Properties

Label 147.6.a.j.1.1
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, 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("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.44622 q^{2} +9.00000 q^{3} +39.3387 q^{4} -36.0000 q^{5} -76.0160 q^{6} -61.9840 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.44622 q^{2} +9.00000 q^{3} +39.3387 q^{4} -36.0000 q^{5} -76.0160 q^{6} -61.9840 q^{8} +81.0000 q^{9} +304.064 q^{10} +295.570 q^{11} +354.048 q^{12} -1148.13 q^{13} -324.000 q^{15} -735.307 q^{16} +1032.38 q^{17} -684.144 q^{18} +2108.51 q^{19} -1416.19 q^{20} -2496.45 q^{22} -640.988 q^{23} -557.856 q^{24} -1829.00 q^{25} +9697.34 q^{26} +729.000 q^{27} +7631.58 q^{29} +2736.58 q^{30} -966.976 q^{31} +8194.05 q^{32} +2660.13 q^{33} -8719.74 q^{34} +3186.43 q^{36} -1773.21 q^{37} -17809.0 q^{38} -10333.2 q^{39} +2231.42 q^{40} -11976.4 q^{41} -19802.9 q^{43} +11627.3 q^{44} -2916.00 q^{45} +5413.93 q^{46} -27966.1 q^{47} -6617.76 q^{48} +15448.1 q^{50} +9291.46 q^{51} -45165.8 q^{52} -7114.33 q^{53} -6157.30 q^{54} -10640.5 q^{55} +18976.6 q^{57} -64458.0 q^{58} -20869.5 q^{59} -12745.7 q^{60} -23868.3 q^{61} +8167.30 q^{62} -45679.0 q^{64} +41332.6 q^{65} -22468.0 q^{66} +34671.5 q^{67} +40612.6 q^{68} -5768.90 q^{69} -28413.2 q^{71} -5020.70 q^{72} -15292.7 q^{73} +14976.9 q^{74} -16461.0 q^{75} +82946.0 q^{76} +87276.1 q^{78} -73059.5 q^{79} +26471.0 q^{80} +6561.00 q^{81} +101155. q^{82} -30340.9 q^{83} -37165.8 q^{85} +167260. q^{86} +68684.2 q^{87} -18320.6 q^{88} -36089.5 q^{89} +24629.2 q^{90} -25215.6 q^{92} -8702.79 q^{93} +236208. q^{94} -75906.4 q^{95} +73746.5 q^{96} -153963. q^{97} +23941.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 18 q^{3} + 37 q^{4} - 72 q^{5} - 27 q^{6} - 249 q^{8} + 162 q^{9} + 108 q^{10} + 480 q^{11} + 333 q^{12} - 1296 q^{13} - 648 q^{15} - 1679 q^{16} - 936 q^{17} - 243 q^{18} + 216 q^{19} - 1332 q^{20} - 1492 q^{22} - 504 q^{23} - 2241 q^{24} - 3658 q^{25} + 8892 q^{26} + 1458 q^{27} + 6372 q^{29} + 972 q^{30} - 9936 q^{31} + 9039 q^{32} + 4320 q^{33} - 19440 q^{34} + 2997 q^{36} + 11124 q^{37} - 28116 q^{38} - 11664 q^{39} + 8964 q^{40} - 20952 q^{41} - 6264 q^{43} + 11196 q^{44} - 5832 q^{45} + 6160 q^{46} - 7920 q^{47} - 15111 q^{48} + 5487 q^{50} - 8424 q^{51} - 44820 q^{52} + 2220 q^{53} - 2187 q^{54} - 17280 q^{55} + 1944 q^{57} - 71318 q^{58} - 29736 q^{59} - 11988 q^{60} + 17280 q^{61} - 40680 q^{62} - 10879 q^{64} + 46656 q^{65} - 13428 q^{66} - 20680 q^{67} + 45216 q^{68} - 4536 q^{69} - 92280 q^{71} - 20169 q^{72} - 56592 q^{73} + 85218 q^{74} - 32922 q^{75} + 87372 q^{76} + 80028 q^{78} - 56096 q^{79} + 60444 q^{80} + 13122 q^{81} + 52272 q^{82} + 71352 q^{83} + 33696 q^{85} + 240996 q^{86} + 57348 q^{87} - 52812 q^{88} - 123192 q^{89} + 8748 q^{90} - 25536 q^{92} - 89424 q^{93} + 345384 q^{94} - 7776 q^{95} + 81351 q^{96} - 35856 q^{97} + 38880 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.44622 −1.49310 −0.746548 0.665332i \(-0.768290\pi\)
−0.746548 + 0.665332i \(0.768290\pi\)
\(3\) 9.00000 0.577350
\(4\) 39.3387 1.22933
\(5\) −36.0000 −0.643988 −0.321994 0.946742i \(-0.604353\pi\)
−0.321994 + 0.946742i \(0.604353\pi\)
\(6\) −76.0160 −0.862039
\(7\) 0 0
\(8\) −61.9840 −0.342416
\(9\) 81.0000 0.333333
\(10\) 304.064 0.961535
\(11\) 295.570 0.736509 0.368255 0.929725i \(-0.379955\pi\)
0.368255 + 0.929725i \(0.379955\pi\)
\(12\) 354.048 0.709756
\(13\) −1148.13 −1.88422 −0.942111 0.335302i \(-0.891162\pi\)
−0.942111 + 0.335302i \(0.891162\pi\)
\(14\) 0 0
\(15\) −324.000 −0.371806
\(16\) −735.307 −0.718073
\(17\) 1032.38 0.866401 0.433200 0.901298i \(-0.357384\pi\)
0.433200 + 0.901298i \(0.357384\pi\)
\(18\) −684.144 −0.497698
\(19\) 2108.51 1.33996 0.669980 0.742379i \(-0.266303\pi\)
0.669980 + 0.742379i \(0.266303\pi\)
\(20\) −1416.19 −0.791675
\(21\) 0 0
\(22\) −2496.45 −1.09968
\(23\) −640.988 −0.252657 −0.126328 0.991988i \(-0.540319\pi\)
−0.126328 + 0.991988i \(0.540319\pi\)
\(24\) −557.856 −0.197694
\(25\) −1829.00 −0.585280
\(26\) 9697.34 2.81332
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 7631.58 1.68508 0.842538 0.538637i \(-0.181060\pi\)
0.842538 + 0.538637i \(0.181060\pi\)
\(30\) 2736.58 0.555142
\(31\) −966.976 −0.180722 −0.0903611 0.995909i \(-0.528802\pi\)
−0.0903611 + 0.995909i \(0.528802\pi\)
\(32\) 8194.05 1.41457
\(33\) 2660.13 0.425224
\(34\) −8719.74 −1.29362
\(35\) 0 0
\(36\) 3186.43 0.409778
\(37\) −1773.21 −0.212939 −0.106470 0.994316i \(-0.533955\pi\)
−0.106470 + 0.994316i \(0.533955\pi\)
\(38\) −17809.0 −2.00069
\(39\) −10333.2 −1.08786
\(40\) 2231.42 0.220512
\(41\) −11976.4 −1.11267 −0.556335 0.830958i \(-0.687793\pi\)
−0.556335 + 0.830958i \(0.687793\pi\)
\(42\) 0 0
\(43\) −19802.9 −1.63327 −0.816636 0.577153i \(-0.804163\pi\)
−0.816636 + 0.577153i \(0.804163\pi\)
\(44\) 11627.3 0.905416
\(45\) −2916.00 −0.214663
\(46\) 5413.93 0.377240
\(47\) −27966.1 −1.84666 −0.923332 0.384002i \(-0.874545\pi\)
−0.923332 + 0.384002i \(0.874545\pi\)
\(48\) −6617.76 −0.414580
\(49\) 0 0
\(50\) 15448.1 0.873879
\(51\) 9291.46 0.500217
\(52\) −45165.8 −2.31634
