\) 0 0
\(229\) 298110.i 0.375654i 0.982202 + 0.187827i \(0.0601444\pi\)
−0.982202 + 0.187827i \(0.939856\pi\)
\(230\) 88256.0 0.110008
\(231\) 0 0
\(232\) 164992.i 0.201253i
\(233\) 611926. 0.738430 0.369215 0.929344i \(-0.379627\pi\)
0.369215 + 0.929344i \(0.379627\pi\)
\(234\) 0 0
\(235\) 83468.0 0.0985940
\(236\) − 222752.i − 0.260340i
\(237\) 0 0
\(238\) −722160. −0.826401
\(239\) 36570.0i 0.0414124i 0.999786 + 0.0207062i \(0.00659146\pi\)
−0.999786 + 0.0207062i \(0.993409\pi\)
\(240\) 0 0
\(241\) − 380922.i − 0.422468i −0.977436 0.211234i \(-0.932252\pi\)
0.977436 0.211234i \(-0.0677482\pi\)
\(242\) − 394204.i − 0.432696i
\(243\) 0 0
\(244\) 526112. 0.565723
\(245\) 169302.i 0.180197i
\(246\) 0 0
\(247\) 0 0
\(248\) 553856. 0.571831
\(249\) 0 0
\(250\) −339024. −0.343068
\(251\) 1.22807e6 1.23038 0.615188 0.788380i \(-0.289080\pi\)
0.615188 + 0.788380i \(0.289080\pi\)
\(252\) 660960.i 0.655653i
\(253\) − 394000.i − 0.386986i
\(254\) 1.02640e6i 0.998234i
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 439278. 0.414865 0.207432 0.978249i \(-0.433489\pi\)
0.207432 + 0.978249i \(0.433489\pi\)
\(258\) 0 0
\(259\) 1.86762e6 1.72997
\(260\) 0 0
\(261\) −626454. −0.569230
\(262\) 1.05094e6i 0.945859i
\(263\) −1.67987e6 −1.49757 −0.748783 0.662816i \(-0.769361\pi\)
−0.748783 + 0.662816i \(0.769361\pi\)
\(264\) 0 0
\(265\) − 406644.i − 0.355713i
\(266\) − 53040.0i − 0.0459621i
\(267\) 0 0
\(268\) 1.11306e6i 0.946629i
\(269\) 1.93840e6 1.63329 0.816645 0.577141i \(-0.195832\pi\)
0.816645 + 0.577141i \(0.195832\pi\)
\(270\) 0 0
\(271\) 695498.i 0.575271i 0.957740 + 0.287636i \(0.0928692\pi\)
−0.957740 + 0.287636i \(0.907131\pi\)
\(272\) −271872. −0.222814
\(273\) 0 0
\(274\) 153144. 0.123232
\(275\) 732250.i 0.583885i
\(276\) 0 0
\(277\) 1.13138e6 0.885948 0.442974 0.896534i \(-0.353923\pi\)
0.442974 + 0.896534i \(0.353923\pi\)
\(278\) 231104.i 0.179348i
\(279\) 2.10292e6i 1.61738i
\(280\) 152320.i 0.116108i
\(281\) − 1.73122e6i − 1.30793i −0.756523 0.653967i \(-0.773103\pi\)
0.756523 0.653967i \(-0.226897\pi\)
\(282\) 0 0
\(283\) 1.47124e6 1.09199 0.545995 0.837788i \(-0.316152\pi\)
0.545995 + 0.837788i \(0.316152\pi\)
\(284\) 808672.i 0.594945i
\(285\) 0 0
\(286\) 0 0
\(287\) −178500. −0.127919
\(288\) 248832.i 0.176777i
\(289\) −292013. −0.205664
\(290\) −144368. −0.100804
\(291\) 0 0
\(292\) − 748000.i − 0.513387i
\(293\) − 2.88855e6i − 1.96567i −0.184491 0.982834i \(-0.559064\pi\)
0.184491 0.982834i \(-0.440936\pi\)
\(294\) 0 0
\(295\) 194908. 0.130399
\(296\) 703104. 0.466434
\(297\) 0 0
\(298\) 115464. 0.0753192
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00300e6i 0.638094i
\(302\) −159480. −0.100621
\(303\) 0 0
\(304\) − 19968.0i − 0.0123923i
\(305\) 460348.i 0.283359i
\(306\) − 1.03226e6i − 0.630213i
\(307\) − 874118.i − 0.529327i −0.964341 0.264664i \(-0.914739\pi\)
0.964341 0.264664i \(-0.0852609\pi\)
\(308\) 680000. 0.408444
\(309\) 0 0
\(310\) 484624.i 0.286418i
\(311\) −2.68224e6 −1.57252 −0.786261 0.617895i \(-0.787986\pi\)
−0.786261 + 0.617895i \(0.787986\pi\)
\(312\) 0 0
\(313\) −1.34459e6 −0.775761 −0.387880 0.921710i \(-0.626793\pi\)
−0.387880 + 0.921710i \(0.626793\pi\)
\(314\) − 644168.i − 0.368702i
\(315\) −578340. −0.328403
\(316\) 309568. 0.174397
\(317\) 1.32074e6i 0.738191i 0.929392 + 0.369095i \(0.120332\pi\)
−0.929392 + 0.369095i \(0.879668\pi\)
\(318\) 0 0
\(319\) 644500.i 0.354606i
\(320\) 57344.0i 0.0313050i
\(321\) 0 0
\(322\) −1.07168e6 −0.576004
\(323\) 82836.0i 0.0441787i
\(324\) −944784. −0.500000
