Properties

Label 441.6.a.q.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
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] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, 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("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(70.7292645375\)
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: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.44622 q^{2} +39.3387 q^{4} -36.0000 q^{5} +61.9840 q^{8} +O(q^{10})\) \(q+8.44622 q^{2} +39.3387 q^{4} -36.0000 q^{5} +61.9840 q^{8} -304.064 q^{10} -295.570 q^{11} +1148.13 q^{13} -735.307 q^{16} +1032.38 q^{17} -2108.51 q^{19} -1416.19 q^{20} -2496.45 q^{22} +640.988 q^{23} -1829.00 q^{25} +9697.34 q^{26} -7631.58 q^{29} +966.976 q^{31} -8194.05 q^{32} +8719.74 q^{34} -1773.21 q^{37} -17809.0 q^{38} -2231.42 q^{40} -11976.4 q^{41} -19802.9 q^{43} -11627.3 q^{44} +5413.93 q^{46} -27966.1 q^{47} -15448.1 q^{50} +45165.8 q^{52} +7114.33 q^{53} +10640.5 q^{55} -64458.0 q^{58} -20869.5 q^{59} +23868.3 q^{61} +8167.30 q^{62} -45679.0 q^{64} -41332.6 q^{65} +34671.5 q^{67} +40612.6 q^{68} +28413.2 q^{71} +15292.7 q^{73} -14976.9 q^{74} -82946.0 q^{76} -73059.5 q^{79} +26471.0 q^{80} -101155. q^{82} -30340.9 q^{83} -37165.8 q^{85} -167260. q^{86} -18320.6 q^{88} -36089.5 q^{89} +25215.6 q^{92} -236208. q^{94} +75906.4 q^{95} +153963. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 37 q^{4} - 72 q^{5} + 249 q^{8} - 108 q^{10} - 480 q^{11} + 1296 q^{13} - 1679 q^{16} - 936 q^{17} - 216 q^{19} - 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} + 8892 q^{26} - 6372 q^{29} + 9936 q^{31} - 9039 q^{32} + 19440 q^{34} + 11124 q^{37} - 28116 q^{38} - 8964 q^{40} - 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} - 7920 q^{47} - 5487 q^{50} + 44820 q^{52} - 2220 q^{53} + 17280 q^{55} - 71318 q^{58} - 29736 q^{59} - 17280 q^{61} - 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} + 45216 q^{68} + 92280 q^{71} + 56592 q^{73} - 85218 q^{74} - 87372 q^{76} - 56096 q^{79} + 60444 q^{80} - 52272 q^{82} + 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} - 123192 q^{89} + 25536 q^{92} - 345384 q^{94} + 7776 q^{95} + 35856 q^{97}+O(q^{100}) \) 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 \(-0.231710\pi\)
0.746548 + 0.665332i \(0.231710\pi\)
\(3\) 0 0
\(4\) 39.3387 1.22933
\(5\) −36.0000 −0.643988 −0.321994 0.946742i \(-0.604353\pi\)
−0.321994 + 0.946742i \(0.604353\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 61.9840 0.342416
\(9\) 0 0
\(10\) −304.064 −0.961535
\(11\) −295.570 −0.736509 −0.368255 0.929725i \(-0.620045\pi\)
−0.368255 + 0.929725i \(0.620045\pi\)
\(12\) 0 0
\(13\) 1148.13 1.88422 0.942111 0.335302i \(-0.108838\pi\)
0.942111 + 0.335302i \(0.108838\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −735.307 −0.718073
\(17\) 1032.38 0.866401 0.433200 0.901298i \(-0.357384\pi\)
0.433200 + 0.901298i \(0.357384\pi\)
\(18\) 0 0
\(19\) −2108.51 −1.33996 −0.669980 0.742379i \(-0.733697\pi\)
−0.669980 + 0.742379i \(0.733697\pi\)
\(20\) −1416.19 −0.791675
\(21\) 0 0
\(22\) −2496.45 −1.09968
\(23\) 640.988 0.252657 0.126328 0.991988i \(-0.459681\pi\)
0.126328 + 0.991988i \(0.459681\pi\)
\(24\) 0 0
\(25\) −1829.00 −0.585280
\(26\) 9697.34 2.81332
\(27\) 0 0
\(28\) 0 0
\(29\) −7631.58 −1.68508 −0.842538 0.538637i \(-0.818940\pi\)
−0.842538 + 0.538637i \(0.818940\pi\)
\(30\) 0 0
\(31\) 966.976 0.180722 0.0903611 0.995909i \(-0.471198\pi\)