\(53\) −7114.33 −0.347892 −0.173946 0.984755i \(-0.555652\pi\)
−0.173946 + 0.984755i \(0.555652\pi\)
\(54\) −6157.30 −0.287346
\(55\) −10640.5 −0.474303
\(56\) 0 0
\(57\) 18976.6 0.773627
\(58\) −64458.0 −2.51598
\(59\) −20869.5 −0.780518 −0.390259 0.920705i \(-0.627614\pi\)
−0.390259 + 0.920705i \(0.627614\pi\)
\(60\) −12745.7 −0.457074
\(61\) −23868.3 −0.821291 −0.410646 0.911795i \(-0.634697\pi\)
−0.410646 + 0.911795i \(0.634697\pi\)
\(62\) 8167.30 0.269835
\(63\) 0 0
\(64\) −45679.0 −1.39401
\(65\) 41332.6 1.21342
\(66\) −22468.0 −0.634900
\(67\) 34671.5 0.943595 0.471798 0.881707i \(-0.343605\pi\)
0.471798 + 0.881707i \(0.343605\pi\)
\(68\) 40612.6 1.06510
\(69\) −5768.90 −0.145871
\(70\) 0 0
\(71\) −28413.2 −0.668921 −0.334461 0.942410i \(-0.608554\pi\)
−0.334461 + 0.942410i \(0.608554\pi\)
\(72\) −5020.70 −0.114139
\(73\) −15292.7 −0.335874 −0.167937 0.985798i \(-0.553711\pi\)
−0.167937 + 0.985798i \(0.553711\pi\)
\(74\) 14976.9 0.317939
\(75\) −16461.0 −0.337912
\(76\) 82946.0 1.64726
\(77\) 0 0
\(78\) 87276.1 1.62427
\(79\) −73059.5 −1.31707 −0.658535 0.752550i \(-0.728824\pi\)
−0.658535 + 0.752550i \(0.728824\pi\)
\(80\) 26471.0 0.462430
\(81\) 6561.00 0.111111
\(82\) 101155. 1.66132
\(83\) −30340.9 −0.483429 −0.241715 0.970347i \(-0.577710\pi\)
−0.241715 + 0.970347i \(0.577710\pi\)
\(84\) 0 0
\(85\) −37165.8 −0.557951
\(86\) 167260. 2.43863
\(87\) 68684.2 0.972879
\(88\) −18320.6 −0.252193
\(89\) −36089.5 −0.482954 −0.241477 0.970407i \(-0.577632\pi\)
−0.241477 + 0.970407i \(0.577632\pi\)
\(90\) 24629.2 0.320512
\(91\) 0 0
\(92\) −25215.6 −0.310599
\(93\) −8702.79 −0.104340
\(94\) 236208. 2.75725
\(95\) −75906.4 −0.862918
\(96\) 73746.5 0.816701
\(97\) −153963. −1.66145 −0.830724 0.556685i \(-0.812073\pi\)
−0.830724 + 0.556685i \(0.812073\pi\)
\(98\) 0 0
\(99\) 23941.2 0.245503
\(100\) −71950.4 −0.719504
\(101\) 139809. 1.36374 0.681869 0.731474i \(-0.261167\pi\)
0.681869 + 0.731474i \(0.261167\pi\)
\(102\) −78477.7 −0.746871
\(103\) −115925. −1.07668 −0.538339 0.842728i \(-0.680948\pi\)
−0.538339 + 0.842728i \(0.680948\pi\)
\(104\) 71165.6 0.645188
\(105\) 0 0
\(106\) 60089.2 0.519436
\(107\) 83061.8 0.701361 0.350681 0.936495i \(-0.385950\pi\)
0.350681 + 0.936495i \(0.385950\pi\)
\(108\) 28677.9 0.236585
\(109\) 45356.2 0.365654 0.182827 0.983145i \(-0.441475\pi\)
0.182827 + 0.983145i \(0.441475\pi\)
\(110\) 89872.1 0.708179
\(111\) −15958.9 −0.122941
\(112\) 0 0
\(113\) −355.533 −0.00261929 −0.00130965 0.999999i \(-0.500417\pi\)
−0.00130965 + 0.999999i \(0.500417\pi\)
\(114\) −160281. −1.15510
\(115\) 23075.6 0.162708
\(116\) 300216. 2.07152
\(117\) −92998.4 −0.628074
\(118\) 176269. 1.16539
\(119\) 0 0
\(120\) 20082.8 0.127313
\(121\) −73689.5 −0.457554
\(122\) 201597. 1.22627
\(123\) −107787. −0.642400
\(124\) −38039.6 −0.222168
\(125\) 178344. 1.02090
\(126\) 0 0
\(127\) 168967. 0.929593 0.464797 0.885417i \(-0.346127\pi\)
0.464797 + 0.885417i \(0.346127\pi\)
\(128\) 123605. 0.666824
\(129\) −178226. −0.942970
\(130\) −349104. −1.81174
\(131\) −173969. −0.885715 −0.442858 0.896592i \(-0.646035\pi\)
−0.442858 + 0.896592i \(0.646035\pi\)
\(132\) 104646. 0.522742
\(133\) 0 0
\(134\) −292843. −1.40888
\(135\) −26244.0 −0.123935
\(136\) −63991.3 −0.296670
\(137\) 367723. 1.67386 0.836931 0.547308i \(-0.184347\pi\)
0.836931 + 0.547308i \(0.184347\pi\)
\(138\) 48725.4 0.217800
\(139\) 217967. 0.956870 0.478435 0.878123i \(-0.341204\pi\)
0.478435 + 0.878123i \(0.341204\pi\)
\(140\) 0 0
\(141\) −251695. −1.06617
\(142\) 239985. 0.998763
\(143\) −339352. −1.38775
\(144\) −59559.8 −0.239358
\(145\) −274737. −1.08517
\(146\) 129165. 0.501492
\(147\) 0 0
\(148\) −69755.7 −0.261773
\(149\) −64906.1 −0.239508 −0.119754 0.992804i \(-0.538211\pi\)
−0.119754 + 0.992804i \(0.538211\pi\)
\(150\) 139033. 0.504534
\(151\) −223777. −0.798681 −0.399341 0.916803i \(-0.630761\pi\)
−0.399341 + 0.916803i \(0.630761\pi\)
\(152\) −130694. −0.458825
\(153\) 83623.1 0.288800
\(154\) 0 0
\(155\) 34811.1 0.116383
\(156\) −406492. −1.33734
\(157\) −459973. −1.48930 −0.744652 0.667453i \(-0.767384\pi\)
−0.744652 + 0.667453i \(0.767384\pi\)
\(158\) 617077. 1.96651
\(159\) −64028.9 −0.200855
\(160\) −294986. −0.910964
\(161\) 0 0
\(162\) −55415.7 −0.165899
\(163\) 91068.6 0.268472 0.134236 0.990949i \(-0.457142\pi\)
0.134236 + 0.990949i \(0.457142\pi\)
\(164\) −471135. −1.36784
\(165\) −95764.6 −0.273839
\(166\) 256266. 0.721806
\(167\) −314772. −0.873384 −0.436692 0.899611i \(-0.643850\pi\)
−0.436692 + 0.899611i \(0.643850\pi\)
\(168\) 0 0
\(169\) 946905. 2.55029
\(170\) 313911. 0.833075
\(171\) 170789. 0.446654
\(172\) −779021. −2.00784
\(173\) −362143. −0.919951 −0.459975 0.887932i \(-0.652142\pi\)
−0.459975 + 0.887932i \(0.652142\pi\)
\(174\) −580122. −1.45260
\(175\) 0 0
\(176\) −217334. −0.528867
\(177\) −187826. −0.450632
\(178\) 304820. 0.721096
\(179\) −173896. −0.405656 −0.202828 0.979214i \(-0.565013\pi\)
−0.202828 + 0.979214i \(0.565013\pi\)
\(180\) −114712. −0.263892
\(181\) 134973. 0.306233 0.153116 0.988208i \(-0.451069\pi\)
0.153116 + 0.988208i \(0.451069\pi\)
\(182\) 0 0