\(325\) 0 0
\(326\) −1.25132e6 −0.652115
\(327\) 0 0
\(328\) −67200.0 −0.0344893
\(329\) −1.01354e6 −0.516239
\(330\) 0 0
\(331\) − 2.05728e6i − 1.03210i −0.856558 0.516051i \(-0.827401\pi\)
0.856558 0.516051i \(-0.172599\pi\)
\(332\) 1.39901e6i 0.696586i
\(333\) 2.66960e6i 1.31927i
\(334\) −2.13170e6 −1.04559
\(335\) −973924. −0.474147
\(336\) 0 0
\(337\) −453398. −0.217473 −0.108736 0.994071i \(-0.534680\pi\)
−0.108736 + 0.994071i \(0.534680\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 237888.i − 0.111603i
\(341\) 2.16350e6 1.00756
\(342\) 75816.0 0.0350506
\(343\) 801380.i 0.367793i
\(344\) 377600.i 0.172043i
\(345\) 0 0
\(346\) − 2.52183e6i − 1.13247i
\(347\) −1.23065e6 −0.548669 −0.274334 0.961634i \(-0.588457\pi\)
−0.274334 + 0.961634i \(0.588457\pi\)
\(348\) 0 0
\(349\) 2.43825e6i 1.07155i 0.844360 + 0.535777i \(0.179981\pi\)
−0.844360 + 0.535777i \(0.820019\pi\)
\(350\) 1.99172e6 0.869077
\(351\) 0 0
\(352\) 256000. 0.110124
\(353\) − 2.68315e6i − 1.14606i −0.819534 0.573031i \(-0.805767\pi\)
0.819534 0.573031i \(-0.194233\pi\)
\(354\) 0 0
\(355\) −707588. −0.297995
\(356\) 1.50672e6i 0.630097i
\(357\) 0 0
\(358\) − 2.69966e6i − 1.11327i
\(359\) − 1.58693e6i − 0.649864i −0.945737 0.324932i \(-0.894659\pi\)
0.945737 0.324932i \(-0.105341\pi\)
\(360\) −217728. −0.0885438
\(361\) 2.47002e6 0.997543
\(362\) 745128.i 0.298854i
\(363\) 0 0
\(364\) 0 0
\(365\) 654500. 0.257145
\(366\) 0 0
\(367\) −60052.0 −0.0232735 −0.0116368 0.999932i \(-0.503704\pi\)
−0.0116368 + 0.999932i \(0.503704\pi\)
\(368\) −403456. −0.155302
\(369\) − 255150.i − 0.0975505i
\(370\) 615216.i 0.233627i
\(371\) 4.93782e6i 1.86252i
\(372\) 0 0
\(373\) −4.01853e6 −1.49553 −0.747766 0.663963i \(-0.768873\pi\)
−0.747766 + 0.663963i \(0.768873\pi\)
\(374\) −1.06200e6 −0.392596
\(375\) 0 0
\(376\) −381568. −0.139188
\(377\) 0 0
\(378\) 0 0
\(379\) 1.67581e6i 0.599276i 0.954053 + 0.299638i \(0.0968659\pi\)
−0.954053 + 0.299638i \(0.903134\pi\)
\(380\) 17472.0 0.00620703
\(381\) 0 0
\(382\) − 3.24872e6i − 1.13908i
\(383\) 687258.i 0.239399i 0.992810 + 0.119700i \(0.0381932\pi\)
−0.992810 + 0.119700i \(0.961807\pi\)
\(384\) 0 0
\(385\) 595000.i 0.204581i
\(386\) 600568. 0.205161
\(387\) −1.43370e6 −0.486610
\(388\) − 2.92458e6i − 0.986242i
\(389\) −1.37611e6 −0.461082 −0.230541 0.973063i \(-0.574050\pi\)
−0.230541 + 0.973063i \(0.574050\pi\)
\(390\) 0 0
\(391\) 1.67371e6 0.553655
\(392\) − 773952.i − 0.254389i
\(393\) 0 0
\(394\) 945576. 0.306871
\(395\) 270872.i 0.0873517i
\(396\) 972000.i 0.311479i
\(397\) 721198.i 0.229656i 0.993385 + 0.114828i \(0.0366317\pi\)
−0.993385 + 0.114828i \(0.963368\pi\)
\(398\) − 157504.i − 0.0498407i
\(399\) 0 0
\(400\) 749824. 0.234320
\(401\) − 2.22681e6i − 0.691548i −0.938318 0.345774i \(-0.887616\pi\)
0.938318 0.345774i \(-0.112384\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −296224. −0.0902957
\(405\) − 826686.i − 0.250440i
\(406\) 1.75304e6 0.527809
\(407\) 2.74650e6 0.821852
\(408\) 0 0
\(409\) 2.00783e6i 0.593496i 0.954956 + 0.296748i \(0.0959021\pi\)
−0.954956 + 0.296748i \(0.904098\pi\)
\(410\) − 58800.0i − 0.0172750i
\(411\) 0 0
\(412\) 1.85690e6 0.538945
\(413\) −2.36674e6 −0.682772
\(414\) − 1.53187e6i − 0.439260i
\(415\) −1.22413e6 −0.348906
\(416\) 0 0
\(417\) 0 0
\(418\) − 78000.0i − 0.0218350i
\(419\) 5.99378e6 1.66788 0.833942 0.551852i \(-0.186079\pi\)
0.833942 + 0.551852i \(0.186079\pi\)
\(420\) 0 0
\(421\) − 5.32737e6i − 1.46490i −0.680822 0.732449i \(-0.738377\pi\)
0.680822 0.732449i \(-0.261623\pi\)
