Properties

Label 245.6.a.m.1.2
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

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: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(10.2693\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.26930 q^{2} +1.27429 q^{3} +53.9199 q^{4} +25.0000 q^{5} -11.8118 q^{6} -203.182 q^{8} -241.376 q^{9} +O(q^{10})\) \(q-9.26930 q^{2} +1.27429 q^{3} +53.9199 q^{4} +25.0000 q^{5} -11.8118 q^{6} -203.182 q^{8} -241.376 q^{9} -231.732 q^{10} +478.980 q^{11} +68.7096 q^{12} -203.039 q^{13} +31.8573 q^{15} +157.917 q^{16} +854.151 q^{17} +2237.39 q^{18} +1385.35 q^{19} +1348.00 q^{20} -4439.81 q^{22} +2414.39 q^{23} -258.913 q^{24} +625.000 q^{25} +1882.03 q^{26} -617.236 q^{27} -4207.61 q^{29} -295.295 q^{30} -7478.19 q^{31} +5038.04 q^{32} +610.360 q^{33} -7917.38 q^{34} -13015.0 q^{36} +9593.09 q^{37} -12841.2 q^{38} -258.731 q^{39} -5079.55 q^{40} -5503.30 q^{41} -5020.82 q^{43} +25826.5 q^{44} -6034.40 q^{45} -22379.7 q^{46} -23703.5 q^{47} +201.232 q^{48} -5793.31 q^{50} +1088.44 q^{51} -10947.9 q^{52} -10841.4 q^{53} +5721.35 q^{54} +11974.5 q^{55} +1765.34 q^{57} +39001.6 q^{58} +10487.3 q^{59} +1717.74 q^{60} +44185.5 q^{61} +69317.5 q^{62} -51752.4 q^{64} -5075.99 q^{65} -5657.61 q^{66} +53131.2 q^{67} +46055.7 q^{68} +3076.64 q^{69} +15897.1 q^{71} +49043.3 q^{72} +77627.1 q^{73} -88921.2 q^{74} +796.432 q^{75} +74697.8 q^{76} +2398.26 q^{78} -86136.0 q^{79} +3947.92 q^{80} +57867.9 q^{81} +51011.7 q^{82} -52126.2 q^{83} +21353.8 q^{85} +46539.5 q^{86} -5361.72 q^{87} -97320.0 q^{88} +23299.1 q^{89} +55934.7 q^{90} +130184. q^{92} -9529.38 q^{93} +219715. q^{94} +34633.7 q^{95} +6419.93 q^{96} +180093. q^{97} -115614. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 58 q^{3} + 182 q^{4} + 250 q^{5} + 144 q^{6} + 270 q^{8} + 700 q^{9} + 250 q^{10} + 794 q^{11} + 2560 q^{12} + 474 q^{13} + 1450 q^{15} + 2394 q^{16} + 802 q^{17} + 3702 q^{18} + 7292 q^{19} + 4550 q^{20} + 3948 q^{22} + 3708 q^{23} + 2092 q^{24} + 6250 q^{25} + 6576 q^{26} + 11818 q^{27} - 8866 q^{29} + 3600 q^{30} + 13292 q^{31} + 2590 q^{32} + 9854 q^{33} + 44468 q^{34} - 10690 q^{36} + 16124 q^{37} + 2180 q^{38} - 24982 q^{39} + 6750 q^{40} + 34836 q^{41} - 28604 q^{43} - 31120 q^{44} + 17500 q^{45} - 39732 q^{46} + 18106 q^{47} + 101788 q^{48} + 6250 q^{50} + 31602 q^{51} - 22480 q^{52} + 36440 q^{53} + 80836 q^{54} + 19850 q^{55} + 126988 q^{57} - 100356 q^{58} + 18644 q^{59} + 64000 q^{60} + 68120 q^{61} + 181052 q^{62} - 59358 q^{64} + 11850 q^{65} + 157780 q^{66} + 92328 q^{67} + 288540 q^{68} + 170888 q^{69} + 5044 q^{71} - 61654 q^{72} + 170160 q^{73} - 216584 q^{74} + 36250 q^{75} + 505180 q^{76} - 158008 q^{78} + 26442 q^{79} + 59850 q^{80} - 56314 q^{81} + 353948 q^{82} + 353360 q^{83} + 20050 q^{85} - 52940 q^{86} - 3190 q^{87} - 114916 q^{88} + 90704 q^{89} + 92550 q^{90} + 183520 q^{92} + 188560 q^{93} - 121388 q^{94} + 182300 q^{95} + 442220 q^{96} + 236382 q^{97} + 109024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.26930 −1.63860 −0.819298 0.573368i \(-0.805637\pi\)
−0.819298 + 0.573368i \(0.805637\pi\)
\(3\) 1.27429 0.0817458 0.0408729 0.999164i \(-0.486986\pi\)
0.0408729 + 0.999164i \(0.486986\pi\)
\(4\) 53.9199 1.68500
\(5\) 25.0000 0.447214
\(6\) −11.8118 −0.133948
\(7\) 0 0
\(8\) −203.182 −1.12243
\(9\) −241.376 −0.993318
\(10\) −231.732 −0.732802
\(11\) 478.980 1.19354 0.596768 0.802414i \(-0.296451\pi\)
0.596768 + 0.802414i \(0.296451\pi\)
\(12\) 68.7096 0.137741
\(13\) −203.039 −0.333213 −0.166607 0.986023i \(-0.553281\pi\)
−0.166607 + 0.986023i \(0.553281\pi\)
\(14\) 0 0
\(15\) 31.8573 0.0365578
\(16\) 157.917 0.154216
\(17\) 854.151 0.716824 0.358412 0.933564i \(-0.383318\pi\)
0.358412 + 0.933564i \(0.383318\pi\)
\(18\) 2237.39 1.62765
\(19\) 1385.35 0.880389 0.440195 0.897902i \(-0.354909\pi\)
0.440195 + 0.897902i \(0.354909\pi\)
\(20\) 1348.00 0.753553
\(21\) 0 0
\(22\) −4439.81 −1.95572
\(23\) 2414.39 0.951674 0.475837 0.879533i \(-0.342145\pi\)
0.475837 + 0.879533i \(0.342145\pi\)
\(24\) −258.913 −0.0917541
\(25\) 625.000 0.200000
\(26\) 1882.03 0.546002
\(27\) −617.236 −0.162945
\(28\) 0 0
\(29\) −4207.61 −0.929053 −0.464526 0.885559i \(-0.653775\pi\)
−0.464526 + 0.885559i \(0.653775\pi\)
\(30\) −295.295 −0.0599035
\(31\) −7478.19 −1.39763 −0.698815 0.715303i \(-0.746289\pi\)
−0.698815 + 0.715303i \(0.746289\pi\)
\(32\) 5038.04 0.869734
\(33\) 610.360 0.0975666
\(34\) −7917.38 −1.17458
\(35\) 0 0
\(36\) −13015.0 −1.67374
\(37\) 9593.09 1.15200 0.576002 0.817448i \(-0.304612\pi\)
0.576002 + 0.817448i \(0.304612\pi\)
\(38\) −12841.2 −1.44260
\(39\) −258.731 −0.0272388
\(40\) −5079.55 −0.501967
\(41\) −5503.30 −0.511285 −0.255643 0.966771i \(-0.582287\pi\)
−0.255643 + 0.966771i \(0.582287\pi\)
\(42\) 0 0
\(43\) −5020.82 −0.414099 −0.207049 0.978331i \(-0.566386\pi\)
−0.207049 + 0.978331i \(0.566386\pi\)
\(44\) 25826.5 2.01110
\(45\) −6034.40 −0.444225
\(46\) −22379.7 −1.55941
\(47\) −23703.5 −1.56520 −0.782598 0.622527i \(-0.786106\pi\)
−0.782598 + 0.622527i \(0.786106\pi\)