\(183\) −214815. −0.474173
\(184\) 39731.0 0.0865138
\(185\) 63835.6 0.137130
\(186\) 73505.7 0.155790
\(187\) 305141. 0.638112
\(188\) −1.10015e6 −2.27017
\(189\) 0 0
\(190\) 641123. 1.28842
\(191\) 181413. 0.359821 0.179910 0.983683i \(-0.442419\pi\)
0.179910 + 0.983683i \(0.442419\pi\)
\(192\) −411111. −0.804833
\(193\) 965999. 1.86674 0.933369 0.358919i \(-0.116855\pi\)
0.933369 + 0.358919i \(0.116855\pi\)
\(194\) 1.30040e6 2.48070
\(195\) 371993. 0.700566
\(196\) 0 0
\(197\) −699058. −1.28336 −0.641679 0.766974i \(-0.721762\pi\)
−0.641679 + 0.766974i \(0.721762\pi\)
\(198\) −202212. −0.366560
\(199\) −416191. −0.745006 −0.372503 0.928031i \(-0.621500\pi\)
−0.372503 + 0.928031i \(0.621500\pi\)
\(200\) 113369. 0.200410
\(201\) 312044. 0.544785
\(202\) −1.18086e6 −2.03619
\(203\) 0 0
\(204\) 365513. 0.614933
\(205\) 431150. 0.716545
\(206\) 979132. 1.60758
\(207\) −51920.1 −0.0842189
\(208\) 844226. 1.35301
\(209\) 623212. 0.986894
\(210\) 0 0
\(211\) −407152. −0.629580 −0.314790 0.949161i \(-0.601934\pi\)
−0.314790 + 0.949161i \(0.601934\pi\)
\(212\) −279868. −0.427675
\(213\) −255719. −0.386202
\(214\) −701558. −1.04720
\(215\) 712906. 1.05181
\(216\) −45186.3 −0.0658981
\(217\) 0 0
\(218\) −383089. −0.545957
\(219\) −137634. −0.193917
\(220\) −418584. −0.583076
\(221\) −1.18531e6 −1.63249
\(222\) 134792. 0.183562
\(223\) 882022. 1.18773 0.593865 0.804565i \(-0.297602\pi\)
0.593865 + 0.804565i \(0.297602\pi\)
\(224\) 0 0
\(225\) −148149. −0.195093
\(226\) 3002.91 0.00391085
\(227\) −1.12650e6 −1.45100 −0.725499 0.688223i \(-0.758391\pi\)
−0.725499 + 0.688223i \(0.758391\pi\)
\(228\) 746514. 0.951045
\(229\) 310084. 0.390743 0.195371 0.980729i \(-0.437409\pi\)
0.195371 + 0.980729i \(0.437409\pi\)
\(230\) −194902. −0.242938
\(231\) 0 0
\(232\) −473036. −0.576998
\(233\) 1.13654e6 1.37149 0.685746 0.727841i \(-0.259476\pi\)
0.685746 + 0.727841i \(0.259476\pi\)
\(234\) 785485. 0.937774
\(235\) 1.00678e6 1.18923
\(236\) −820980. −0.959516
\(237\) −657536. −0.760411
\(238\) 0 0
\(239\) 87506.8 0.0990940 0.0495470 0.998772i \(-0.484222\pi\)
0.0495470 + 0.998772i \(0.484222\pi\)
\(240\) 238239. 0.266984
\(241\) 537768. 0.596421 0.298210 0.954500i \(-0.403610\pi\)
0.298210 + 0.954500i \(0.403610\pi\)
\(242\) 622398. 0.683171
\(243\) 59049.0 0.0641500
\(244\) −938948. −1.00964
\(245\) 0 0
\(246\) 910397. 0.959164
\(247\) −2.42084e6 −2.52478
\(248\) 59937.1 0.0618823
\(249\) −273068. −0.279108
\(250\) −1.50633e6 −1.52430
\(251\) 1.35353e6 1.35607 0.678036 0.735028i \(-0.262831\pi\)
0.678036 + 0.735028i \(0.262831\pi\)
\(252\) 0 0
\(253\) −189457. −0.186084
\(254\) −1.42713e6 −1.38797
\(255\) −334492. −0.322133
\(256\) 417731. 0.398380
\(257\) −976900. −0.922608 −0.461304 0.887242i \(-0.652618\pi\)
−0.461304 + 0.887242i \(0.652618\pi\)
\(258\) 1.50534e6 1.40794
\(259\) 0 0
\(260\) 1.62597e6 1.49169
\(261\) 618158. 0.561692
\(262\) 1.46938e6 1.32246
\(263\) −1.24375e6 −1.10877 −0.554387 0.832259i \(-0.687047\pi\)
−0.554387 + 0.832259i \(0.687047\pi\)
\(264\) −164885. −0.145604
\(265\) 256116. 0.224038
\(266\) 0 0
\(267\) −324805. −0.278833
\(268\) 1.36393e6 1.15999
\(269\) −1.08408e6 −0.913445 −0.456722 0.889609i \(-0.650977\pi\)
−0.456722 + 0.889609i \(0.650977\pi\)
\(270\) 221663. 0.185047
\(271\) −2.16627e6 −1.79180 −0.895900 0.444256i \(-0.853468\pi\)
−0.895900 + 0.444256i \(0.853468\pi\)
\(272\) −759119. −0.622139
\(273\) 0 0
\(274\) −3.10587e6 −2.49924
\(275\) −540597. −0.431064
\(276\) −226941. −0.179324
\(277\) 253859. 0.198789 0.0993946 0.995048i \(-0.468309\pi\)
0.0993946 + 0.995048i \(0.468309\pi\)
\(278\) −1.84099e6 −1.42870
\(279\) −78325.1 −0.0602407
\(280\) 0 0
\(281\) 1.14116e6 0.862143 0.431072 0.902318i \(-0.358136\pi\)
0.431072 + 0.902318i \(0.358136\pi\)
\(282\) 2.12587e6 1.59190
\(283\) −609918. −0.452694 −0.226347 0.974047i \(-0.572678\pi\)
−0.226347 + 0.974047i \(0.572678\pi\)
\(284\) −1.11774e6 −0.822327
\(285\) −683158. −0.498206
\(286\) 2.86624e6 2.07204
\(287\) 0 0
\(288\) 663718. 0.471523
\(289\) −354040. −0.249349
\(290\) 2.32049e6 1.62026
\(291\) −1.38567e6 −0.959237
\(292\) −601593. −0.412901
\(293\) 156438. 0.106457 0.0532283 0.998582i \(-0.483049\pi\)
0.0532283 + 0.998582i \(0.483049\pi\)
\(294\) 0 0
\(295\) 751303. 0.502644
\(296\) 109911. 0.0729139
\(297\) 215470. 0.141741
\(298\) 548211. 0.357608
\(299\) 735937. 0.476061
\(300\) −647554. −0.415406
\(301\) 0 0
\(302\) 1.89007e6 1.19251
\(303\) 1.25828e6 0.787354
\(304\) −1.55040e6 −0.962189
\(305\) 859259. 0.528901
\(306\) −706299. −0.431206
\(307\) 293229. 0.177566 0.0887831 0.996051i \(-0.471702\pi\)
0.0887831 + 0.996051i \(0.471702\pi\)
\(308\) 0 0
\(309\) −1.04333e6 −0.621620
\(310\) −294023. −0.173771
\(311\) 2.45216e6 1.43763 0.718816 0.695200i \(-0.244684\pi\)
0.718816 + 0.695200i \(0.244684\pi\)
\(312\) 640490. 0.372500
\(313\) 1.83541e6 1.05894 0.529471 0.848328i \(-0.322390\pi\)
0.529471 + 0.848328i \(0.322390\pi\)
\(314\) 3.88503e6 2.22367
\(315\) 0 0
\(316\) −2.87406e6 −1.61912
\(317\) 589960. 0.329742 0.164871 0.986315i \(-0.447279\pi\)
0.164871 + 0.986315i \(0.447279\pi\)
\(318\) 540803. 0.299896