\(422\) 1.64310e6i 0.449142i
\(423\) − 1.44877e6i − 0.393684i
\(424\) 1.85894e6i 0.502171i
\(425\) −3.11060e6 −0.835356
\(426\) 0 0
\(427\) − 5.58994e6i − 1.48367i
\(428\) −2.45632e6 −0.648150
\(429\) 0 0
\(430\) −330400. −0.0861725
\(431\) − 5.42972e6i − 1.40794i −0.710230 0.703970i \(-0.751409\pi\)
0.710230 0.703970i \(-0.248591\pi\)
\(432\) 0 0
\(433\) −7.43979e6 −1.90696 −0.953479 0.301459i \(-0.902526\pi\)
−0.953479 + 0.301459i \(0.902526\pi\)
\(434\) − 5.88472e6i − 1.49969i
\(435\) 0 0
\(436\) 2.85795e6i 0.720010i
\(437\) 122928.i 0.0307927i
\(438\) 0 0
\(439\) −6.86418e6 −1.69991 −0.849957 0.526852i \(-0.823372\pi\)
−0.849957 + 0.526852i \(0.823372\pi\)
\(440\) 224000.i 0.0551590i
\(441\) 2.93860e6 0.719522
\(442\) 0 0
\(443\) −3.46630e6 −0.839182 −0.419591 0.907713i \(-0.637827\pi\)
−0.419591 + 0.907713i \(0.637827\pi\)
\(444\) 0 0
\(445\) −1.31838e6 −0.315603
\(446\) 4.34753e6 1.03492
\(447\) 0 0
\(448\) − 696320.i − 0.163913i
\(449\) 1.40426e6i 0.328725i 0.986400 + 0.164362i \(0.0525566\pi\)
−0.986400 + 0.164362i \(0.947443\pi\)
\(450\) 2.84699e6i 0.662757i
\(451\) −262500. −0.0607698
\(452\) 3.91606e6 0.901579
\(453\) 0 0
\(454\) −1.02588e6 −0.233591
\(455\) 0 0
\(456\) 0 0
\(457\) − 5.95072e6i − 1.33284i −0.745575 0.666421i \(-0.767825\pi\)
0.745575 0.666421i \(-0.232175\pi\)
\(458\) 1.19244e6 0.265627
\(459\) 0 0
\(460\) − 353024.i − 0.0777875i
\(461\) − 6.25465e6i − 1.37073i −0.728202 0.685363i \(-0.759644\pi\)
0.728202 0.685363i \(-0.240356\pi\)
\(462\) 0 0
\(463\) 1.55055e6i 0.336149i 0.985774 + 0.168075i \(0.0537550\pi\)
−0.985774 + 0.168075i \(0.946245\pi\)
\(464\) 659968. 0.142308
\(465\) 0 0
\(466\) − 2.44770e6i − 0.522149i
\(467\) 1.80480e6 0.382945 0.191472 0.981498i \(-0.438674\pi\)
0.191472 + 0.981498i \(0.438674\pi\)
\(468\) 0 0
\(469\) 1.18262e7 2.48264
\(470\) − 333872.i − 0.0697165i
\(471\) 0 0
\(472\) −891008. −0.184088
\(473\) 1.47500e6i 0.303137i
\(474\) 0 0
\(475\) − 228462.i − 0.0464601i
\(476\) 2.88864e6i 0.584354i
\(477\) −7.05818e6 −1.42035
\(478\) 146280. 0.0292830
\(479\) − 2.21809e6i − 0.441712i −0.975306 0.220856i \(-0.929115\pi\)
0.975306 0.220856i \(-0.0708851\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.52369e6 −0.298730
\(483\) 0 0
\(484\) −1.57682e6 −0.305962
\(485\) 2.55900e6 0.493988
\(486\) 0 0
\(487\) − 6.14268e6i − 1.17364i −0.809717 0.586821i \(-0.800379\pi\)
0.809717 0.586821i \(-0.199621\pi\)
\(488\) − 2.10445e6i − 0.400026i
\(489\) 0 0
\(490\) 677208. 0.127418
\(491\) −6.44486e6 −1.20645 −0.603226 0.797571i \(-0.706118\pi\)
−0.603226 + 0.797571i \(0.706118\pi\)
\(492\) 0 0
\(493\) −2.73784e6 −0.507330
\(494\) 0 0
\(495\) −850500. −0.156013
\(496\) − 2.21542e6i − 0.404346i
\(497\) 8.59214e6 1.56031
\(498\) 0 0
\(499\) 4.25838e6i 0.765584i 0.923835 + 0.382792i \(0.125037\pi\)
−0.923835 + 0.382792i \(0.874963\pi\)
\(500\) 1.35610e6i 0.242586i
\(501\) 0 0
\(502\) − 4.91227e6i − 0.870008i
\(503\) −3.56242e6 −0.627806 −0.313903 0.949455i \(-0.601637\pi\)
−0.313903 + 0.949455i \(0.601637\pi\)
\(504\) 2.64384e6 0.463616
\(505\) − 259196.i − 0.0452272i
\(506\) −1.57600e6 −0.273640
\(507\) 0 0
\(508\) 4.10560e6 0.705858
\(509\) 4.23936e6i 0.725281i 0.931929 + 0.362640i \(0.118125\pi\)
−0.931929 + 0.362640i \(0.881875\pi\)
\(510\) 0 0
\(511\) −7.94750e6 −1.34641
\(512\) − 262144.i − 0.0441942i
\(513\) 0 0
\(514\) − 1.75711e6i − 0.293354i
\(515\) 1.62478e6i 0.269946i
\(516\) 0 0
\(517\) −1.49050e6 −0.245248
\(518\) − 7.47048e6i − 1.22328i
\(519\) 0 0
\(520\) 0 0