\(48\) 201.232 0.0126065
\(49\) 0 0
\(50\) −5793.31 −0.327719
\(51\) 1088.44 0.0585973
\(52\) −10947.9 −0.561463
\(53\) −10841.4 −0.530144 −0.265072 0.964229i \(-0.585396\pi\)
−0.265072 + 0.964229i \(0.585396\pi\)
\(54\) 5721.35 0.267002
\(55\) 11974.5 0.533766
\(56\) 0 0
\(57\) 1765.34 0.0719681
\(58\) 39001.6 1.52234
\(59\) 10487.3 0.392224 0.196112 0.980582i \(-0.437168\pi\)
0.196112 + 0.980582i \(0.437168\pi\)
\(60\) 1717.74 0.0615998
\(61\) 44185.5 1.52039 0.760196 0.649694i \(-0.225103\pi\)
0.760196 + 0.649694i \(0.225103\pi\)
\(62\) 69317.5 2.29015
\(63\) 0 0
\(64\) −51752.4 −1.57936
\(65\) −5075.99 −0.149017
\(66\) −5657.61 −0.159872
\(67\) 53131.2 1.44598 0.722991 0.690857i \(-0.242767\pi\)
0.722991 + 0.690857i \(0.242767\pi\)
\(68\) 46055.7 1.20785
\(69\) 3076.64 0.0777954
\(70\) 0 0
\(71\) 15897.1 0.374258 0.187129 0.982335i \(-0.440082\pi\)
0.187129 + 0.982335i \(0.440082\pi\)
\(72\) 49043.3 1.11493
\(73\) 77627.1 1.70493 0.852464 0.522786i \(-0.175107\pi\)
0.852464 + 0.522786i \(0.175107\pi\)
\(74\) −88921.2 −1.88767
\(75\) 796.432 0.0163492
\(76\) 74697.8 1.48345
\(77\) 0 0
\(78\) 2398.26 0.0446333
\(79\) −86136.0 −1.55280 −0.776402 0.630238i \(-0.782957\pi\)
−0.776402 + 0.630238i \(0.782957\pi\)
\(80\) 3947.92 0.0689674
\(81\) 57867.9 0.979998
\(82\) 51011.7 0.837790
\(83\) −52126.2 −0.830541 −0.415271 0.909698i \(-0.636313\pi\)
−0.415271 + 0.909698i \(0.636313\pi\)
\(84\) 0 0
\(85\) 21353.8 0.320573
\(86\) 46539.5 0.678540
\(87\) −5361.72 −0.0759461
\(88\) −97320.0 −1.33966
\(89\) 23299.1 0.311791 0.155896 0.987774i \(-0.450174\pi\)
0.155896 + 0.987774i \(0.450174\pi\)
\(90\) 55934.7 0.727905
\(91\) 0 0
\(92\) 130184. 1.60357
\(93\) −9529.38 −0.114250
\(94\) 219715. 2.56472
\(95\) 34633.7 0.393722
\(96\) 6419.93 0.0710971
\(97\) 180093. 1.94343 0.971714 0.236161i \(-0.0758893\pi\)
0.971714 + 0.236161i \(0.0758893\pi\)
\(98\) 0 0
\(99\) −115614. −1.18556
\(100\) 33699.9 0.336999
\(101\) 107548. 1.04906 0.524531 0.851392i \(-0.324241\pi\)
0.524531 + 0.851392i \(0.324241\pi\)
\(102\) −10089.0 −0.0960173
\(103\) 172551. 1.60259 0.801297 0.598266i \(-0.204144\pi\)
0.801297 + 0.598266i \(0.204144\pi\)
\(104\) 41253.9 0.374009
\(105\) 0 0
\(106\) 100492. 0.868691
\(107\) 14341.5 0.121098 0.0605488 0.998165i \(-0.480715\pi\)
0.0605488 + 0.998165i \(0.480715\pi\)
\(108\) −33281.3 −0.274562
\(109\) −190351. −1.53458 −0.767290 0.641300i \(-0.778395\pi\)
−0.767290 + 0.641300i \(0.778395\pi\)
\(110\) −110995. −0.874626
\(111\) 12224.4 0.0941715
\(112\) 0 0
\(113\) 66701.8 0.491407 0.245704 0.969345i \(-0.420981\pi\)
0.245704 + 0.969345i \(0.420981\pi\)
\(114\) −16363.4 −0.117927
\(115\) 60359.8 0.425602
\(116\) −226874. −1.56545
\(117\) 49008.9 0.330986
\(118\) −97209.9 −0.642696
\(119\) 0 0
\(120\) −6472.82 −0.0410337
\(121\) 68370.8 0.424529
\(122\) −409569. −2.49131
\(123\) −7012.80 −0.0417954
\(124\) −403223. −2.35500
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 156626. 0.861698 0.430849 0.902424i \(-0.358214\pi\)
0.430849 + 0.902424i \(0.358214\pi\)
\(128\) 318491. 1.71820
\(129\) −6397.99 −0.0338508
\(130\) 47050.8 0.244179
\(131\) 71582.9 0.364444 0.182222 0.983257i \(-0.441671\pi\)
0.182222 + 0.983257i \(0.441671\pi\)
\(132\) 32910.5 0.164399
\(133\) 0 0
\(134\) −492489. −2.36938
\(135\) −15430.9 −0.0728714
\(136\) −173548. −0.804586
\(137\) 299786. 1.36462 0.682308 0.731065i \(-0.260976\pi\)
0.682308 + 0.731065i \(0.260976\pi\)
\(138\) −28518.3 −0.127475
\(139\) 52166.8 0.229012 0.114506 0.993423i \(-0.463472\pi\)
0.114506 + 0.993423i \(0.463472\pi\)
\(140\) 0 0
\(141\) −30205.2 −0.127948
\(142\) −147355. −0.613258
\(143\) −97251.8 −0.397702
\(144\) −38117.4 −0.153185
\(145\) −105190. −0.415485
\(146\) −719549. −2.79369
\(147\) 0 0
\(148\) 517258. 1.94112
\(149\) −432166. −1.59472 −0.797361 0.603502i \(-0.793772\pi\)
−0.797361 + 0.603502i \(0.793772\pi\)
\(150\) −7382.36 −0.0267897
\(151\) 224235. 0.800315 0.400157 0.916446i \(-0.368955\pi\)
0.400157 + 0.916446i \(0.368955\pi\)
\(152\) −281477. −0.988177
\(153\) −206172. −0.712034
\(154\) 0 0
\(155\) −186955. −0.625039
\(156\) −13950.8 −0.0458972
\(157\) 235433. 0.762287 0.381143 0.924516i \(-0.375530\pi\)
0.381143 + 0.924516i \(0.375530\pi\)
\(158\) 798420. 2.54442
\(159\) −13815.0 −0.0433370
\(160\) 125951. 0.388957
\(161\) 0 0
\(162\) −536395. −1.60582
\(163\) 2875.40 0.00847675 0.00423837 0.999991i \(-0.498651\pi\)
0.00423837 + 0.999991i \(0.498651\pi\)
\(164\) −296737. −0.861513
\(165\) 15259.0 0.0436331
\(166\) 483174. 1.36092
\(167\) −402505. −1.11681 −0.558406 0.829568i \(-0.688587\pi\)
−0.558406 + 0.829568i \(0.688587\pi\)
\(168\) 0 0
\(169\) −330068. −0.888969
\(170\) −197935. −0.525290
\(171\) −334390. −0.874506
\(172\) −270722. −0.697755
\(173\) 599804. 1.52368 0.761841 0.647764i \(-0.224296\pi\)
0.761841 + 0.647764i \(0.224296\pi\)
\(174\) 49699.3 0.124445
\(175\) 0 0
\(176\) 75639.0 0.184062
\(177\) 13363.9 0.0320626
\(178\) −215966. −0.510900
\(179\) 640796. 1.49481 0.747407 0.664366i \(-0.231298\pi\)
0.747407 + 0.664366i \(0.231298\pi\)
\(180\) −325374. −0.748518