\(319\) 2.25567e6 1.24107
\(320\) 1.64444e6 0.897726
\(321\) 747556. 0.404931
\(322\) 0 0
\(323\) 2.17679e6 1.16094
\(324\) 258101. 0.136593
\(325\) 2.09993e6 1.10280
\(326\) −769186. −0.400855
\(327\) 408206. 0.211111
\(328\) 742344. 0.380996
\(329\) 0 0
\(330\) 808849. 0.408868
\(331\) 177318. 0.0889577 0.0444789 0.999010i \(-0.485837\pi\)
0.0444789 + 0.999010i \(0.485837\pi\)
\(332\) −1.19357e6 −0.594296
\(333\) −143630. −0.0709798
\(334\) 2.65864e6 1.30405
\(335\) −1.24817e6 −0.607664
\(336\) 0 0
\(337\) −3.04781e6 −1.46189 −0.730943 0.682438i \(-0.760920\pi\)
−0.730943 + 0.682438i \(0.760920\pi\)
\(338\) −7.99777e6 −3.80783
\(339\) −3199.80 −0.00151225
\(340\) −1.46205e6 −0.685908
\(341\) −285809. −0.133104
\(342\) −1.44253e6 −0.666896
\(343\) 0 0
\(344\) 1.22747e6 0.559259
\(345\) 207680. 0.0939393
\(346\) 3.05874e6 1.37357
\(347\) −2.42361e6 −1.08054 −0.540268 0.841493i \(-0.681677\pi\)
−0.540268 + 0.841493i \(0.681677\pi\)
\(348\) 2.70195e6 1.19599
\(349\) −2.67690e6 −1.17644 −0.588218 0.808702i \(-0.700170\pi\)
−0.588218 + 0.808702i \(0.700170\pi\)
\(350\) 0 0
\(351\) −836985. −0.362619
\(352\) 2.42191e6 1.04184
\(353\) 950412. 0.405953 0.202976 0.979184i \(-0.434939\pi\)
0.202976 + 0.979184i \(0.434939\pi\)
\(354\) 1.58642e6 0.672837
\(355\) 1.02288e6 0.430777
\(356\) −1.41971e6 −0.593711
\(357\) 0 0
\(358\) 1.46877e6 0.605683
\(359\) 2.78881e6 1.14204 0.571022 0.820935i \(-0.306547\pi\)
0.571022 + 0.820935i \(0.306547\pi\)
\(360\) 180745. 0.0735040
\(361\) 1.96972e6 0.795495
\(362\) −1.14001e6 −0.457235
\(363\) −663206. −0.264169
\(364\) 0 0
\(365\) 550536. 0.216299
\(366\) 1.81437e6 0.707985
\(367\) 153881. 0.0596377 0.0298189 0.999555i \(-0.490507\pi\)
0.0298189 + 0.999555i \(0.490507\pi\)
\(368\) 471323. 0.181426
\(369\) −970087. −0.370890
\(370\) −539169. −0.204749
\(371\) 0 0
\(372\) −342356. −0.128269
\(373\) −2.38381e6 −0.887156 −0.443578 0.896236i \(-0.646291\pi\)
−0.443578 + 0.896236i \(0.646291\pi\)
\(374\) −2.57729e6 −0.952763
\(375\) 1.60510e6 0.589417
\(376\) 1.73345e6 0.632328
\(377\) −8.76203e6 −3.17506
\(378\) 0 0
\(379\) 3.65191e6 1.30594 0.652969 0.757385i \(-0.273523\pi\)
0.652969 + 0.757385i \(0.273523\pi\)
\(380\) −2.98606e6 −1.06081
\(381\) 1.52070e6 0.536701
\(382\) −1.53226e6 −0.537246
\(383\) 2.15730e6 0.751472 0.375736 0.926727i \(-0.377390\pi\)
0.375736 + 0.926727i \(0.377390\pi\)
\(384\) 1.11245e6 0.384991
\(385\) 0 0
\(386\) −8.15904e6 −2.78722
\(387\) −1.60404e6 −0.544424
\(388\) −6.05669e6 −2.04247
\(389\) −3.66471e6 −1.22791 −0.613954 0.789342i \(-0.710422\pi\)
−0.613954 + 0.789342i \(0.710422\pi\)
\(390\) −3.14194e6 −1.04601
\(391\) −661746. −0.218902
\(392\) 0 0
\(393\) −1.56572e6 −0.511368
\(394\) 5.90440e6 1.91617
\(395\) 2.63014e6 0.848177
\(396\) 941813. 0.301805
\(397\) −3.94648e6 −1.25671 −0.628353 0.777928i \(-0.716271\pi\)
−0.628353 + 0.777928i \(0.716271\pi\)
\(398\) 3.51524e6 1.11236
\(399\) 0 0
\(400\) 1.34488e6 0.420274
\(401\) 25016.1 0.00776887 0.00388444 0.999992i \(-0.498764\pi\)
0.00388444 + 0.999992i \(0.498764\pi\)
\(402\) −2.63559e6 −0.813416
\(403\) 1.11021e6 0.340521
\(404\) 5.49989e6 1.67649
\(405\) −236196. −0.0715542
\(406\) 0 0
\(407\) −524107. −0.156832
\(408\) −575922. −0.171282
\(409\) 832700. 0.246139 0.123069 0.992398i \(-0.460726\pi\)
0.123069 + 0.992398i \(0.460726\pi\)
\(410\) −3.64159e6 −1.06987
\(411\) 3.30951e6 0.966405
\(412\) −4.56035e6 −1.32360
\(413\) 0 0
\(414\) 438528. 0.125747
\(415\) 1.09227e6 0.311323
\(416\) −9.40782e6 −2.66536
\(417\) 1.96170e6 0.552449
\(418\) −5.26379e6 −1.47353
\(419\) 3.95178e6 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(420\) 0 0
\(421\) 4.72285e6 1.29867 0.649336 0.760502i \(-0.275047\pi\)
0.649336 + 0.760502i \(0.275047\pi\)
\(422\) 3.43890e6 0.940023
\(423\) −2.26526e6 −0.615555
\(424\) 440974. 0.119124
\(425\) −1.88823e6 −0.507087
\(426\) 2.15986e6 0.576636
\(427\) 0 0
\(428\) 3.26754e6 0.862207
\(429\) −3.05417e6 −0.801216
\(430\) −6.02136e6 −1.57045
\(431\) 4.07810e6 1.05746 0.528731 0.848790i \(-0.322668\pi\)
0.528731 + 0.848790i \(0.322668\pi\)
\(432\) −536039. −0.138193
\(433\) −1.79927e6 −0.461186 −0.230593 0.973050i \(-0.574067\pi\)
−0.230593 + 0.973050i \(0.574067\pi\)
\(434\) 0 0
\(435\) −2.47263e6 −0.626522
\(436\) 1.78425e6 0.449511
\(437\) −1.35153e6 −0.338550
\(438\) 1.16249e6 0.289536
\(439\) 4.51827e6 1.11895 0.559475 0.828847i \(-0.311003\pi\)
0.559475 + 0.828847i \(0.311003\pi\)
\(440\) 659542. 0.162409
\(441\) 0 0
\(442\) 1.00114e7 2.43746
\(443\) −2.85256e6 −0.690597 −0.345299 0.938493i \(-0.612222\pi\)
−0.345299 + 0.938493i \(0.612222\pi\)
\(444\) −627802. −0.151135
\(445\) 1.29922e6 0.311016
\(446\) −7.44976e6 −1.77339
\(447\) −584155. −0.138280
\(448\) 0 0
\(449\) −1.90246e6 −0.445348 −0.222674 0.974893i \(-0.571478\pi\)
−0.222674 + 0.974893i \(0.571478\pi\)
\(450\) 1.25130e6 0.291293
\(451\) −3.53986e6 −0.819491
\(452\) −13986.2 −0.00321998
\(453\) −2.01400e6 −0.461119
\(454\) 9.51468e6 2.16648
\(455\) 0 0
\(456\) −1.17625e6 −0.264903
\(457\) 2.64834e6 0.593176 0.296588 0.955006i \(-0.404151\pi\)