0.0903611 + 0.995909i \(0.471198\pi\)
\(32\) −8194.05 −1.41457
\(33\) 0 0
\(34\) 8719.74 1.29362
\(35\) 0 0
\(36\) 0 0
\(37\) −1773.21 −0.212939 −0.106470 0.994316i \(-0.533955\pi\)
−0.106470 + 0.994316i \(0.533955\pi\)
\(38\) −17809.0 −2.00069
\(39\) 0 0
\(40\) −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
\(45\) 0 0
\(46\) 5413.93 0.377240
\(47\) −27966.1 −1.84666 −0.923332 0.384002i \(-0.874545\pi\)
−0.923332 + 0.384002i \(0.874545\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −15448.1 −0.873879
\(51\) 0 0
\(52\) 45165.8 2.31634
\(53\) 7114.33 0.347892 0.173946 0.984755i \(-0.444348\pi\)
0.173946 + 0.984755i \(0.444348\pi\)
\(54\) 0 0
\(55\) 10640.5 0.474303
\(56\) 0 0
\(57\) 0 0
\(58\) −64458.0 −2.51598
\(59\) −20869.5 −0.780518 −0.390259 0.920705i \(-0.627614\pi\)
−0.390259 + 0.920705i \(0.627614\pi\)
\(60\) 0 0
\(61\) 23868.3 0.821291 0.410646 0.911795i \(-0.365303\pi\)
0.410646 + 0.911795i \(0.365303\pi\)
\(62\) 8167.30 0.269835
\(63\) 0 0
\(64\) −45679.0 −1.39401
\(65\) −41332.6 −1.21342
\(66\) 0 0
\(67\) 34671.5 0.943595 0.471798 0.881707i \(-0.343605\pi\)
0.471798 + 0.881707i \(0.343605\pi\)
\(68\) 40612.6 1.06510
\(69\) 0 0
\(70\) 0 0
\(71\) 28413.2 0.668921 0.334461 0.942410i \(-0.391446\pi\)
0.334461 + 0.942410i \(0.391446\pi\)
\(72\) 0 0
\(73\) 15292.7 0.335874 0.167937 0.985798i \(-0.446289\pi\)
0.167937 + 0.985798i \(0.446289\pi\)
\(74\) −14976.9 −0.317939
\(75\) 0 0
\(76\) −82946.0 −1.64726
\(77\) 0 0
\(78\) 0 0
\(79\) −73059.5 −1.31707 −0.658535 0.752550i \(-0.728824\pi\)
−0.658535 + 0.752550i \(0.728824\pi\)
\(80\) 26471.0 0.462430
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) −18320.6 −0.252193
\(89\) −36089.5 −0.482954 −0.241477 0.970407i \(-0.577632\pi\)
−0.241477 + 0.970407i \(0.577632\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 25215.6 0.310599
\(93\) 0 0
\(94\) −236208. −2.75725
\(95\) 75906.4 0.862918
\(96\) 0 0
\(97\) 153963. 1.66145 0.830724 0.556685i \(-0.187927\pi\)
0.830724 + 0.556685i \(0.187927\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −71950.4 −0.719504
\(101\) 139809. 1.36374 0.681869 0.731474i \(-0.261167\pi\)
0.681869 + 0.731474i \(0.261167\pi\)
\(102\) 0 0
\(103\) 115925. 1.07668 0.538339 0.842728i \(-0.319052\pi\)
0.538339 + 0.842728i \(0.319052\pi\)
\(104\) 71165.6 0.645188
\(105\) 0 0
\(106\) 60089.2 0.519436
\(107\) −83061.8 −0.701361 −0.350681 0.936495i \(-0.614050\pi\)
−0.350681 + 0.936495i \(0.614050\pi\)
\(108\) 0 0
\(109\) 45356.2 0.365654 0.182827 0.983145i \(-0.441475\pi\)
0.182827 + 0.983145i \(0.441475\pi\)
\(110\) 89872.1 0.708179
\(111\) 0 0
\(112\) 0 0
\(113\) 355.533 0.00261929 0.00130965 0.999999i \(-0.499583\pi\)
0.00130965 + 0.999999i \(0.499583\pi\)
\(114\) 0 0
\(115\) −23075.6 −0.162708
\(116\) −300216. −2.07152
\(117\) 0 0
\(118\) −176269. −1.16539
\(119\) 0 0
\(120\) 0 0
\(121\) −73689.5 −0.457554
\(122\) 201597. 1.22627
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) −349104. −1.81174
\(131\) −173969. −0.885715 −0.442858 0.896592i \(-0.646035\pi\)
−0.442858 + 0.896592i \(0.646035\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 292843. 1.40888
\(135\) 0 0
\(136\) 63991.3 0.296670
\(137\) −367723. −1.67386 −0.836931 0.547308i \(-0.815653\pi\)
−0.836931 + 0.547308i \(0.815653\pi\)
\(138\) 0 0
\(139\) −217967. −0.956870 −0.478435 0.878123i \(-0.658796\pi\)