\(181\) 640997. 1.45432 0.727160 0.686468i \(-0.240840\pi\)
0.727160 + 0.686468i \(0.240840\pi\)
\(182\) 0 0
\(183\) 56305.2 0.124286
\(184\) −490561. −1.06819
\(185\) 239827. 0.515192
\(186\) 88330.7 0.187210
\(187\) 409121. 0.855555
\(188\) −1.27809e6 −2.63735
\(189\) 0 0
\(190\) −321030. −0.645151
\(191\) −795460. −1.57774 −0.788870 0.614561i \(-0.789333\pi\)
−0.788870 + 0.614561i \(0.789333\pi\)
\(192\) −65947.7 −0.129106
\(193\) 164019. 0.316958 0.158479 0.987362i \(-0.449341\pi\)
0.158479 + 0.987362i \(0.449341\pi\)
\(194\) −1.66934e6 −3.18449
\(195\) −6468.28 −0.0121816
\(196\) 0 0
\(197\) −79137.4 −0.145283 −0.0726417 0.997358i \(-0.523143\pi\)
−0.0726417 + 0.997358i \(0.523143\pi\)
\(198\) 1.07166e6 1.94265
\(199\) 990048. 1.77224 0.886122 0.463451i \(-0.153389\pi\)
0.886122 + 0.463451i \(0.153389\pi\)
\(200\) −126989. −0.224486
\(201\) 67704.6 0.118203
\(202\) −996899. −1.71899
\(203\) 0 0
\(204\) 58688.4 0.0987363
\(205\) −137582. −0.228654
\(206\) −1.59942e6 −2.62600
\(207\) −582777. −0.945315
\(208\) −32063.4 −0.0513867
\(209\) 663554. 1.05078
\(210\) 0 0
\(211\) 983192. 1.52031 0.760155 0.649742i \(-0.225123\pi\)
0.760155 + 0.649742i \(0.225123\pi\)
\(212\) −584564. −0.893290
\(213\) 20257.5 0.0305940
\(214\) −132936. −0.198430
\(215\) −125521. −0.185191
\(216\) 125411. 0.182895
\(217\) 0 0
\(218\) 1.76442e6 2.51456
\(219\) 98919.5 0.139371
\(220\) 645663. 0.899393
\(221\) −173426. −0.238855
\(222\) −113311. −0.154309
\(223\) 285245. 0.384110 0.192055 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(224\) 0 0
\(225\) −150860. −0.198664
\(226\) −618279. −0.805218
\(227\) 949950. 1.22359 0.611795 0.791016i \(-0.290448\pi\)
0.611795 + 0.791016i \(0.290448\pi\)
\(228\) 95186.7 0.121266
\(229\) −1.05143e6 −1.32493 −0.662463 0.749094i \(-0.730489\pi\)
−0.662463 + 0.749094i \(0.730489\pi\)
\(230\) −559493. −0.697389
\(231\) 0 0
\(232\) 854910. 1.04280
\(233\) −168689. −0.203562 −0.101781 0.994807i \(-0.532454\pi\)
−0.101781 + 0.994807i \(0.532454\pi\)
\(234\) −454278. −0.542353
\(235\) −592589. −0.699977
\(236\) 565474. 0.660895
\(237\) −109762. −0.126935
\(238\) 0 0
\(239\) 658618. 0.745829 0.372914 0.927866i \(-0.378359\pi\)
0.372914 + 0.927866i \(0.378359\pi\)
\(240\) 5030.80 0.00563779
\(241\) 314422. 0.348715 0.174358 0.984682i \(-0.444215\pi\)
0.174358 + 0.984682i \(0.444215\pi\)
\(242\) −633749. −0.695631
\(243\) 223729. 0.243056
\(244\) 2.38248e6 2.56185
\(245\) 0 0
\(246\) 65003.7 0.0684858
\(247\) −281280. −0.293357
\(248\) 1.51943e6 1.56874
\(249\) −66424.0 −0.0678932
\(250\) −144833. −0.146560
\(251\) −18719.5 −0.0187547 −0.00937736 0.999956i \(-0.502985\pi\)
−0.00937736 + 0.999956i \(0.502985\pi\)
\(252\) 0 0
\(253\) 1.15645e6 1.13586
\(254\) −1.45181e6 −1.41197
\(255\) 27210.9 0.0262055
\(256\) −1.29611e6 −1.23607
\(257\) 638749. 0.603251 0.301625 0.953427i \(-0.402471\pi\)
0.301625 + 0.953427i \(0.402471\pi\)
\(258\) 59304.9 0.0554678
\(259\) 0 0
\(260\) −273697. −0.251094
\(261\) 1.01562e6 0.922844
\(262\) −663524. −0.597177
\(263\) 32605.9 0.0290674 0.0145337 0.999894i \(-0.495374\pi\)
0.0145337 + 0.999894i \(0.495374\pi\)
\(264\) −124014. −0.109512
\(265\) −271034. −0.237088
\(266\) 0 0
\(267\) 29689.8 0.0254876
\(268\) 2.86483e6 2.43647
\(269\) −137381. −0.115757 −0.0578785 0.998324i \(-0.518434\pi\)
−0.0578785 + 0.998324i \(0.518434\pi\)
\(270\) 143034. 0.119407
\(271\) 1.08503e6 0.897467 0.448733 0.893666i \(-0.351875\pi\)
0.448733 + 0.893666i \(0.351875\pi\)
\(272\) 134885. 0.110545
\(273\) 0 0
\(274\) −2.77881e6 −2.23605
\(275\) 299362. 0.238707
\(276\) 165892. 0.131085
\(277\) −1.19753e6 −0.937753 −0.468877 0.883264i \(-0.655341\pi\)
−0.468877 + 0.883264i \(0.655341\pi\)
\(278\) −483550. −0.375257
\(279\) 1.80506e6 1.38829
\(280\) 0 0
\(281\) 864879. 0.653415 0.326708 0.945125i \(-0.394061\pi\)
0.326708 + 0.945125i \(0.394061\pi\)
\(282\) 279981. 0.209655
\(283\) 1.34159e6 0.995757 0.497879 0.867247i \(-0.334112\pi\)
0.497879 + 0.867247i \(0.334112\pi\)
\(284\) 857168. 0.630623
\(285\) 44133.4 0.0321851
\(286\) 901456. 0.651673
\(287\) 0 0
\(288\) −1.21606e6 −0.863923
\(289\) −690283. −0.486164
\(290\) 975039. 0.680812
\(291\) 229491. 0.158867
\(292\) 4.18564e6 2.87280
\(293\) −669371. −0.455510 −0.227755 0.973718i \(-0.573139\pi\)
−0.227755 + 0.973718i \(0.573139\pi\)
\(294\) 0 0
\(295\) 262183. 0.175408
\(296\) −1.94914e6 −1.29305
\(297\) −295644. −0.194481
\(298\) 4.00588e6 2.61311
\(299\) −490217. −0.317110
\(300\) 42943.5 0.0275483
\(301\) 0 0
\(302\) −2.07850e6 −1.31139
\(303\) 137048. 0.0857563
\(304\) 218770. 0.135770
\(305\) 1.10464e6 0.679940
\(306\) 1.91107e6 1.16674
\(307\) 134224. 0.0812804 0.0406402 0.999174i \(-0.487060\pi\)
0.0406402 + 0.999174i \(0.487060\pi\)
\(308\) 0 0
\(309\) 219880. 0.131005
\(310\) 1.73294e6 1.02419
\(311\) −395022. −0.231590 −0.115795 0.993273i \(-0.536942\pi\)
−0.115795 + 0.993273i \(0.536942\pi\)
\(312\) 52569.5 0.0305737
\(313\) −1.37654e6 −0.794195 −0.397098 0.917776i \(-0.629983\pi\)
−0.397098 + 0.917776i \(0.629983\pi\)
\(314\) −2.18230e6 −1.24908
\(315\) 0 0