\(521\) 2.38657e6 0.385194 0.192597 0.981278i \(-0.438309\pi\)
0.192597 + 0.981278i \(0.438309\pi\)
\(522\) 2.50582e6i 0.402507i
\(523\) −8.84129e6 −1.41339 −0.706694 0.707519i \(-0.749814\pi\)
−0.706694 + 0.707519i \(0.749814\pi\)
\(524\) 4.20378e6 0.668823
\(525\) 0 0
\(526\) 6.71947e6i 1.05894i
\(527\) 9.19055e6i 1.44150i
\(528\) 0 0
\(529\) −3.95257e6 −0.614101
\(530\) −1.62658e6 −0.251527
\(531\) − 3.38305e6i − 0.520681i
\(532\) −212160. −0.0325001
\(533\) 0 0
\(534\) 0 0
\(535\) − 2.14928e6i − 0.324645i
\(536\) 4.45222e6 0.669368
\(537\) 0 0
\(538\) − 7.75361e6i − 1.15491i
\(539\) − 3.02325e6i − 0.448231i
\(540\) 0 0
\(541\) − 70058.0i − 0.0102912i −0.999987 0.00514558i \(-0.998362\pi\)
0.999987 0.00514558i \(-0.00163790\pi\)
\(542\) 2.78199e6 0.406778
\(543\) 0 0
\(544\) 1.08749e6i 0.157553i
\(545\) −2.50071e6 −0.360638
\(546\) 0 0
\(547\) −6.60752e6 −0.944213 −0.472107 0.881541i \(-0.656506\pi\)
−0.472107 + 0.881541i \(0.656506\pi\)
\(548\) − 612576.i − 0.0871382i
\(549\) 7.99033e6 1.13145
\(550\) 2.92900e6 0.412869
\(551\) − 201084.i − 0.0282162i
\(552\) 0 0
\(553\) − 3.28916e6i − 0.457375i
\(554\) − 4.52551e6i − 0.626460i
\(555\) 0 0
\(556\) 924416. 0.126818
\(557\) 1.10726e7i 1.51221i 0.654448 + 0.756107i \(0.272901\pi\)
−0.654448 + 0.756107i \(0.727099\pi\)
\(558\) 8.41169e6 1.14366
\(559\) 0 0
\(560\) 609280. 0.0821007
\(561\) 0 0
\(562\) −6.92487e6 −0.924849
\(563\) 1.43532e6 0.190843 0.0954216 0.995437i \(-0.469580\pi\)
0.0954216 + 0.995437i \(0.469580\pi\)
\(564\) 0 0
\(565\) 3.42656e6i 0.451582i
\(566\) − 5.88498e6i − 0.772153i
\(567\) 1.00383e7i 1.31131i
\(568\) 3.23469e6 0.420689
\(569\) 1.17051e7 1.51564 0.757818 0.652466i \(-0.226266\pi\)
0.757818 + 0.652466i \(0.226266\pi\)
\(570\) 0 0
\(571\) −4.81885e6 −0.618519 −0.309260 0.950978i \(-0.600081\pi\)
−0.309260 + 0.950978i \(0.600081\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 714000.i 0.0904521i
\(575\) −4.61610e6 −0.582245
\(576\) 995328. 0.125000
\(577\) − 1.35572e6i − 0.169523i −0.996401 0.0847617i \(-0.972987\pi\)
0.996401 0.0847617i \(-0.0270129\pi\)
\(578\) 1.16805e6i 0.145426i
\(579\) 0 0
\(580\) 577472.i 0.0712789i
\(581\) 1.48645e7 1.82687
\(582\) 0 0
\(583\) 7.26150e6i 0.884820i
\(584\) −2.99200e6 −0.363019
\(585\) 0 0
\(586\) −1.15542e7 −1.38994
\(587\) 5.03941e6i 0.603649i 0.953364 + 0.301824i \(0.0975957\pi\)
−0.953364 + 0.301824i \(0.902404\pi\)
\(588\) 0 0
\(589\) −675012. −0.0801721
\(590\) − 779632.i − 0.0922061i
\(591\) 0 0
\(592\) − 2.81242e6i − 0.329819i
\(593\) − 9.16124e6i − 1.06984i −0.844904 0.534919i \(-0.820342\pi\)
0.844904 0.534919i \(-0.179658\pi\)
\(594\) 0 0
\(595\) −2.52756e6 −0.292691
\(596\) − 461856.i − 0.0532587i
\(597\) 0 0
\(598\) 0 0
\(599\) −6.46635e6 −0.736363 −0.368182 0.929754i \(-0.620020\pi\)
−0.368182 + 0.929754i \(0.620020\pi\)
\(600\) 0 0
\(601\) −1.18021e7 −1.33282 −0.666411 0.745585i \(-0.732170\pi\)
−0.666411 + 0.745585i \(0.732170\pi\)
\(602\) 4.01200e6 0.451201
\(603\) 1.69045e7i 1.89326i
\(604\) 637920.i 0.0711498i
\(605\) − 1.37971e6i − 0.153250i
\(606\) 0 0
\(607\) 2.25748e6 0.248686 0.124343 0.992239i \(-0.460318\pi\)
0.124343 + 0.992239i \(0.460318\pi\)
\(608\) −79872.0 −0.00876265
\(609\) 0 0
\(610\) 1.84139e6 0.200365
\(611\) 0 0
\(612\) −4.12906e6 −0.445628
\(613\) − 2.75378e6i − 0.295991i −0.988988 0.147995i \(-0.952718\pi\)
0.988988 0.147995i \(-0.0472821\pi\)
\(614\) −3.49647e6 −0.374291
\(615\) 0 0
\(616\) − 2.72000e6i − 0.288813i
\(617\) 3.41607e6i 0.361255i 0.983552 + 0.180627i \(0.0578128\pi\)
−0.983552 + 0.180627i \(0.942187\pi\)