0.296588 + 0.955006i \(0.404151\pi\)
\(458\) −2.61904e6 −0.583416
\(459\) 752608. 0.166739
\(460\) 907763. 0.200022
\(461\) 1.09031e6 0.238944 0.119472 0.992838i \(-0.461880\pi\)
0.119472 + 0.992838i \(0.461880\pi\)
\(462\) 0 0
\(463\) −2.50851e6 −0.543831 −0.271916 0.962321i \(-0.587657\pi\)
−0.271916 + 0.962321i \(0.587657\pi\)
\(464\) −5.61155e6 −1.21001
\(465\) 313300. 0.0671937
\(466\) −9.59943e6 −2.04777
\(467\) −3.20935e6 −0.680966 −0.340483 0.940251i \(-0.610591\pi\)
−0.340483 + 0.940251i \(0.610591\pi\)
\(468\) −3.65843e6 −0.772112
\(469\) 0 0
\(470\) −8.50350e6 −1.77563
\(471\) −4.13976e6 −0.859850
\(472\) 1.29358e6 0.267262
\(473\) −5.85315e6 −1.20292
\(474\) 5.55369e6 1.13537
\(475\) −3.85647e6 −0.784252
\(476\) 0 0
\(477\) −576260. −0.115964
\(478\) −739102. −0.147957
\(479\) 2.31462e6 0.460936 0.230468 0.973080i \(-0.425974\pi\)
0.230468 + 0.973080i \(0.425974\pi\)
\(480\) −2.65487e6 −0.525945
\(481\) 2.03587e6 0.401225
\(482\) −4.54211e6 −0.890513
\(483\) 0 0
\(484\) −2.89885e6 −0.562486
\(485\) 5.54266e6 1.06995
\(486\) −498741. −0.0957821
\(487\) −4.63735e6 −0.886028 −0.443014 0.896515i \(-0.646091\pi\)
−0.443014 + 0.896515i \(0.646091\pi\)
\(488\) 1.47945e6 0.281224
\(489\) 819618. 0.155003
\(490\) 0 0
\(491\) 5.02151e6 0.940007 0.470003 0.882665i \(-0.344253\pi\)
0.470003 + 0.882665i \(0.344253\pi\)
\(492\) −4.24021e6 −0.789724
\(493\) 7.87872e6 1.45995
\(494\) 2.04470e7 3.76974
\(495\) −861881. −0.158101
\(496\) 711024. 0.129772
\(497\) 0 0
\(498\) 2.30639e6 0.416735
\(499\) 3.37822e6 0.607347 0.303673 0.952776i \(-0.401787\pi\)
0.303673 + 0.952776i \(0.401787\pi\)
\(500\) 7.01582e6 1.25503
\(501\) −2.83295e6 −0.504249
\(502\) −1.14322e7 −2.02475
\(503\) 5.03743e6 0.887747 0.443873 0.896090i \(-0.353604\pi\)
0.443873 + 0.896090i \(0.353604\pi\)
\(504\) 0 0
\(505\) −5.03311e6 −0.878230
\(506\) 1.60019e6 0.277841
\(507\) 8.52214e6 1.47241
\(508\) 6.64694e6 1.14278
\(509\) 6.72466e6 1.15047 0.575236 0.817988i \(-0.304910\pi\)
0.575236 + 0.817988i \(0.304910\pi\)
\(510\) 2.82520e6 0.480976
\(511\) 0 0
\(512\) −7.48361e6 −1.26164
\(513\) 1.53711e6 0.257876
\(514\) 8.25112e6 1.37754
\(515\) 4.17332e6 0.693367
\(516\) −7.01119e6 −1.15922
\(517\) −8.26595e6 −1.36009
\(518\) 0 0
\(519\) −3.25929e6 −0.531134
\(520\) −2.56196e6 −0.415493
\(521\) 4.42770e6 0.714635 0.357317 0.933983i \(-0.383691\pi\)
0.357317 + 0.933983i \(0.383691\pi\)
\(522\) −5.22110e6 −0.838660
\(523\) 8.95911e6 1.43222 0.716111 0.697986i \(-0.245920\pi\)
0.716111 + 0.697986i \(0.245920\pi\)
\(524\) −6.84372e6 −1.08884
\(525\) 0 0
\(526\) 1.05050e7 1.65551
\(527\) −998291. −0.156578
\(528\) −1.95601e6 −0.305342
\(529\) −6.02548e6 −0.936165
\(530\) −2.16321e6 −0.334510
\(531\) −1.69043e6 −0.260173
\(532\) 0 0
\(533\) 1.37504e7 2.09652
\(534\) 2.74338e6 0.416325
\(535\) −2.99022e6 −0.451668
\(536\) −2.14908e6 −0.323103
\(537\) −1.56507e6 −0.234205
\(538\) 9.15641e6 1.36386
\(539\) 0 0
\(540\) −1.03240e6 −0.152358
\(541\) −1.00467e7 −1.47581 −0.737907 0.674902i \(-0.764186\pi\)
−0.737907 + 0.674902i \(0.764186\pi\)
\(542\) 1.82968e7 2.67533
\(543\) 1.21476e6 0.176804
\(544\) 8.45941e6 1.22558
\(545\) −1.63282e6 −0.235477
\(546\) 0 0
\(547\) −1.31426e7 −1.87808 −0.939039 0.343811i \(-0.888282\pi\)
−0.939039 + 0.343811i \(0.888282\pi\)
\(548\) 1.44657e7 2.05774
\(549\) −1.93333e6 −0.273764
\(550\) 4.56600e6 0.643620
\(551\) 1.60913e7 2.25794
\(552\) 357579. 0.0499487
\(553\) 0 0
\(554\) −2.14415e6 −0.296811
\(555\) 574520. 0.0791722
\(556\) 8.57452e6 1.17631
\(557\) 9.06752e6 1.23837 0.619185 0.785245i \(-0.287463\pi\)
0.619185 + 0.785245i \(0.287463\pi\)
\(558\) 661551. 0.0899452
\(559\) 2.27363e7 3.07744
\(560\) 0 0
\(561\) 2.74627e6 0.368414
\(562\) −9.63846e6 −1.28726
\(563\) −1.05180e7 −1.39849 −0.699247 0.714880i \(-0.746481\pi\)
−0.699247 + 0.714880i \(0.746481\pi\)
\(564\) −9.90136e6 −1.31068
\(565\) 12799.2 0.00168679
\(566\) 5.15150e6 0.675916
\(567\) 0 0
\(568\) 1.76117e6 0.229050
\(569\) 7.32307e6 0.948227 0.474114 0.880464i \(-0.342769\pi\)
0.474114 + 0.880464i \(0.342769\pi\)
\(570\) 5.77010e6 0.743869
\(571\) −6.97981e6 −0.895887 −0.447943 0.894062i \(-0.647843\pi\)
−0.447943 + 0.894062i \(0.647843\pi\)
\(572\) −1.33497e7 −1.70600
\(573\) 1.63272e6 0.207743
\(574\) 0 0
\(575\) 1.17237e6 0.147875
\(576\) −3.70000e6 −0.464670
\(577\) 5.81210e6 0.726765 0.363382 0.931640i \(-0.381622\pi\)
0.363382 + 0.931640i \(0.381622\pi\)
\(578\) 2.99030e6 0.372302
\(579\) 8.69399e6 1.07776
\(580\) −1.08078e7 −1.33403
\(581\) 0 0
\(582\) 1.17036e7 1.43223
\(583\) −2.10278e6 −0.256226
\(584\) 947901. 0.115009
\(585\) 3.34794e6 0.404472
\(586\) −1.32131e6 −0.158950
\(587\) 7.37446e6 0.883355 0.441677 0.897174i \(-0.354384\pi\)
0.441677 + 0.897174i \(0.354384\pi\)
\(588\) 0 0
\(589\) −2.03888e6 −0.242161
\(590\) −6.34567e6 −0.750495
\(591\) −6.29152e6 −0.740947
\(592\) 1.30385e6 0.152906
\(593\) −9.46528e6 −1.10534 −0.552671 0.833399i \(-0.686391\pi\)
−0.552671 + 0.833399i \(0.686391\pi\)
\(594\) −1.81991e6 −0.211633
\(595\) 0 0
\(596\) −2.55332e6 −0.294435
\(597\) −3.74571e6 −0.430129
\(598\) −6.21589e6 −0.710804