\(316\) −4.64444e6 −2.61647
\(317\) −1.68888e6 −0.943951 −0.471976 0.881612i \(-0.656459\pi\)
−0.471976 + 0.881612i \(0.656459\pi\)
\(318\) 128056. 0.0710119
\(319\) −2.01536e6 −1.10886
\(320\) −1.29381e6 −0.706311
\(321\) 18275.3 0.00989922
\(322\) 0 0
\(323\) 1.18330e6 0.631084
\(324\) 3.12023e6 1.65129
\(325\) −126900. −0.0666426
\(326\) −26652.9 −0.0138900
\(327\) −242563. −0.125445
\(328\) 1.11817e6 0.573883
\(329\) 0 0
\(330\) −141440. −0.0714970
\(331\) 1.59017e6 0.797760 0.398880 0.917003i \(-0.369399\pi\)
0.398880 + 0.917003i \(0.369399\pi\)
\(332\) −2.81064e6 −1.39946
\(333\) −2.31554e6 −1.14431
\(334\) 3.73094e6 1.83000
\(335\) 1.32828e6 0.646663
\(336\) 0 0
\(337\) −823436. −0.394962 −0.197481 0.980307i \(-0.563276\pi\)
−0.197481 + 0.980307i \(0.563276\pi\)
\(338\) 3.05950e6 1.45666
\(339\) 84997.5 0.0401705
\(340\) 1.15139e6 0.540165
\(341\) −3.58190e6 −1.66812
\(342\) 3.09956e6 1.43296
\(343\) 0 0
\(344\) 1.02014e6 0.464797
\(345\) 76915.9 0.0347911
\(346\) −5.55977e6 −2.49670
\(347\) −2.62964e6 −1.17239 −0.586195 0.810170i \(-0.699375\pi\)
−0.586195 + 0.810170i \(0.699375\pi\)
\(348\) −289103. −0.127969
\(349\) −3.26813e6 −1.43627 −0.718134 0.695905i \(-0.755003\pi\)
−0.718134 + 0.695905i \(0.755003\pi\)
\(350\) 0 0
\(351\) 125323. 0.0542955
\(352\) 2.41312e6 1.03806
\(353\) −621440. −0.265438 −0.132719 0.991154i \(-0.542371\pi\)
−0.132719 + 0.991154i \(0.542371\pi\)
\(354\) −123874. −0.0525377
\(355\) 397427. 0.167373
\(356\) 1.25628e6 0.525367
\(357\) 0 0
\(358\) −5.93973e6 −2.44940
\(359\) 1.34486e6 0.550734 0.275367 0.961339i \(-0.411201\pi\)
0.275367 + 0.961339i \(0.411201\pi\)
\(360\) 1.22608e6 0.498612
\(361\) −556911. −0.224915
\(362\) −5.94159e6 −2.38304
\(363\) 87124.3 0.0347034
\(364\) 0 0
\(365\) 1.94068e6 0.762467
\(366\) −521910. −0.203654
\(367\) −2.75801e6 −1.06888 −0.534442 0.845205i \(-0.679478\pi\)
−0.534442 + 0.845205i \(0.679478\pi\)
\(368\) 381273. 0.146763
\(369\) 1.32836e6 0.507869
\(370\) −2.22303e6 −0.844192
\(371\) 0 0
\(372\) −513823. −0.192511
\(373\) 3.29037e6 1.22454 0.612269 0.790649i \(-0.290257\pi\)
0.612269 + 0.790649i \(0.290257\pi\)
\(374\) −3.79227e6 −1.40191
\(375\) 19910.8 0.00731157
\(376\) 4.81613e6 1.75683
\(377\) 854311. 0.309573
\(378\) 0 0
\(379\) −1.33527e6 −0.477496 −0.238748 0.971081i \(-0.576737\pi\)
−0.238748 + 0.971081i \(0.576737\pi\)
\(380\) 1.86744e6 0.663420
\(381\) 199587. 0.0704402
\(382\) 7.37336e6 2.58528
\(383\) −3.37282e6 −1.17489 −0.587443 0.809266i \(-0.699865\pi\)
−0.587443 + 0.809266i \(0.699865\pi\)
\(384\) 405851. 0.140455
\(385\) 0 0
\(386\) −1.52034e6 −0.519366
\(387\) 1.21191e6 0.411332
\(388\) 9.71061e6 3.27467
\(389\) −3.46519e6 −1.16105 −0.580527 0.814241i \(-0.697154\pi\)
−0.580527 + 0.814241i \(0.697154\pi\)
\(390\) 59956.4 0.0199606
\(391\) 2.06226e6 0.682183
\(392\) 0 0
\(393\) 91217.5 0.0297918
\(394\) 733548. 0.238061
\(395\) −2.15340e6 −0.694435
\(396\) −6.23391e6 −1.99766
\(397\) −1.75508e6 −0.558882 −0.279441 0.960163i \(-0.590149\pi\)
−0.279441 + 0.960163i \(0.590149\pi\)
\(398\) −9.17705e6 −2.90399
\(399\) 0 0
\(400\) 98698.0 0.0308431
\(401\) −5.12451e6 −1.59145 −0.795723 0.605661i \(-0.792909\pi\)
−0.795723 + 0.605661i \(0.792909\pi\)
\(402\) −627574. −0.193687
\(403\) 1.51837e6 0.465708
\(404\) 5.79900e6 1.76766
\(405\) 1.44670e6 0.438268
\(406\) 0 0
\(407\) 4.59490e6 1.37496
\(408\) −221151. −0.0657715
\(409\) 879358. 0.259931 0.129965 0.991519i \(-0.458513\pi\)
0.129965 + 0.991519i \(0.458513\pi\)
\(410\) 1.27529e6 0.374671
\(411\) 382015. 0.111552
\(412\) 9.30391e6 2.70037
\(413\) 0 0
\(414\) 5.40193e6 1.54899
\(415\) −1.30316e6 −0.371429
\(416\) −1.02292e6 −0.289807
\(417\) 66475.7 0.0187207
\(418\) −6.15068e6 −1.72180
\(419\) −5.68610e6 −1.58227 −0.791133 0.611644i \(-0.790508\pi\)
−0.791133 + 0.611644i \(0.790508\pi\)
\(420\) 0 0
\(421\) 627448. 0.172533 0.0862666 0.996272i \(-0.472506\pi\)
0.0862666 + 0.996272i \(0.472506\pi\)
\(422\) −9.11350e6 −2.49117
\(423\) 5.72147e6 1.55474
\(424\) 2.20277e6 0.595050
\(425\) 533844. 0.143365
\(426\) −187773. −0.0501312
\(427\) 0 0
\(428\) 773292. 0.204049
\(429\) −123927. −0.0325105
\(430\) 1.16349e6 0.303452
\(431\) −953877. −0.247343 −0.123671 0.992323i \(-0.539467\pi\)
−0.123671 + 0.992323i \(0.539467\pi\)
\(432\) −97472.0 −0.0251287
\(433\) 4.52050e6 1.15869 0.579344 0.815083i \(-0.303309\pi\)
0.579344 + 0.815083i \(0.303309\pi\)
\(434\) 0 0
\(435\) −134043. −0.0339641
\(436\) −1.02637e7 −2.58576
\(437\) 3.34477e6 0.837844
\(438\) −916914. −0.228372
\(439\) 6.99034e6 1.73116 0.865579 0.500772i \(-0.166950\pi\)
0.865579 + 0.500772i \(0.166950\pi\)
\(440\) −2.43300e6 −0.599115
\(441\) 0 0
\(442\) 1.60754e6 0.391387
\(443\) −4.66063e6 −1.12833 −0.564164 0.825663i \(-0.690801\pi\)
−0.564164 + 0.825663i \(0.690801\pi\)
\(444\) 659137. 0.158679
\(445\) 582478. 0.139437
\(446\) −2.64402e6 −0.629401
\(447\) −550706. −0.130362
\(448\) 0 0
\(449\) 6.40084e6 1.49838 0.749189 0.662357i \(-0.230444\pi\)
0.749189 + 0.662357i \(0.230444\pi\)
\(450\) 1.39837e6 0.325529
\(451\) −2.63597e6 −0.610237
\(452\) 3.59655e6 0.828020