\(618\) 0 0
\(619\) − 9.43169e6i − 0.989379i −0.869070 0.494690i \(-0.835282\pi\)
0.869070 0.494690i \(-0.164718\pi\)
\(620\) 1.93850e6 0.202528
\(621\) 0 0
\(622\) 1.07290e7i 1.11194i
\(623\) 1.60089e7 1.65250
\(624\) 0 0
\(625\) 7.96654e6 0.815774
\(626\) 5.37834e6i 0.548546i
\(627\) 0 0
\(628\) −2.57667e6 −0.260711
\(629\) 1.16671e7i 1.17581i
\(630\) 2.31336e6i 0.232216i
\(631\) 4.87474e6i 0.487391i 0.969852 + 0.243696i \(0.0783598\pi\)
−0.969852 + 0.243696i \(0.921640\pi\)
\(632\) − 1.23827e6i − 0.123317i
\(633\) 0 0
\(634\) 5.28295e6 0.521980
\(635\) 3.59240e6i 0.353550i
\(636\) 0 0
\(637\) 0 0
\(638\) 2.57800e6 0.250744
\(639\) 1.22817e7i 1.18989i
\(640\) 229376. 0.0221359
\(641\) −9.74279e6 −0.936566 −0.468283 0.883579i \(-0.655127\pi\)
−0.468283 + 0.883579i \(0.655127\pi\)
\(642\) 0 0
\(643\) 1.63894e6i 0.156327i 0.996941 + 0.0781637i \(0.0249057\pi\)
−0.996941 + 0.0781637i \(0.975094\pi\)
\(644\) 4.28672e6i 0.407296i
\(645\) 0 0
\(646\) 331344. 0.0312390
\(647\) 1.59069e6 0.149391 0.0746955 0.997206i \(-0.476202\pi\)
0.0746955 + 0.997206i \(0.476202\pi\)
\(648\) 3.77914e6i 0.353553i
\(649\) −3.48050e6 −0.324362
\(650\) 0 0
\(651\) 0 0
\(652\) 5.00528e6i 0.461115i
\(653\) 1.59778e7 1.46634 0.733170 0.680045i \(-0.238040\pi\)
0.733170 + 0.680045i \(0.238040\pi\)
\(654\) 0 0
\(655\) 3.67830e6i 0.335000i
\(656\) 268800.i 0.0243876i
\(657\) − 1.13602e7i − 1.02677i
\(658\) 4.05416e6i 0.365036i
\(659\) −6.02458e6 −0.540397 −0.270199 0.962805i \(-0.587089\pi\)
−0.270199 + 0.962805i \(0.587089\pi\)
\(660\) 0 0
\(661\) 2.00705e7i 1.78671i 0.449352 + 0.893355i \(0.351655\pi\)
−0.449352 + 0.893355i \(0.648345\pi\)
\(662\) −8.22911e6 −0.729807
\(663\) 0 0
\(664\) 5.59603e6 0.492561
\(665\) − 185640.i − 0.0162786i
\(666\) 1.06784e7 0.932868
\(667\) −4.06293e6 −0.353610
\(668\) 8.52682e6i 0.739343i
\(669\) 0 0
\(670\) 3.89570e6i 0.335273i
\(671\) − 8.22050e6i − 0.704842i
\(672\) 0 0
\(673\) 5.48575e6 0.466873 0.233436 0.972372i \(-0.425003\pi\)
0.233436 + 0.972372i \(0.425003\pi\)
\(674\) 1.81359e6i 0.153776i
\(675\) 0 0
\(676\) 0 0
\(677\) −4.74926e6 −0.398248 −0.199124 0.979974i \(-0.563810\pi\)
−0.199124 + 0.979974i \(0.563810\pi\)
\(678\) 0 0
\(679\) −3.10736e7 −2.58653
\(680\) −951552. −0.0789151
\(681\) 0 0
\(682\) − 8.65400e6i − 0.712453i
\(683\) 6.13964e6i 0.503606i 0.967778 + 0.251803i \(0.0810236\pi\)
−0.967778 + 0.251803i \(0.918976\pi\)
\(684\) − 303264.i − 0.0247845i
\(685\) 536004. 0.0436457
\(686\) 3.20552e6 0.260069
\(687\) 0 0
\(688\) 1.51040e6 0.121652
\(689\) 0 0
\(690\) 0 0
\(691\) 1.57617e7i 1.25577i 0.778308 + 0.627883i \(0.216078\pi\)
−0.778308 + 0.627883i \(0.783922\pi\)
\(692\) −1.00873e7 −0.800776
\(693\) 1.03275e7 0.816887
\(694\) 4.92259e6i 0.387967i
\(695\) 808864.i 0.0635204i
\(696\) 0 0
\(697\) − 1.11510e6i − 0.0869424i
\(698\) 9.75298e6 0.757703
\(699\) 0 0
\(700\) − 7.96688e6i − 0.614530i
\(701\) 1.42036e7 1.09170 0.545851 0.837882i \(-0.316207\pi\)
0.545851 + 0.837882i \(0.316207\pi\)
\(702\) 0 0
\(703\) −856908. −0.0653952
\(704\) − 1.02400e6i − 0.0778697i
\(705\) 0 0
\(706\) −1.07326e7 −0.810388
\(707\) 3.14738e6i 0.236810i
\(708\) 0 0
\(709\) − 1.60718e7i − 1.20074i −0.799723 0.600369i \(-0.795020\pi\)
0.799723 0.600369i \(-0.204980\pi\)
\(710\) 2.83035e6i 0.210715i
\(711\) 4.70156e6 0.348793
\(712\) 6.02688e6 0.445546
\(713\) 1.36387e7i 1.00473i
\(714\) 0 0
\(715\) 0 0
\(716\) −1.07987e7 −0.787204
\(717\) 0 0
\(718\) −6.34774e6 −0.459524
\(719\) 2.07078e7 1.49387 0.746933 0.664900i \(-0.231526\pi\)