\(599\) −8.52195e6 −0.970448 −0.485224 0.874390i \(-0.661262\pi\)
−0.485224 + 0.874390i \(0.661262\pi\)
\(600\) 1.02032e6 0.115706
\(601\) 657065. 0.0742031 0.0371016 0.999311i \(-0.488187\pi\)
0.0371016 + 0.999311i \(0.488187\pi\)
\(602\) 0 0
\(603\) 2.80839e6 0.314532
\(604\) −8.80310e6 −0.981846
\(605\) 2.65282e6 0.294659
\(606\) −1.06277e7 −1.17560
\(607\) −5.86885e6 −0.646519 −0.323260 0.946310i \(-0.604779\pi\)
−0.323260 + 0.946310i \(0.604779\pi\)
\(608\) 1.72773e7 1.89547
\(609\) 0 0
\(610\) −7.25750e6 −0.789700
\(611\) 3.21087e7 3.47952
\(612\) 3.28962e6 0.355032
\(613\) 3.84402e6 0.413175 0.206588 0.978428i \(-0.433764\pi\)
0.206588 + 0.978428i \(0.433764\pi\)
\(614\) −2.47667e6 −0.265123
\(615\) 3.88035e6 0.413698
\(616\) 0 0
\(617\) 6.44660e6 0.681739 0.340869 0.940111i \(-0.389279\pi\)
0.340869 + 0.940111i \(0.389279\pi\)
\(618\) 8.81219e6 0.928138
\(619\) −6.73740e6 −0.706749 −0.353375 0.935482i \(-0.614966\pi\)
−0.353375 + 0.935482i \(0.614966\pi\)
\(620\) 1.36942e6 0.143073
\(621\) −467281. −0.0486238
\(622\) −2.07115e7 −2.14652
\(623\) 0 0
\(624\) 7.59804e6 0.781160
\(625\) −704759. −0.0721673
\(626\) −1.55023e7 −1.58110
\(627\) 5.60891e6 0.569783
\(628\) −1.80947e7 −1.83085
\(629\) −1.83063e6 −0.184491
\(630\) 0 0
\(631\) −9.14514e6 −0.914360 −0.457180 0.889374i \(-0.651140\pi\)
−0.457180 + 0.889374i \(0.651140\pi\)
\(632\) 4.52852e6 0.450987
\(633\) −3.66437e6 −0.363488
\(634\) −4.98293e6 −0.492336
\(635\) −6.08282e6 −0.598647
\(636\) −2.51881e6 −0.246918
\(637\) 0 0
\(638\) −1.90518e7 −1.85304
\(639\) −2.30147e6 −0.222974
\(640\) −4.44978e6 −0.429426
\(641\) −1.04088e6 −0.100059 −0.0500296 0.998748i \(-0.515932\pi\)
−0.0500296 + 0.998748i \(0.515932\pi\)
\(642\) −6.31403e6 −0.604601
\(643\) 9.08713e6 0.866761 0.433381 0.901211i \(-0.357321\pi\)
0.433381 + 0.901211i \(0.357321\pi\)
\(644\) 0 0
\(645\) 6.41615e6 0.607261
\(646\) −1.83857e7 −1.73340
\(647\) −2.20711e6 −0.207283 −0.103641 0.994615i \(-0.533049\pi\)
−0.103641 + 0.994615i \(0.533049\pi\)
\(648\) −406677. −0.0380463
\(649\) −6.16840e6 −0.574859
\(650\) −1.77364e7 −1.64658
\(651\) 0 0
\(652\) 3.58252e6 0.330042
\(653\) −1.83610e7 −1.68505 −0.842524 0.538658i \(-0.818931\pi\)
−0.842524 + 0.538658i \(0.818931\pi\)
\(654\) −3.44780e6 −0.315208
\(655\) 6.26289e6 0.570390
\(656\) 8.80632e6 0.798978
\(657\) −1.23871e6 −0.111958
\(658\) 0 0
\(659\) 6.21208e6 0.557216 0.278608 0.960405i \(-0.410127\pi\)
0.278608 + 0.960405i \(0.410127\pi\)
\(660\) −3.76725e6 −0.336639
\(661\) −1.54230e7 −1.37298 −0.686491 0.727138i \(-0.740850\pi\)
−0.686491 + 0.727138i \(0.740850\pi\)
\(662\) −1.49767e6 −0.132822
\(663\) −1.06678e7 −0.942519
\(664\) 1.88065e6 0.165534
\(665\) 0 0
\(666\) 1.21313e6 0.105980
\(667\) −4.89176e6 −0.425746
\(668\) −1.23827e7 −1.07368
\(669\) 7.93820e6 0.685736
\(670\) 1.05424e7 0.907300
\(671\) −7.05475e6 −0.604889
\(672\) 0 0
\(673\) −2.27201e7 −1.93362 −0.966811 0.255491i \(-0.917763\pi\)
−0.966811 + 0.255491i \(0.917763\pi\)
\(674\) 2.57425e7 2.18274
\(675\) −1.33334e6 −0.112637
\(676\) 3.72500e7 3.13516
\(677\) −1.36173e7 −1.14188 −0.570940 0.820992i \(-0.693421\pi\)
−0.570940 + 0.820992i \(0.693421\pi\)
\(678\) 27026.2 0.00225793
\(679\) 0 0
\(680\) 2.30369e6 0.191052
\(681\) −1.01385e7 −0.837734
\(682\) 2.41401e6 0.198736
\(683\) 2.39985e6 0.196849 0.0984245 0.995145i \(-0.468620\pi\)
0.0984245 + 0.995145i \(0.468620\pi\)
\(684\) 6.71863e6 0.549086
\(685\) −1.32380e7 −1.07795
\(686\) 0 0
\(687\) 2.79076e6 0.225595
\(688\) 1.45612e7 1.17281
\(689\) 8.16816e6 0.655505
\(690\) −1.75411e6 −0.140260
\(691\) 1.40365e6 0.111831 0.0559156 0.998435i \(-0.482192\pi\)
0.0559156 + 0.998435i \(0.482192\pi\)
\(692\) −1.42462e7 −1.13093
\(693\) 0 0
\(694\) 2.04703e7 1.61334
\(695\) −7.84680e6 −0.616212
\(696\) −4.25732e6 −0.333130
\(697\) −1.23642e7 −0.964018
\(698\) 2.26097e7 1.75653
\(699\) 1.02288e7 0.791831
\(700\) 0 0
\(701\) −5.78991e6 −0.445017 −0.222509 0.974931i \(-0.571425\pi\)
−0.222509 + 0.974931i \(0.571425\pi\)
\(702\) 7.06936e6 0.541424
\(703\) −3.73884e6 −0.285330
\(704\) −1.35013e7 −1.02670
\(705\) 9.06103e6 0.686602
\(706\) −8.02739e6 −0.606126
\(707\) 0 0
\(708\) −7.38882e6 −0.553977
\(709\) −1.13143e7 −0.845304 −0.422652 0.906292i \(-0.638901\pi\)
−0.422652 + 0.906292i \(0.638901\pi\)
\(710\) −8.63944e6 −0.643191
\(711\) −5.91782e6 −0.439024
\(712\) 2.23697e6 0.165371
\(713\) 619821. 0.0456607
\(714\) 0 0
\(715\) 1.22167e7 0.893692
\(716\) −6.84085e6 −0.498686
\(717\) 787562. 0.0572119
\(718\) −2.35549e7 −1.70518
\(719\) 2.73780e7 1.97506 0.987529 0.157437i \(-0.0503231\pi\)
0.987529 + 0.157437i \(0.0503231\pi\)
\(720\) 2.14415e6 0.154143
\(721\) 0 0
\(722\) −1.66367e7 −1.18775
\(723\) 4.83992e6 0.344344
\(724\) 5.30967e6 0.376462
\(725\) −1.39582e7 −0.986241
\(726\) 5.60158e6 0.394429
\(727\) 9.86471e6 0.692226 0.346113 0.938193i \(-0.387501\pi\)
0.346113 + 0.938193i \(0.387501\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −4.64995e6 −0.322954
\(731\) −2.04442e7 −1.41507
\(732\) −8.45053e6 −0.582916
\(733\) −3.87876e6 −0.266645 −0.133322 0.991073i \(-0.542565\pi\)