\(453\) 285740. 0.0654224
\(454\) −8.80537e6 −2.00497
\(455\) 0 0
\(456\) −358684. −0.0807793
\(457\) 4.09410e6 0.916997 0.458498 0.888695i \(-0.348387\pi\)
0.458498 + 0.888695i \(0.348387\pi\)
\(458\) 9.74602e6 2.17102
\(459\) −527213. −0.116803
\(460\) 3.25459e6 0.717137
\(461\) 7.91448e6 1.73448 0.867242 0.497887i \(-0.165891\pi\)
0.867242 + 0.497887i \(0.165891\pi\)
\(462\) 0 0
\(463\) −4.60789e6 −0.998963 −0.499482 0.866324i \(-0.666476\pi\)
−0.499482 + 0.866324i \(0.666476\pi\)
\(464\) −664452. −0.143275
\(465\) −238235. −0.0510943
\(466\) 1.56363e6 0.333555
\(467\) 3.90317e6 0.828181 0.414090 0.910236i \(-0.364100\pi\)
0.414090 + 0.910236i \(0.364100\pi\)
\(468\) 2.64255e6 0.557711
\(469\) 0 0
\(470\) 5.49288e6 1.14698
\(471\) 300010. 0.0623137
\(472\) −2.13083e6 −0.440244
\(473\) −2.40487e6 −0.494242
\(474\) 1.01742e6 0.207996
\(475\) 865842. 0.176078
\(476\) 0 0
\(477\) 2.61684e6 0.526601
\(478\) −6.10493e6 −1.22211
\(479\) 4.42629e6 0.881457 0.440728 0.897640i \(-0.354720\pi\)
0.440728 + 0.897640i \(0.354720\pi\)
\(480\) 160498. 0.0317956
\(481\) −1.94778e6 −0.383863
\(482\) −2.91447e6 −0.571403
\(483\) 0 0
\(484\) 3.68654e6 0.715329
\(485\) 4.50233e6 0.869127
\(486\) −2.07381e6 −0.398271
\(487\) −2.22924e6 −0.425926 −0.212963 0.977060i \(-0.568311\pi\)
−0.212963 + 0.977060i \(0.568311\pi\)
\(488\) −8.97770e6 −1.70654
\(489\) 3664.10 0.000692939 0
\(490\) 0 0
\(491\) 627714. 0.117505 0.0587527 0.998273i \(-0.481288\pi\)
0.0587527 + 0.998273i \(0.481288\pi\)
\(492\) −378129. −0.0704251
\(493\) −3.59393e6 −0.665967
\(494\) 2.60727e6 0.480694
\(495\) −2.89036e6 −0.530199
\(496\) −1.18093e6 −0.215536
\(497\) 0 0
\(498\) 615704. 0.111250
\(499\) 1.00327e6 0.180370 0.0901850 0.995925i \(-0.471254\pi\)
0.0901850 + 0.995925i \(0.471254\pi\)
\(500\) 842498. 0.150711
\(501\) −512908. −0.0912946
\(502\) 173517. 0.0307314
\(503\) −904051. −0.159321 −0.0796605 0.996822i \(-0.525384\pi\)
−0.0796605 + 0.996822i \(0.525384\pi\)
\(504\) 0 0
\(505\) 2.68871e6 0.469154
\(506\) −1.07194e7 −1.86121
\(507\) −420603. −0.0726695
\(508\) 8.44526e6 1.45196
\(509\) −3.02145e6 −0.516917 −0.258458 0.966022i \(-0.583215\pi\)
−0.258458 + 0.966022i \(0.583215\pi\)
\(510\) −252226. −0.0429403
\(511\) 0 0
\(512\) 1.82234e6 0.307223
\(513\) −855087. −0.143455
\(514\) −5.92076e6 −0.988484
\(515\) 4.31377e6 0.716702
\(516\) −344979. −0.0570385
\(517\) −1.13535e7 −1.86812
\(518\) 0 0
\(519\) 764325. 0.124555
\(520\) 1.03135e6 0.167262
\(521\) 8882.28 0.00143361 0.000716804 1.00000i \(-0.499772\pi\)
0.000716804 1.00000i \(0.499772\pi\)
\(522\) −9.41405e6 −1.51217
\(523\) −9.60588e6 −1.53562 −0.767809 0.640679i \(-0.778653\pi\)
−0.767809 + 0.640679i \(0.778653\pi\)
\(524\) 3.85974e6 0.614087
\(525\) 0 0
\(526\) −302234. −0.0476298
\(527\) −6.38750e6 −1.00185
\(528\) 96386.1 0.0150463
\(529\) −607053. −0.0943165
\(530\) 2.51229e6 0.388491
\(531\) −2.53138e6 −0.389603
\(532\) 0 0
\(533\) 1.11739e6 0.170367
\(534\) −275204. −0.0417639
\(535\) 358538. 0.0541565
\(536\) −1.07953e7 −1.62302
\(537\) 816561. 0.122195
\(538\) 1.27343e6 0.189679
\(539\) 0 0
\(540\) −832032. −0.122788
\(541\) 2.47643e6 0.363775 0.181887 0.983319i \(-0.441779\pi\)
0.181887 + 0.983319i \(0.441779\pi\)
\(542\) −1.00575e7 −1.47058
\(543\) 816817. 0.118884
\(544\) 4.30325e6 0.623446
\(545\) −4.75878e6 −0.686285
\(546\) 0 0
\(547\) −1.10660e7 −1.58132 −0.790662 0.612253i \(-0.790263\pi\)
−0.790662 + 0.612253i \(0.790263\pi\)
\(548\) 1.61644e7 2.29937
\(549\) −1.06653e7 −1.51023
\(550\) −2.77488e6 −0.391145
\(551\) −5.82900e6 −0.817928
\(552\) −625117. −0.0873200
\(553\) 0 0
\(554\) 1.11003e7 1.53660
\(555\) 305610. 0.0421148
\(556\) 2.81283e6 0.385884
\(557\) −4.36913e6 −0.596702 −0.298351 0.954456i \(-0.596437\pi\)
−0.298351 + 0.954456i \(0.596437\pi\)
\(558\) −1.67316e7 −2.27485
\(559\) 1.01943e6 0.137983
\(560\) 0 0
\(561\) 521339. 0.0699380
\(562\) −8.01682e6 −1.07068
\(563\) −5.02223e6 −0.667768 −0.333884 0.942614i \(-0.608359\pi\)
−0.333884 + 0.942614i \(0.608359\pi\)
\(564\) −1.62866e6 −0.215592
\(565\) 1.66755e6 0.219764
\(566\) −1.24356e7 −1.63164
\(567\) 0 0
\(568\) −3.23000e6 −0.420079
\(569\) −348871. −0.0451735 −0.0225868 0.999745i \(-0.507190\pi\)
−0.0225868 + 0.999745i \(0.507190\pi\)
\(570\) −409086. −0.0527384
\(571\) 1.53588e7 1.97136 0.985680 0.168627i \(-0.0539334\pi\)
0.985680 + 0.168627i \(0.0539334\pi\)
\(572\) −5.24381e6 −0.670126
\(573\) −1.01365e6 −0.128974
\(574\) 0 0
\(575\) 1.50900e6 0.190335
\(576\) 1.24918e7 1.56881
\(577\) 1.05696e7 1.32166 0.660830 0.750536i \(-0.270204\pi\)
0.660830 + 0.750536i \(0.270204\pi\)
\(578\) 6.39844e6 0.796626
\(579\) 209008. 0.0259100
\(580\) −5.67184e6 −0.700091
\(581\) 0 0
\(582\) −2.12722e6 −0.260319
\(583\) −5.19279e6 −0.632746
\(584\) −1.57724e7 −1.91367
\(585\) 1.22522e6 0.148022
\(586\) 6.20460e6 0.746397
\(587\) −4.57417e6 −0.547920 −0.273960 0.961741i \(-0.588334\pi\)
−0.273960 + 0.961741i \(0.588334\pi\)
\(588\) 0 0
\(589\) −1.03599e7 −1.23046
\(590\) −2.43025e6 −0.287422
\(591\) −100844. −0.0118763
\(592\) 1.51491e6 0.177657