0.746933 + 0.664900i \(0.231526\pi\)
\(720\) 870912.i 0.0626099i
\(721\) − 1.97295e7i − 1.41344i
\(722\) − 9.88006e6i − 0.705369i
\(723\) 0 0
\(724\) 2.98051e6 0.211322
\(725\) 7.55096e6 0.533528
\(726\) 0 0
\(727\) −5.04803e6 −0.354231 −0.177115 0.984190i \(-0.556677\pi\)
−0.177115 + 0.984190i \(0.556677\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) − 2.61800e6i − 0.181829i
\(731\) −6.26580e6 −0.433694
\(732\) 0 0
\(733\) − 2.10377e7i − 1.44623i −0.690728 0.723115i \(-0.742710\pi\)
0.690728 0.723115i \(-0.257290\pi\)
\(734\) 240208.i 0.0164569i
\(735\) 0 0
\(736\) 1.61382e6i 0.109815i
\(737\) 1.73915e7 1.17942
\(738\) −1.02060e6 −0.0689786
\(739\) − 1.38992e7i − 0.936218i −0.883671 0.468109i \(-0.844935\pi\)
0.883671 0.468109i \(-0.155065\pi\)
\(740\) 2.46086e6 0.165199
\(741\) 0 0
\(742\) 1.97513e7 1.31700
\(743\) 1.23267e6i 0.0819169i 0.999161 + 0.0409584i \(0.0130411\pi\)
−0.999161 + 0.0409584i \(0.986959\pi\)
\(744\) 0 0
\(745\) 404124. 0.0266762
\(746\) 1.60741e7i 1.05750i
\(747\) 2.12474e7i 1.39317i
\(748\) 4.24800e6i 0.277607i
\(749\) 2.60984e7i 1.69985i
\(750\) 0 0
\(751\) 1.62624e6 0.105217 0.0526084 0.998615i \(-0.483247\pi\)
0.0526084 + 0.998615i \(0.483247\pi\)
\(752\) 1.52627e6i 0.0984209i
\(753\) 0 0
\(754\) 0 0
\(755\) −558180. −0.0356375
\(756\) 0 0
\(757\) 3.49882e6 0.221913 0.110956 0.993825i \(-0.464609\pi\)
0.110956 + 0.993825i \(0.464609\pi\)
\(758\) 6.70324e6 0.423752
\(759\) 0 0
\(760\) − 69888.0i − 0.00438903i
\(761\) − 2.21713e7i − 1.38781i −0.720067 0.693905i \(-0.755889\pi\)
0.720067 0.693905i \(-0.244111\pi\)
\(762\) 0 0
\(763\) 3.03657e7 1.88831
\(764\) −1.29949e7 −0.805451
\(765\) − 3.61292e6i − 0.223206i
\(766\) 2.74903e6 0.169281
\(767\) 0 0
\(768\) 0 0
\(769\) 1.08955e6i 0.0664400i 0.999448 + 0.0332200i \(0.0105762\pi\)
−0.999448 + 0.0332200i \(0.989424\pi\)
\(770\) 2.38000e6 0.144661
\(771\) 0 0
\(772\) − 2.40227e6i − 0.145070i
\(773\) 1.95219e6i 0.117510i 0.998272 + 0.0587549i \(0.0187130\pi\)
−0.998272 + 0.0587549i \(0.981287\pi\)
\(774\) 5.73480e6i 0.344085i
\(775\) − 2.53476e7i − 1.51594i
\(776\) −1.16983e7 −0.697379
\(777\) 0 0
\(778\) 5.50442e6i 0.326034i
\(779\) 81900.0 0.00483549
\(780\) 0 0
\(781\) 1.26355e7 0.741250
\(782\) − 6.69485e6i − 0.391493i
\(783\) 0 0
\(784\) −3.09581e6 −0.179880
\(785\) − 2.25459e6i − 0.130585i
\(786\) 0 0
\(787\) 1.44531e7i 0.831809i 0.909408 + 0.415904i \(0.136535\pi\)
−0.909408 + 0.415904i \(0.863465\pi\)
\(788\) − 3.78230e6i − 0.216991i
\(789\) 0 0
\(790\) 1.08349e6 0.0617670
\(791\) − 4.16082e7i − 2.36449i
\(792\) 3.88800e6 0.220249
\(793\) 0 0
\(794\) 2.88479e6 0.162391
\(795\) 0 0
\(796\) −630016. −0.0352427
\(797\) −1.23500e7 −0.688685 −0.344343 0.938844i \(-0.611898\pi\)
−0.344343 + 0.938844i \(0.611898\pi\)
\(798\) 0 0
\(799\) − 6.33164e6i − 0.350873i
\(800\) − 2.99930e6i − 0.165689i
\(801\) 2.28833e7i 1.26019i
\(802\) −8.90724e6 −0.488998
\(803\) −1.16875e7 −0.639636
\(804\) 0 0
\(805\) −3.75088e6 −0.204006
\(806\) 0 0
\(807\) 0 0
\(808\) 1.18490e6i 0.0638487i
\(809\) 1.15968e7 0.622970 0.311485 0.950251i \(-0.399174\pi\)
0.311485 + 0.950251i \(0.399174\pi\)
\(810\) −3.30674e6 −0.177088
\(811\) − 2.47534e7i − 1.32155i −0.750585 0.660774i \(-0.770228\pi\)
0.750585 0.660774i \(-0.229772\pi\)
\(812\) − 7.01216e6i − 0.373217i
\(813\) 0 0
\(814\) − 1.09860e7i − 0.581137i
\(815\) −4.37962e6 −0.230963
\(816\) 0 0
\(817\) − 460200.i − 0.0241208i
\(818\) 8.03130e6 0.419665
\(819\) 0 0
\(820\) −235200. −0.0122153