−0.133322 + 0.991073i \(0.542565\pi\)
\(734\) −1.29972e6 −0.0890448
\(735\) 0 0
\(736\) −5.25229e6 −0.357400
\(737\) 1.02479e7 0.694967
\(738\) 8.19357e6 0.553774
\(739\) −7.95498e6 −0.535831 −0.267916 0.963442i \(-0.586335\pi\)
−0.267916 + 0.963442i \(0.586335\pi\)
\(740\) 2.51121e6 0.168579
\(741\) −2.17876e7 −1.45768
\(742\) 0 0
\(743\) 1.65977e7 1.10300 0.551500 0.834175i \(-0.314056\pi\)
0.551500 + 0.834175i \(0.314056\pi\)
\(744\) 539433. 0.0357277
\(745\) 2.33662e6 0.154240
\(746\) 2.01342e7 1.32461
\(747\) −2.45761e6 −0.161143
\(748\) 1.20039e7 0.784453
\(749\) 0 0
\(750\) −1.35570e7 −0.880056
\(751\) 1.51072e7 0.977426 0.488713 0.872445i \(-0.337467\pi\)
0.488713 + 0.872445i \(0.337467\pi\)
\(752\) 2.05637e7 1.32604
\(753\) 1.21818e7 0.782929
\(754\) 7.40061e7 4.74066
\(755\) 8.05598e6 0.514341
\(756\) 0 0
\(757\) 5.80923e6 0.368450 0.184225 0.982884i \(-0.441022\pi\)
0.184225 + 0.982884i \(0.441022\pi\)
\(758\) −3.08449e7 −1.94989
\(759\) −1.70511e6 −0.107436
\(760\) 4.70498e6 0.295477
\(761\) −2.54270e7 −1.59160 −0.795799 0.605561i \(-0.792949\pi\)
−0.795799 + 0.605561i \(0.792949\pi\)
\(762\) −1.28442e7 −0.801346
\(763\) 0 0
\(764\) 7.13656e6 0.442340
\(765\) −3.01043e6 −0.185984
\(766\) −1.82210e7 −1.12202
\(767\) 2.39609e7 1.47067
\(768\) 3.75958e6 0.230005
\(769\) −1.53909e7 −0.938532 −0.469266 0.883057i \(-0.655481\pi\)
−0.469266 + 0.883057i \(0.655481\pi\)
\(770\) 0 0
\(771\) −8.79210e6 −0.532668
\(772\) 3.80011e7 2.29484
\(773\) −905393. −0.0544990 −0.0272495 0.999629i \(-0.508675\pi\)
−0.0272495 + 0.999629i \(0.508675\pi\)
\(774\) 1.35481e7 0.812877
\(775\) 1.76860e6 0.105773
\(776\) 9.54323e6 0.568907
\(777\) 0 0
\(778\) 3.09530e7 1.83338
\(779\) −2.52523e7 −1.49093
\(780\) 1.46337e7 0.861229
\(781\) −8.39810e6 −0.492667
\(782\) 5.58926e6 0.326841
\(783\) 5.56342e6 0.324293
\(784\) 0 0
\(785\) 1.65590e7 0.959093
\(786\) 1.32244e7 0.763521
\(787\) 2.79334e7 1.60763 0.803817 0.594877i \(-0.202799\pi\)
0.803817 + 0.594877i \(0.202799\pi\)
\(788\) −2.75000e7 −1.57767
\(789\) −1.11937e7 −0.640151
\(790\) −2.22148e7 −1.26641
\(791\) 0 0
\(792\) −1.48397e6 −0.0840643
\(793\) 2.74039e7 1.54749
\(794\) 3.33328e7 1.87638
\(795\) 2.30504e6 0.129348
\(796\) −1.63724e7 −0.915860
\(797\) −2.18824e7 −1.22025 −0.610126 0.792304i \(-0.708881\pi\)
−0.610126 + 0.792304i \(0.708881\pi\)
\(798\) 0 0
\(799\) −2.88718e7 −1.59995
\(800\) −1.49869e7 −0.827918
\(801\) −2.92325e6 −0.160985
\(802\) −211291. −0.0115997
\(803\) −4.52005e6 −0.247374
\(804\) 1.22754e7 0.669722
\(805\) 0 0
\(806\) −9.37710e6 −0.508430
\(807\) −9.75676e6 −0.527377
\(808\) −8.66590e6 −0.466966
\(809\) 2.44194e7 1.31179 0.655893 0.754854i \(-0.272292\pi\)
0.655893 + 0.754854i \(0.272292\pi\)
\(810\) 1.99496e6 0.106837
\(811\) −4.46711e6 −0.238492 −0.119246 0.992865i \(-0.538048\pi\)
−0.119246 + 0.992865i \(0.538048\pi\)
\(812\) 0 0
\(813\) −1.94964e7 −1.03450
\(814\) 4.42673e6 0.234165
\(815\) −3.27847e6 −0.172893
\(816\) −6.83207e6 −0.359192
\(817\) −4.17547e7 −2.18852
\(818\) −7.03317e6 −0.367509
\(819\) 0 0
\(820\) 1.69609e7 0.880873
\(821\) 1.34708e7 0.697485 0.348743 0.937219i \(-0.386609\pi\)
0.348743 + 0.937219i \(0.386609\pi\)
\(822\) −2.79529e7 −1.44293
\(823\) −4.03958e6 −0.207891 −0.103946 0.994583i \(-0.533147\pi\)
−0.103946 + 0.994583i \(0.533147\pi\)
\(824\) 7.18552e6 0.368672
\(825\) −4.86537e6 −0.248875
\(826\) 0 0
\(827\) 1.24927e7 0.635175 0.317588 0.948229i \(-0.397127\pi\)
0.317588 + 0.948229i \(0.397127\pi\)
\(828\) −2.04247e6 −0.103533
\(829\) 1.45980e7 0.737749 0.368874 0.929479i \(-0.379743\pi\)
0.368874 + 0.929479i \(0.379743\pi\)
\(830\) −9.22557e6 −0.464834
\(831\) 2.28473e6 0.114771
\(832\) 5.24453e7 2.62663
\(833\) 0 0
\(834\) −1.65690e7 −0.824859
\(835\) 1.13318e7 0.562449
\(836\) 2.45163e7 1.21322
\(837\) −704926. −0.0347800
\(838\) −3.33776e7 −1.64189
\(839\) 9.15983e6 0.449244 0.224622 0.974446i \(-0.427885\pi\)
0.224622 + 0.974446i \(0.427885\pi\)
\(840\) 0 0
\(841\) 3.77299e7 1.83948
\(842\) −3.98903e7 −1.93904
\(843\) 1.02704e7 0.497759
\(844\) −1.60168e7 −0.773964
\(845\) −3.40886e7 −1.64236
\(846\) 1.91329e7 0.919082
\(847\) 0 0
\(848\) 5.23121e6 0.249812
\(849\) −5.48926e6 −0.261363
\(850\) 1.59484e7 0.757129
\(851\) 1.13661e6 0.0538005
\(852\) −1.00597e7 −0.474771
\(853\) 8.68253e6 0.408577 0.204289 0.978911i \(-0.434512\pi\)
0.204289 + 0.978911i \(0.434512\pi\)
\(854\) 0 0
\(855\) −6.14842e6 −0.287639
\(856\) −5.14850e6 −0.240158
\(857\) −1.04988e7 −0.488302 −0.244151 0.969737i \(-0.578509\pi\)
−0.244151 + 0.969737i \(0.578509\pi\)
\(858\) 2.57962e7 1.19629
\(859\) −9.03780e6 −0.417907 −0.208954 0.977926i \(-0.567006\pi\)
−0.208954 + 0.977926i \(0.567006\pi\)
\(860\) 2.80448e7 1.29302
\(861\) 0 0
\(862\) −3.44445e7 −1.57889
\(863\) −1.59858e7 −0.730645 −0.365322 0.930881i \(-0.619041\pi\)
−0.365322 + 0.930881i \(0.619041\pi\)
\(864\) 5.97346e6 0.272234
\(865\) 1.30371e7 0.592437
\(866\) 1.51970e7 0.688594
\(867\) −3.18636e6 −0.143962
\(868\) 0 0
\(869\) −2.15942e7 −0.970035
\(870\) 2.08844e7 0.935457
\(871\) −3.98073e7 −1.77794
\(872\) −2.81136e6 −0.125206
\(873\) −1.24710e7 −0.553816