\(593\) 1.46686e7 1.71297 0.856487 0.516169i \(-0.172642\pi\)
0.856487 + 0.516169i \(0.172642\pi\)
\(594\) 2.74041e6 0.318676
\(595\) 0 0
\(596\) −2.33024e7 −2.68710
\(597\) 1.26161e6 0.144874
\(598\) 4.54397e6 0.519616
\(599\) 1.42374e7 1.62130 0.810649 0.585532i \(-0.199114\pi\)
0.810649 + 0.585532i \(0.199114\pi\)
\(600\) −161820. −0.0183508
\(601\) 8.21658e6 0.927908 0.463954 0.885859i \(-0.346430\pi\)
0.463954 + 0.885859i \(0.346430\pi\)
\(602\) 0 0
\(603\) −1.28246e7 −1.43632
\(604\) 1.20907e7 1.34853
\(605\) 1.70927e6 0.189855
\(606\) −1.27034e6 −0.140520
\(607\) 1.69772e7 1.87023 0.935114 0.354348i \(-0.115297\pi\)
0.935114 + 0.354348i \(0.115297\pi\)
\(608\) 6.97944e6 0.765705
\(609\) 0 0
\(610\) −1.02392e7 −1.11415
\(611\) 4.81276e6 0.521544
\(612\) −1.11168e7 −1.19977
\(613\) 5.22841e6 0.561977 0.280988 0.959711i \(-0.409338\pi\)
0.280988 + 0.959711i \(0.409338\pi\)
\(614\) −1.24417e6 −0.133186
\(615\) −175320. −0.0186915
\(616\) 0 0
\(617\) −1.71179e7 −1.81025 −0.905123 0.425150i \(-0.860221\pi\)
−0.905123 + 0.425150i \(0.860221\pi\)
\(618\) −2.03813e6 −0.214665
\(619\) 7.20377e6 0.755672 0.377836 0.925872i \(-0.376668\pi\)
0.377836 + 0.925872i \(0.376668\pi\)
\(620\) −1.00806e7 −1.05319
\(621\) −1.49025e6 −0.155071
\(622\) 3.66158e6 0.379483
\(623\) 0 0
\(624\) −40858.0 −0.00420065
\(625\) 390625. 0.0400000
\(626\) 1.27595e7 1.30136
\(627\) 845560. 0.0858966
\(628\) 1.26945e7 1.28445
\(629\) 8.19395e6 0.825784
\(630\) 0 0
\(631\) 8.56923e6 0.856779 0.428389 0.903594i \(-0.359081\pi\)
0.428389 + 0.903594i \(0.359081\pi\)
\(632\) 1.75013e7 1.74292
\(633\) 1.25287e6 0.124279
\(634\) 1.56547e7 1.54675
\(635\) 3.91565e6 0.385363
\(636\) −744905. −0.0730227
\(637\) 0 0
\(638\) 1.86810e7 1.81697
\(639\) −3.83717e6 −0.371757
\(640\) 7.96229e6 0.768401
\(641\) 9.91653e6 0.953267 0.476633 0.879102i \(-0.341857\pi\)
0.476633 + 0.879102i \(0.341857\pi\)
\(642\) −169399. −0.0162208
\(643\) 3.70762e6 0.353645 0.176822 0.984243i \(-0.443418\pi\)
0.176822 + 0.984243i \(0.443418\pi\)
\(644\) 0 0
\(645\) −159950. −0.0151385
\(646\) −1.09683e7 −1.03409
\(647\) 6.36378e6 0.597660 0.298830 0.954306i \(-0.403404\pi\)
0.298830 + 0.954306i \(0.403404\pi\)
\(648\) −1.17577e7 −1.09998
\(649\) 5.02321e6 0.468133
\(650\) 1.17627e6 0.109200
\(651\) 0 0
\(652\) 155041. 0.0142833
\(653\) 7.58179e6 0.695807 0.347903 0.937530i \(-0.386894\pi\)
0.347903 + 0.937530i \(0.386894\pi\)
\(654\) 2.24839e6 0.205554
\(655\) 1.78957e6 0.162984
\(656\) −869063. −0.0788482
\(657\) −1.87373e7 −1.69354
\(658\) 0 0
\(659\) −9.44484e6 −0.847191 −0.423595 0.905851i \(-0.639232\pi\)
−0.423595 + 0.905851i \(0.639232\pi\)
\(660\) 822763. 0.0735216
\(661\) −2.23762e7 −1.99197 −0.995985 0.0895192i \(-0.971467\pi\)
−0.995985 + 0.0895192i \(0.971467\pi\)
\(662\) −1.47397e7 −1.30721
\(663\) −220996. −0.0195254
\(664\) 1.05911e7 0.932226
\(665\) 0 0
\(666\) 2.14635e7 1.87506
\(667\) −1.01588e7 −0.884155
\(668\) −2.17030e7 −1.88182
\(669\) 363485. 0.0313994
\(670\) −1.23122e7 −1.05962
\(671\) 2.11640e7 1.81464
\(672\) 0 0
\(673\) 121935. 0.0103775 0.00518875 0.999987i \(-0.498348\pi\)
0.00518875 + 0.999987i \(0.498348\pi\)
\(674\) 7.63267e6 0.647183
\(675\) −385773. −0.0325891
\(676\) −1.77972e7 −1.49791
\(677\) −4.12171e6 −0.345625 −0.172813 0.984955i \(-0.555286\pi\)
−0.172813 + 0.984955i \(0.555286\pi\)
\(678\) −787868. −0.0658232
\(679\) 0 0
\(680\) −4.33870e6 −0.359822
\(681\) 1.21051e6 0.100023
\(682\) 3.32017e7 2.73338
\(683\) −4.30505e6 −0.353123 −0.176562 0.984290i \(-0.556497\pi\)
−0.176562 + 0.984290i \(0.556497\pi\)
\(684\) −1.80303e7 −1.47354
\(685\) 7.49466e6 0.610275
\(686\) 0 0
\(687\) −1.33983e6 −0.108307
\(688\) −792873. −0.0638605
\(689\) 2.20122e6 0.176651
\(690\) −712957. −0.0570086
\(691\) 1.54709e7 1.23260 0.616298 0.787513i \(-0.288632\pi\)
0.616298 + 0.787513i \(0.288632\pi\)
\(692\) 3.23414e7 2.56740
\(693\) 0 0
\(694\) 2.43749e7 1.92107
\(695\) 1.30417e6 0.102417
\(696\) 1.08940e6 0.0852444
\(697\) −4.70065e6 −0.366501
\(698\) 3.02932e7 2.35346
\(699\) −214958. −0.0166403
\(700\) 0 0
\(701\) 631976. 0.0485742 0.0242871 0.999705i \(-0.492268\pi\)
0.0242871 + 0.999705i \(0.492268\pi\)
\(702\) −1.16166e6 −0.0889684
\(703\) 1.32898e7 1.01421
\(704\) −2.47884e7 −1.88502
\(705\) −755130. −0.0572202
\(706\) 5.76032e6 0.434945
\(707\) 0 0
\(708\) 720578. 0.0540254
\(709\) −1.13182e7 −0.845594 −0.422797 0.906224i \(-0.638952\pi\)
−0.422797 + 0.906224i \(0.638952\pi\)
\(710\) −3.68387e6 −0.274257
\(711\) 2.07912e7 1.54243
\(712\) −4.73396e6 −0.349965
\(713\) −1.80553e7 −1.33009
\(714\) 0 0
\(715\) −2.43130e6 −0.177858
\(716\) 3.45516e7 2.51876
\(717\) 839271. 0.0609684
\(718\) −1.24659e7 −0.902431
\(719\) −1.35453e7 −0.977162 −0.488581 0.872519i \(-0.662485\pi\)
−0.488581 + 0.872519i \(0.662485\pi\)
\(720\) −952934. −0.0685065
\(721\) 0 0
\(722\) 5.16218e6 0.368545
\(723\) 400666. 0.0285060
\(724\) 3.45625e7 2.45052
\(725\) −2.62976e6 −0.185811
\(726\) −807581. −0.0568649
\(727\) −1.41927e7 −0.995932 −0.497966 0.867197i \(-0.665920\pi\)
−0.497966 + 0.867197i \(0.665920\pi\)