\(821\) − 2.47470e6i − 0.128134i −0.997946 0.0640671i \(-0.979593\pi\)
0.997946 0.0640671i \(-0.0204072\pi\)
\(822\) 0 0
\(823\) 7.84754e6 0.403863 0.201932 0.979400i \(-0.435278\pi\)
0.201932 + 0.979400i \(0.435278\pi\)
\(824\) − 7.42758e6i − 0.381092i
\(825\) 0 0
\(826\) 9.46696e6i 0.482792i
\(827\) 2.26192e7i 1.15004i 0.818140 + 0.575020i \(0.195006\pi\)
−0.818140 + 0.575020i \(0.804994\pi\)
\(828\) −6.12749e6 −0.310604
\(829\) 1.73912e7 0.878907 0.439454 0.898265i \(-0.355172\pi\)
0.439454 + 0.898265i \(0.355172\pi\)
\(830\) 4.89653e6i 0.246714i
\(831\) 0 0
\(832\) 0 0
\(833\) 1.28428e7 0.641278
\(834\) 0 0
\(835\) −7.46096e6 −0.370321
\(836\) −312000. −0.0154397
\(837\) 0 0
\(838\) − 2.39751e7i − 1.17937i
\(839\) 3.43825e7i 1.68629i 0.537684 + 0.843147i \(0.319299\pi\)
−0.537684 + 0.843147i \(0.680701\pi\)
\(840\) 0 0
\(841\) −1.38651e7 −0.675977
\(842\) −2.13095e7 −1.03584
\(843\) 0 0
\(844\) 6.57242e6 0.317592
\(845\) 0 0
\(846\) −5.79506e6 −0.278376
\(847\) 1.67537e7i 0.802419i
\(848\) 7.43578e6 0.355089
\(849\) 0 0
\(850\) 1.24424e7i 0.590686i
\(851\) 1.73139e7i 0.819543i
\(852\) 0 0
\(853\) − 2.31007e7i − 1.08706i −0.839391 0.543528i \(-0.817088\pi\)
0.839391 0.543528i \(-0.182912\pi\)
\(854\) −2.23598e7 −1.04911
\(855\) 265356. 0.0124141
\(856\) 9.82528e6i 0.458311i
\(857\) 7.02305e6 0.326643 0.163322 0.986573i \(-0.447779\pi\)
0.163322 + 0.986573i \(0.447779\pi\)
\(858\) 0 0
\(859\) 8.82135e6 0.407899 0.203949 0.978981i \(-0.434622\pi\)
0.203949 + 0.978981i \(0.434622\pi\)
\(860\) 1.32160e6i 0.0609332i
\(861\) 0 0
\(862\) −2.17189e7 −0.995564
\(863\) 2.39560e7i 1.09493i 0.836828 + 0.547466i \(0.184408\pi\)
−0.836828 + 0.547466i \(0.815592\pi\)
\(864\) 0 0
\(865\) − 8.82641e6i − 0.401092i
\(866\) 2.97592e7i 1.34842i
\(867\) 0 0
\(868\) −2.35389e7 −1.06044
\(869\) − 4.83700e6i − 0.217283i
\(870\) 0 0
\(871\) 0 0
\(872\) 1.14318e7 0.509124
\(873\) − 4.44170e7i − 1.97248i
\(874\) 491712. 0.0217737
\(875\) 1.44085e7 0.636208
\(876\) 0 0
\(877\) − 5.79805e6i − 0.254556i −0.991867 0.127278i \(-0.959376\pi\)
0.991867 0.127278i \(-0.0406240\pi\)
\(878\) 2.74567e7i 1.20202i
\(879\) 0 0
\(880\) 896000. 0.0390033
\(881\) 1.30527e7 0.566580 0.283290 0.959034i \(-0.408574\pi\)
0.283290 + 0.959034i \(0.408574\pi\)
\(882\) − 1.17544e7i − 0.508779i
\(883\) −4.73009e6 −0.204159 −0.102079 0.994776i \(-0.532550\pi\)
−0.102079 + 0.994776i \(0.532550\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.38652e7i 0.593392i
\(887\) 2.80737e7 1.19809 0.599046 0.800714i \(-0.295547\pi\)
0.599046 + 0.800714i \(0.295547\pi\)
\(888\) 0 0
\(889\) − 4.36220e7i − 1.85119i
\(890\) 5.27352e6i 0.223165i
\(891\) 1.47622e7i 0.622957i
\(892\) − 1.73901e7i − 0.731796i
\(893\) 465036. 0.0195145
\(894\) 0 0
\(895\) − 9.44882e6i − 0.394294i
\(896\) −2.78528e6 −0.115904
\(897\) 0 0
\(898\) 5.61705e6 0.232443
\(899\) − 2.23100e7i − 0.920663i
\(900\) 1.13880e7 0.468640
\(901\) −3.08469e7 −1.26590
\(902\) 1.05000e6i 0.0429708i
\(903\) 0 0
\(904\) − 1.56643e7i − 0.637512i
\(905\) 2.60795e6i 0.105847i
\(906\) 0 0
\(907\) −2.28552e7 −0.922500 −0.461250 0.887270i \(-0.652599\pi\)
−0.461250 + 0.887270i \(0.652599\pi\)
\(908\) 4.10352e6i 0.165174i
\(909\) −4.49890e6 −0.180591
\(910\) 0 0
\(911\) −3.27335e7 −1.30676 −0.653381 0.757029i \(-0.726650\pi\)
−0.653381 + 0.757029i \(0.726650\pi\)
\(912\) 0 0
\(913\) 2.18595e7 0.867887
\(914\) −2.38029e7 −0.942462
\(915\) 0 0
\(916\) − 4.76976e6i − 0.187827i
\(917\) − 4.46651e7i − 1.75406i
\(918\) 0 0
\(919\) −1.27717e7 −0.498839 −0.249419 0.968396i \(-0.580240\pi\)