\(874\) 1.14153e7 0.505487
\(875\) 0 0
\(876\) −5.41434e6 −0.238388
\(877\) 2.67453e7 1.17422 0.587109 0.809508i \(-0.300266\pi\)
0.587109 + 0.809508i \(0.300266\pi\)
\(878\) −3.81623e7 −1.67070
\(879\) 1.40794e6 0.0614628
\(880\) 7.82404e6 0.340584
\(881\) 6.40715e6 0.278115 0.139058 0.990284i \(-0.455593\pi\)
0.139058 + 0.990284i \(0.455593\pi\)
\(882\) 0 0
\(883\) 4.96462e6 0.214281 0.107141 0.994244i \(-0.465831\pi\)
0.107141 + 0.994244i \(0.465831\pi\)
\(884\) −4.66285e7 −2.00688
\(885\) 6.76173e6 0.290201
\(886\) 2.40933e7 1.03113
\(887\) −8.71625e6 −0.371981 −0.185990 0.982552i \(-0.559549\pi\)
−0.185990 + 0.982552i \(0.559549\pi\)
\(888\) 989196. 0.0420969
\(889\) 0 0
\(890\) −1.09735e7 −0.464377
\(891\) 1.93923e6 0.0818344
\(892\) 3.46976e7 1.46011
\(893\) −5.89669e7 −2.47446
\(894\) 4.93390e6 0.206465
\(895\) 6.26026e6 0.261237
\(896\) 0 0
\(897\) 6.62343e6 0.274854
\(898\) 1.60686e7 0.664946
\(899\) −7.37956e6 −0.304531
\(900\) −5.82798e6 −0.239835
\(901\) −7.34472e6 −0.301414
\(902\) 2.98984e7 1.22358
\(903\) 0 0
\(904\) 22037.4 0.000896888 0
\(905\) −4.85904e6 −0.197210
\(906\) 1.70107e7 0.688494
\(907\) −2.36255e7 −0.953594 −0.476797 0.879014i \(-0.658202\pi\)
−0.476797 + 0.879014i \(0.658202\pi\)
\(908\) −4.43151e7 −1.78376
\(909\) 1.13245e7 0.454579
\(910\) 0 0
\(911\) −1.69360e7 −0.676105 −0.338053 0.941127i \(-0.609768\pi\)
−0.338053 + 0.941127i \(0.609768\pi\)
\(912\) −1.39536e7 −0.555520
\(913\) −8.96785e6 −0.356050
\(914\) −2.23685e7 −0.885668
\(915\) 7.73334e6 0.305361
\(916\) 1.21983e7 0.480353
\(917\) 0 0
\(918\) −6.35669e6 −0.248957
\(919\) −3.13804e7 −1.22566 −0.612829 0.790216i \(-0.709968\pi\)
−0.612829 + 0.790216i \(0.709968\pi\)
\(920\) −1.43032e6 −0.0557138
\(921\) 2.63906e6 0.102518
\(922\) −9.20896e6 −0.356766
\(923\) 3.26220e7 1.26040
\(924\) 0 0
\(925\) 3.24320e6 0.124629
\(926\) 2.11875e7 0.811992
\(927\) −9.38996e6 −0.358893
\(928\) 6.25336e7 2.38365
\(929\) 1.43089e7 0.543959 0.271980 0.962303i \(-0.412322\pi\)
0.271980 + 0.962303i \(0.412322\pi\)
\(930\) −2.64620e6 −0.100327
\(931\) 0 0
\(932\) 4.47098e7 1.68602
\(933\) 2.20694e7 0.830017
\(934\) 2.71069e7 1.01675
\(935\) −1.09851e7 −0.410937
\(936\) 5.76441e6 0.215063
\(937\) −2.81206e7 −1.04635 −0.523173 0.852227i \(-0.675252\pi\)
−0.523173 + 0.852227i \(0.675252\pi\)
\(938\) 0 0
\(939\) 1.65187e7 0.611381
\(940\) 3.96054e7 1.46196
\(941\) 3.29569e7 1.21331 0.606656 0.794964i \(-0.292510\pi\)
0.606656 + 0.794964i \(0.292510\pi\)
\(942\) 3.49653e7 1.28384
\(943\) 7.67672e6 0.281123
\(944\) 1.53455e7 0.560469
\(945\) 0 0
\(946\) 4.94370e7 1.79607
\(947\) 1.62975e7 0.590535 0.295268 0.955415i \(-0.404591\pi\)
0.295268 + 0.955415i \(0.404591\pi\)
\(948\) −2.58666e7 −0.934799
\(949\) 1.75579e7 0.632861
\(950\) 3.25726e7 1.17096
\(951\) 5.30964e6 0.190377
\(952\) 0 0
\(953\) −3.03230e7 −1.08153 −0.540767 0.841172i \(-0.681866\pi\)
−0.540767 + 0.841172i \(0.681866\pi\)
\(954\) 4.86722e6 0.173145
\(955\) −6.53088e6 −0.231720
\(956\) 3.44240e6 0.121820
\(957\) 2.03010e7 0.716535
\(958\) −1.95498e7 −0.688221
\(959\) 0 0
\(960\) 1.48000e7 0.518302
\(961\) −2.76941e7 −0.967339
\(962\) −1.71954e7 −0.599067
\(963\) 6.72801e6 0.233787
\(964\) 2.11551e7 0.733200
\(965\) −3.47759e7 −1.20216
\(966\) 0 0
\(967\) −1.49059e7 −0.512616 −0.256308 0.966595i \(-0.582506\pi\)
−0.256308 + 0.966595i \(0.582506\pi\)
\(968\) 4.56757e6 0.156674
\(969\) 1.95911e7 0.670271
\(970\) −4.68145e7 −1.59754
\(971\) −1.65608e7 −0.563679 −0.281840 0.959462i \(-0.590945\pi\)
−0.281840 + 0.959462i \(0.590945\pi\)
\(972\) 2.32291e6 0.0788618
\(973\) 0 0
\(974\) 3.91681e7 1.32292
\(975\) 1.88993e7 0.636700
\(976\) 1.75505e7 0.589747
\(977\) 7.56862e6 0.253676 0.126838 0.991923i \(-0.459517\pi\)
0.126838 + 0.991923i \(0.459517\pi\)
\(978\) −6.92267e6 −0.231434
\(979\) −1.06670e7 −0.355700
\(980\) 0 0
\(981\) 3.67386e6 0.121885
\(982\) −4.24128e7 −1.40352
\(983\) 3.32042e7 1.09600 0.547998 0.836480i \(-0.315390\pi\)
0.547998 + 0.836480i \(0.315390\pi\)
\(984\) 6.68110e6 0.219968
\(985\) 2.51661e7 0.826466
\(986\) −6.65454e7 −2.17985
\(987\) 0 0
\(988\) −9.52327e7 −3.10380
\(989\) 1.26935e7 0.412657
\(990\) 7.27964e6 0.236060
\(991\) 3.48003e7 1.12564 0.562819 0.826580i \(-0.309717\pi\)
0.562819 + 0.826580i \(0.309717\pi\)
\(992\) −7.92345e6 −0.255644
\(993\) 1.59586e6 0.0513598
\(994\) 0 0
\(995\) 1.49829e7 0.479774
\(996\) −1.07421e7 −0.343117
\(997\) −9.40852e6 −0.299767 −0.149883 0.988704i \(-0.547890\pi\)
−0.149883 + 0.988704i \(0.547890\pi\)
\(998\) −2.85332e7 −0.906826
\(999\) −1.29267e6 −0.0409802
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 147.6.a.j.1.1 yes 2
3.2 odd 2 441.6.a.r.1.2 2
7.2 even 3 147.6.e.m.67.2 4
7.3 odd 6 147.6.e.n.79.2 4
7.4 even 3 147.6.e.m.79.2 4
7.5 odd 6 147.6.e.n.67.2 4
7.6 odd 2 147.6.a.h.1.1 2
21.20 even 2 441.6.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.1 2 7.6 odd 2
147.6.a.j.1.1 yes 2 1.1 even 1 trivial
147.6.e.m.67.2 4 7.2 even 3
147.6.e.m.79.2 4 7.4 even 3
147.6.e.n.67.2 4 7.5 odd 6
147.6.e.n.79.2 4 7.3 odd 6
441.6.a.q.1.2 2 21.20 even 2
441.6.a.r.1.2 2 3.2 odd 2