\(728\) 0 0
\(729\) −1.37768e7 −0.960129
\(730\) −1.79887e7 −1.24938
\(731\) −4.28854e6 −0.296836
\(732\) 3.03597e6 0.209421
\(733\) −1.55238e6 −0.106718 −0.0533591 0.998575i \(-0.516993\pi\)
−0.0533591 + 0.998575i \(0.516993\pi\)
\(734\) 2.55648e7 1.75147
\(735\) 0 0
\(736\) 1.21638e7 0.827704
\(737\) 2.54488e7 1.72583
\(738\) −1.23130e7 −0.832191
\(739\) −1.20108e7 −0.809026 −0.404513 0.914532i \(-0.632559\pi\)
−0.404513 + 0.914532i \(0.632559\pi\)
\(740\) 1.29315e7 0.868097
\(741\) −358433. −0.0239807
\(742\) 0 0
\(743\) 7.84220e6 0.521154 0.260577 0.965453i \(-0.416087\pi\)
0.260577 + 0.965453i \(0.416087\pi\)
\(744\) 1.93620e6 0.128238
\(745\) −1.08042e7 −0.713182
\(746\) −3.04994e7 −2.00652
\(747\) 1.25820e7 0.824991
\(748\) 2.20598e7 1.44161
\(749\) 0 0
\(750\) −184559. −0.0119807
\(751\) −3.95564e6 −0.255927 −0.127964 0.991779i \(-0.540844\pi\)
−0.127964 + 0.991779i \(0.540844\pi\)
\(752\) −3.74319e6 −0.241378
\(753\) −23854.1 −0.00153312
\(754\) −7.91886e6 −0.507264
\(755\) 5.60587e6 0.357912
\(756\) 0 0
\(757\) 2.28287e7 1.44791 0.723956 0.689846i \(-0.242322\pi\)
0.723956 + 0.689846i \(0.242322\pi\)
\(758\) 1.23770e7 0.782424
\(759\) 1.47365e6 0.0928516
\(760\) −7.03694e6 −0.441926
\(761\) −2.59221e7 −1.62259 −0.811295 0.584638i \(-0.801237\pi\)
−0.811295 + 0.584638i \(0.801237\pi\)
\(762\) −1.85003e6 −0.115423
\(763\) 0 0
\(764\) −4.28911e7 −2.65848
\(765\) −5.15429e6 −0.318431
\(766\) 3.12636e7 1.92516
\(767\) −2.12934e6 −0.130694
\(768\) −1.65163e6 −0.101044
\(769\) −381532. −0.0232656 −0.0116328 0.999932i \(-0.503703\pi\)
−0.0116328 + 0.999932i \(0.503703\pi\)
\(770\) 0 0
\(771\) 813953. 0.0493132
\(772\) 8.84390e6 0.534073
\(773\) −9.18755e6 −0.553033 −0.276516 0.961009i \(-0.589180\pi\)
−0.276516 + 0.961009i \(0.589180\pi\)
\(774\) −1.12335e7 −0.674006
\(775\) −4.67387e6 −0.279526
\(776\) −3.65917e7 −2.18136
\(777\) 0 0
\(778\) 3.21198e7 1.90250
\(779\) −7.62398e6 −0.450130
\(780\) −348769. −0.0205259
\(781\) 7.61438e6 0.446691
\(782\) −1.91157e7 −1.11782
\(783\) 2.59709e6 0.151385
\(784\) 0 0
\(785\) 5.88583e6 0.340905
\(786\) −845522. −0.0488167
\(787\) −6.94528e6 −0.399717 −0.199859 0.979825i \(-0.564048\pi\)
−0.199859 + 0.979825i \(0.564048\pi\)
\(788\) −4.26708e6 −0.244802
\(789\) 41549.4 0.00237614
\(790\) 1.99605e7 1.13790
\(791\) 0 0
\(792\) 2.34907e7 1.33071
\(793\) −8.97141e6 −0.506614
\(794\) 1.62683e7 0.915782
\(795\) −345376. −0.0193809
\(796\) 5.33833e7 2.98623
\(797\) 5.81421e6 0.324224 0.162112 0.986772i \(-0.448169\pi\)
0.162112 + 0.986772i \(0.448169\pi\)
\(798\) 0 0
\(799\) −2.02464e7 −1.12197
\(800\) 3.14878e6 0.173947
\(801\) −5.62385e6 −0.309708
\(802\) 4.75007e7 2.60774
\(803\) 3.71818e7 2.03489
\(804\) 3.65063e6 0.199172
\(805\) 0 0
\(806\) −1.40742e7 −0.763108
\(807\) −175064. −0.00946265
\(808\) −2.18519e7 −1.17750
\(809\) −2.76000e7 −1.48265 −0.741324 0.671147i \(-0.765802\pi\)
−0.741324 + 0.671147i \(0.765802\pi\)
\(810\) −1.34099e7 −0.718144
\(811\) −5.51268e6 −0.294314 −0.147157 0.989113i \(-0.547012\pi\)
−0.147157 + 0.989113i \(0.547012\pi\)
\(812\) 0 0
\(813\) 1.38264e6 0.0733641
\(814\) −4.25915e7 −2.25300
\(815\) 71885.0 0.00379092
\(816\) 171883. 0.00903663
\(817\) −6.95559e6 −0.364568
\(818\) −8.15103e6 −0.425921
\(819\) 0 0
\(820\) −7.41842e6 −0.385281
\(821\) 4.69730e6 0.243215 0.121607 0.992578i \(-0.461195\pi\)
0.121607 + 0.992578i \(0.461195\pi\)
\(822\) −3.54101e6 −0.182788
\(823\) 1.68012e7 0.864652 0.432326 0.901717i \(-0.357693\pi\)
0.432326 + 0.901717i \(0.357693\pi\)
\(824\) −3.50592e7 −1.79880
\(825\) 381475. 0.0195133
\(826\) 0 0
\(827\) 1.91479e6 0.0973548 0.0486774 0.998815i \(-0.484499\pi\)
0.0486774 + 0.998815i \(0.484499\pi\)
\(828\) −3.14233e7 −1.59285
\(829\) 7.32861e6 0.370369 0.185185 0.982704i \(-0.440712\pi\)
0.185185 + 0.982704i \(0.440712\pi\)
\(830\) 1.20793e7 0.608622
\(831\) −1.52601e6 −0.0766574
\(832\) 1.05078e7 0.526263
\(833\) 0 0
\(834\) −616183. −0.0306757
\(835\) −1.00626e7 −0.499453
\(836\) 3.57787e7 1.77055
\(837\) 4.61581e6 0.227737
\(838\) 5.27062e7 2.59269
\(839\) −1.77241e7 −0.869281 −0.434641 0.900604i \(-0.643125\pi\)
−0.434641 + 0.900604i \(0.643125\pi\)
\(840\) 0 0
\(841\) −2.80718e6 −0.136861
\(842\) −5.81601e6 −0.282712
\(843\) 1.10211e6 0.0534140
\(844\) 5.30136e7 2.56172
\(845\) −8.25170e6 −0.397559
\(846\) −5.30340e7 −2.54759
\(847\) 0 0
\(848\) −1.71203e6 −0.0817565
\(849\) 1.70958e6 0.0813990
\(850\) −4.94836e6 −0.234917
\(851\) 2.31615e7 1.09633
\(852\) 1.09228e6 0.0515508
\(853\) −2.61947e7 −1.23265 −0.616327 0.787490i \(-0.711380\pi\)
−0.616327 + 0.787490i \(0.711380\pi\)
\(854\) 0 0
\(855\) −8.35975e6 −0.391091
\(856\) −2.91393e6 −0.135924
\(857\) −3.21925e7 −1.49728 −0.748639 0.662977i \(-0.769293\pi\)
−0.748639 + 0.662977i \(0.769293\pi\)
\(858\) 1.14872e6 0.0532715
\(859\) −2.22096e7 −1.02697 −0.513485 0.858098i \(-0.671646\pi\)
−0.513485 + 0.858098i \(0.671646\pi\)
\(860\) −6.76805e6 −0.312045
\(861\) 0 0
\(862\) 8.84177e6 0.405295
\(863\) 2.10546e7 0.962322 0.481161 0.876632i \(-0.340215\pi\)
0.481161 + 0.876632i \(0.340215\pi\)