−0.249419 + 0.968396i \(0.580240\pi\)
\(920\) −1.41210e6 −0.0550040
\(921\) 0 0
\(922\) −2.50186e7 −0.969249
\(923\) 0 0
\(924\) 0 0
\(925\) − 3.21780e7i − 1.23653i
\(926\) 6.20218e6 0.237693
\(927\) 2.82016e7 1.07789
\(928\) − 2.63987e6i − 0.100627i
\(929\) 3.48297e7i 1.32407i 0.749473 + 0.662034i \(0.230307\pi\)
−0.749473 + 0.662034i \(0.769693\pi\)
\(930\) 0 0
\(931\) 943254.i 0.0356660i
\(932\) −9.79082e6 −0.369215
\(933\) 0 0
\(934\) − 7.21918e6i − 0.270783i
\(935\) −3.71700e6 −0.139048
\(936\) 0 0
\(937\) 3.00172e7 1.11692 0.558459 0.829532i \(-0.311393\pi\)
0.558459 + 0.829532i \(0.311393\pi\)
\(938\) − 4.73049e7i − 1.75549i
\(939\) 0 0
\(940\) −1.33549e6 −0.0492970
\(941\) − 4.50649e7i − 1.65907i −0.558457 0.829534i \(-0.688606\pi\)
0.558457 0.829534i \(-0.311394\pi\)
\(942\) 0 0
\(943\) − 1.65480e6i − 0.0605991i
\(944\) 3.56403e6i 0.130170i
\(945\) 0 0
\(946\) 5.90000e6 0.214350
\(947\) − 2.99276e7i − 1.08442i −0.840243 0.542210i \(-0.817588\pi\)
0.840243 0.542210i \(-0.182412\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −913848. −0.0328522
\(951\) 0 0
\(952\) 1.15546e7 0.413201
\(953\) 4.25147e7 1.51638 0.758188 0.652036i \(-0.226085\pi\)
0.758188 + 0.652036i \(0.226085\pi\)
\(954\) 2.82327e7i 1.00434i
\(955\) − 1.13705e7i − 0.403433i
\(956\) − 585120.i − 0.0207062i
\(957\) 0 0
\(958\) −8.87234e6 −0.312338
\(959\) −6.50862e6 −0.228530
\(960\) 0 0
\(961\) −4.62626e7 −1.61593
\(962\) 0 0
\(963\) −3.73054e7 −1.29630
\(964\) 6.09475e6i 0.211234i
\(965\) 2.10199e6 0.0726628
\(966\) 0 0
\(967\) − 3.00251e7i − 1.03257i −0.856417 0.516284i \(-0.827315\pi\)
0.856417 0.516284i \(-0.172685\pi\)
\(968\) 6.30726e6i 0.216348i
\(969\) 0 0
\(970\) − 1.02360e7i − 0.349302i
\(971\) −4.00864e7 −1.36442 −0.682211 0.731155i \(-0.738982\pi\)
−0.682211 + 0.731155i \(0.738982\pi\)
\(972\) 0 0
\(973\) − 9.82192e6i − 0.332594i
\(974\) −2.45707e7 −0.829890
\(975\) 0 0
\(976\) −8.41779e6 −0.282861
\(977\) 5.12151e7i 1.71657i 0.513174 + 0.858284i \(0.328469\pi\)
−0.513174 + 0.858284i \(0.671531\pi\)
\(978\) 0 0
\(979\) 2.35425e7 0.785047
\(980\) − 2.70883e6i − 0.0900984i
\(981\) 4.34051e7i 1.44002i
\(982\) 2.57794e7i 0.853090i
\(983\) − 1.82382e7i − 0.602004i −0.953624 0.301002i \(-0.902679\pi\)
0.953624 0.301002i \(-0.0973211\pi\)
\(984\) 0 0
\(985\) 3.30952e6 0.108686
\(986\) 1.09513e7i 0.358736i
\(987\) 0 0
\(988\) 0 0
\(989\) −9.29840e6 −0.302286
\(990\) 3.40200e6i 0.110318i
\(991\) −3.24103e7 −1.04833 −0.524166 0.851616i \(-0.675623\pi\)
−0.524166 + 0.851616i \(0.675623\pi\)
\(992\) −8.86170e6 −0.285915
\(993\) 0 0
\(994\) − 3.43686e7i − 1.10330i
\(995\) − 551264.i − 0.0176523i
\(996\) 0 0
\(997\) −2.07867e7 −0.662289 −0.331145 0.943580i \(-0.607435\pi\)
−0.331145 + 0.943580i \(0.607435\pi\)
\(998\) 1.70335e7 0.541350
\(999\) 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 338.6.b.a.337.1 2
13.5 odd 4 26.6.a.a.1.1 1
13.8 odd 4 338.6.a.d.1.1 1
13.12 even 2 inner 338.6.b.a.337.2 2
39.5 even 4 234.6.a.g.1.1 1
52.31 even 4 208.6.a.b.1.1 1
65.18 even 4 650.6.b.a.599.2 2
65.44 odd 4 650.6.a.a.1.1 1
65.57 even 4 650.6.b.a.599.1 2
104.5 odd 4 832.6.a.d.1.1 1
104.83 even 4 832.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.a.a.1.1 1 13.5 odd 4
208.6.a.b.1.1 1 52.31 even 4
234.6.a.g.1.1 1 39.5 even 4
338.6.a.d.1.1 1 13.8 odd 4
338.6.b.a.337.1 2 1.1 even 1 trivial
338.6.b.a.337.2 2 13.12 even 2 inner
650.6.a.a.1.1 1 65.44 odd 4
650.6.b.a.599.1 2 65.57 even 4
650.6.b.a.599.2 2 65.18 even 4
832.6.a.d.1.1 1 104.5 odd 4
832.6.a.e.1.1 1 104.83 even 4