\(864\) −3.10966e6 −0.141719
\(865\) 1.49951e7 0.681411
\(866\) −4.19018e7 −1.89862
\(867\) −879621. −0.0397418
\(868\) 0 0
\(869\) −4.12574e7 −1.85333
\(870\) 1.24248e6 0.0556535
\(871\) −1.07877e7 −0.481820
\(872\) 3.86759e7 1.72246
\(873\) −4.34703e7 −1.93044
\(874\) −3.10037e7 −1.37289
\(875\) 0 0
\(876\) 5.33373e6 0.234839
\(877\) 1.49233e7 0.655189 0.327594 0.944818i \(-0.393762\pi\)
0.327594 + 0.944818i \(0.393762\pi\)
\(878\) −6.47955e7 −2.83667
\(879\) −852974. −0.0372360
\(880\) 1.89098e6 0.0823150
\(881\) 3.85987e7 1.67546 0.837729 0.546087i \(-0.183883\pi\)
0.837729 + 0.546087i \(0.183883\pi\)
\(882\) 0 0
\(883\) −3.35117e7 −1.44642 −0.723210 0.690628i \(-0.757334\pi\)
−0.723210 + 0.690628i \(0.757334\pi\)
\(884\) −9.35113e6 −0.402470
\(885\) 334097. 0.0143388
\(886\) 4.32008e7 1.84887
\(887\) 4.18929e7 1.78785 0.893927 0.448213i \(-0.147939\pi\)
0.893927 + 0.448213i \(0.147939\pi\)
\(888\) −2.48377e6 −0.105701
\(889\) 0 0
\(890\) −5.39916e6 −0.228482
\(891\) 2.77176e7 1.16966
\(892\) 1.53804e7 0.647224
\(893\) −3.28377e7 −1.37798
\(894\) 5.10465e6 0.213610
\(895\) 1.60199e7 0.668501
\(896\) 0 0
\(897\) −624679. −0.0259224
\(898\) −5.93313e7 −2.45524
\(899\) 3.14653e7 1.29847
\(900\) −8.13436e6 −0.334747
\(901\) −9.26015e6 −0.380020
\(902\) 2.44336e7 0.999932
\(903\) 0 0
\(904\) −1.35526e7 −0.551571
\(905\) 1.60249e7 0.650391
\(906\) −2.64861e6 −0.107201
\(907\) 2.30904e6 0.0931993 0.0465997 0.998914i \(-0.485161\pi\)
0.0465997 + 0.998914i \(0.485161\pi\)
\(908\) 5.12212e7 2.06175
\(909\) −2.59596e7 −1.04205
\(910\) 0 0
\(911\) 1.48503e7 0.592843 0.296421 0.955057i \(-0.404207\pi\)
0.296421 + 0.955057i \(0.404207\pi\)
\(912\) 278776. 0.0110986
\(913\) −2.49674e7 −0.991281
\(914\) −3.79494e7 −1.50259
\(915\) 1.40763e6 0.0555822
\(916\) −5.66930e7 −2.23250
\(917\) 0 0
\(918\) 4.88689e6 0.191393
\(919\) 1.02358e7 0.399793 0.199896 0.979817i \(-0.435939\pi\)
0.199896 + 0.979817i \(0.435939\pi\)
\(920\) −1.22640e7 −0.477709
\(921\) 171041. 0.00664433
\(922\) −7.33617e7 −2.84212
\(923\) −3.22773e6 −0.124708
\(924\) 0 0
\(925\) 5.99568e6 0.230401
\(926\) 4.27119e7 1.63690
\(927\) −4.16496e7 −1.59189
\(928\) −2.11981e7 −0.808029
\(929\) −2.00459e7 −0.762054 −0.381027 0.924564i \(-0.624429\pi\)
−0.381027 + 0.924564i \(0.624429\pi\)
\(930\) 2.20827e6 0.0837229
\(931\) 0 0
\(932\) −9.09567e6 −0.343001
\(933\) −503373. −0.0189315
\(934\) −3.61796e7 −1.35705
\(935\) 1.02280e7 0.382616
\(936\) −9.95772e6 −0.371510
\(937\) −3.63602e7 −1.35293 −0.676467 0.736473i \(-0.736490\pi\)
−0.676467 + 0.736473i \(0.736490\pi\)
\(938\) 0 0
\(939\) −1.75411e6 −0.0649221
\(940\) −3.19523e7 −1.17946
\(941\) 1.57487e7 0.579789 0.289895 0.957059i \(-0.406380\pi\)
0.289895 + 0.957059i \(0.406380\pi\)
\(942\) −2.78088e6 −0.102107
\(943\) −1.32871e7 −0.486577
\(944\) 1.65612e6 0.0604870
\(945\) 0 0
\(946\) 2.22915e7 0.809862
\(947\) −2.19938e7 −0.796938 −0.398469 0.917182i \(-0.630458\pi\)
−0.398469 + 0.917182i \(0.630458\pi\)
\(948\) −5.91837e6 −0.213885
\(949\) −1.57614e7 −0.568105
\(950\) −8.02575e6 −0.288520
\(951\) −2.15212e6 −0.0771640
\(952\) 0 0
\(953\) 3.79156e7 1.35234 0.676170 0.736746i \(-0.263639\pi\)
0.676170 + 0.736746i \(0.263639\pi\)
\(954\) −2.42563e7 −0.862887
\(955\) −1.98865e7 −0.705586
\(956\) 3.55126e7 1.25672
\(957\) −2.56815e6 −0.0906445
\(958\) −4.10286e7 −1.44435
\(959\) 0 0
\(960\) −1.64869e6 −0.0577379
\(961\) 2.72941e7 0.953368
\(962\) 1.80545e7 0.628996
\(963\) −3.46170e6 −0.120288
\(964\) 1.69536e7 0.587584
\(965\) 4.10048e6 0.141748
\(966\) 0 0
\(967\) −2.20041e7 −0.756724 −0.378362 0.925658i \(-0.623513\pi\)
−0.378362 + 0.925658i \(0.623513\pi\)
\(968\) −1.38917e7 −0.476505
\(969\) 1.50786e6 0.0515885
\(970\) −4.17335e7 −1.42415
\(971\) −3.57397e7 −1.21647 −0.608237 0.793755i \(-0.708123\pi\)
−0.608237 + 0.793755i \(0.708123\pi\)
\(972\) 1.20634e7 0.409548
\(973\) 0 0
\(974\) 2.06635e7 0.697921
\(975\) −161707. −0.00544775
\(976\) 6.97764e6 0.234468
\(977\) −5.52739e7 −1.85261 −0.926305 0.376774i \(-0.877033\pi\)
−0.926305 + 0.376774i \(0.877033\pi\)
\(978\) −33963.6 −0.00113545
\(979\) 1.11598e7 0.372134
\(980\) 0 0
\(981\) 4.59463e7 1.52433
\(982\) −5.81847e6 −0.192544
\(983\) −22923.2 −0.000756644 0 −0.000378322 1.00000i \(-0.500120\pi\)
−0.000378322 1.00000i \(0.500120\pi\)
\(984\) 1.42487e6 0.0469125
\(985\) −1.97843e6 −0.0649727
\(986\) 3.33132e7 1.09125
\(987\) 0 0
\(988\) −1.51666e7 −0.494306
\(989\) −1.21222e7 −0.394087
\(990\) 2.67916e7 0.868782
\(991\) −8.30772e6 −0.268719 −0.134359 0.990933i \(-0.542898\pi\)
−0.134359 + 0.990933i \(0.542898\pi\)
\(992\) −3.76754e7 −1.21557
\(993\) 2.02633e6 0.0652136
\(994\) 0 0
\(995\) 2.47512e7 0.792572
\(996\) −3.58157e6 −0.114400
\(997\) 3.35426e7 1.06871 0.534353 0.845261i \(-0.320555\pi\)
0.534353 + 0.845261i \(0.320555\pi\)
\(998\) −9.29956e6 −0.295554
\(999\) −5.92120e6 −0.187714
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 245.6.a.m.1.2 yes 10
7.6 odd 2 245.6.a.l.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.6.a.l.1.2 10 7.6 odd 2
245.6.a.m.1.2 yes 10 1.1 even 1 trivial