−0.478435 + 0.878123i \(0.658796\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 239985. 0.998763
\(143\) −339352. −1.38775
\(144\) 0 0
\(145\) 274737. 1.08517
\(146\) 129165. 0.501492
\(147\) 0 0
\(148\) −69755.7 −0.261773
\(149\) 64906.1 0.239508 0.119754 0.992804i \(-0.461789\pi\)
0.119754 + 0.992804i \(0.461789\pi\)
\(150\) 0 0
\(151\) −223777. −0.798681 −0.399341 0.916803i \(-0.630761\pi\)
−0.399341 + 0.916803i \(0.630761\pi\)
\(152\) −130694. −0.458825
\(153\) 0 0
\(154\) 0 0
\(155\) −34811.1 −0.116383
\(156\) 0 0
\(157\) 459973. 1.48930 0.744652 0.667453i \(-0.232616\pi\)
0.744652 + 0.667453i \(0.232616\pi\)
\(158\) −617077. −1.96651
\(159\) 0 0
\(160\) 294986. 0.910964
\(161\) 0 0
\(162\) 0 0
\(163\) 91068.6 0.268472 0.134236 0.990949i \(-0.457142\pi\)
0.134236 + 0.990949i \(0.457142\pi\)
\(164\) −471135. −1.36784
\(165\) 0 0
\(166\) −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
\(171\) 0 0
\(172\) −779021. −2.00784
\(173\) −362143. −0.919951 −0.459975 0.887932i \(-0.652142\pi\)
−0.459975 + 0.887932i \(0.652142\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 217334. 0.528867
\(177\) 0 0
\(178\) −304820. −0.721096
\(179\) 173896. 0.405656 0.202828 0.979214i \(-0.434987\pi\)
0.202828 + 0.979214i \(0.434987\pi\)
\(180\) 0 0
\(181\) −134973. −0.306233 −0.153116 0.988208i \(-0.548931\pi\)
−0.153116 + 0.988208i \(0.548931\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 39731.0 0.0865138
\(185\) 63835.6 0.137130
\(186\) 0 0
\(187\) −305141. −0.638112
\(188\) −1.10015e6 −2.27017
\(189\) 0 0
\(190\) 641123. 1.28842
\(191\) −181413. −0.359821 −0.179910 0.983683i \(-0.557581\pi\)
−0.179910 + 0.983683i \(0.557581\pi\)
\(192\) 0 0
\(193\) 965999. 1.86674 0.933369 0.358919i \(-0.116855\pi\)
0.933369 + 0.358919i \(0.116855\pi\)
\(194\) 1.30040e6 2.48070
\(195\) 0 0
\(196\) 0 0
\(197\) 699058. 1.28336 0.641679 0.766974i \(-0.278238\pi\)
0.641679 + 0.766974i \(0.278238\pi\)
\(198\) 0 0
\(199\) 416191. 0.745006 0.372503 0.928031i \(-0.378500\pi\)
0.372503 + 0.928031i \(0.378500\pi\)
\(200\) −113369. −0.200410
\(201\) 0 0
\(202\) 1.18086e6 2.03619
\(203\) 0 0
\(204\) 0 0
\(205\) 431150. 0.716545
\(206\) 979132. 1.60758
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −701558. −1.04720
\(215\) 712906. 1.05181
\(216\) 0 0
\(217\) 0 0
\(218\) 383089. 0.545957
\(219\) 0 0
\(220\) 418584. 0.583076
\(221\) 1.18531e6 1.63249
\(222\) 0 0
\(223\) −882022. −1.18773 −0.593865 0.804565i \(-0.702398\pi\)
−0.593865 + 0.804565i \(0.702398\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 3002.91 0.00391085
\(227\) −1.12650e6 −1.45100 −0.725499 0.688223i \(-0.758391\pi\)
−0.725499 + 0.688223i \(0.758391\pi\)
\(228\) 0 0
\(229\) −310084. −0.390743 −0.195371 0.980729i \(-0.562591\pi\)
−0.195371 + 0.980729i \(0.562591\pi\)
\(230\) −194902. −0.242938
\(231\) 0 0
\(232\) −473036. −0.576998
\(233\) −1.13654e6 −1.37149 −0.685746 0.727841i \(-0.740524\pi\)
−0.685746 + 0.727841i \(0.740524\pi\)
\(234\) 0 0
\(235\) 1.00678e6 1.18923
\(236\) −820980. −0.959516
\(237\) 0 0
\(238\) 0 0
\(239\) −87506.8 −0.0990940 −0.0495470 0.998772i \(-0.515778\pi\)
−0.0495470 + 0.998772i \(0.515778\pi\)
\(240\) 0 0
\(241\) −537768. −0.596421 −0.298210 0.954500i \(-0.596390\pi\)
−0.298210 + 0.954500i \(0.596390\pi\)
\(242\) −622398. −0.683171
\(243\) 0 0
\(244\) 938948. 1.00964
\(245\) 0 0
\(246\) 0 0
\(247\) −2.42084e6 −2.52478
\(248\) 59937.1 0.0618823
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) 417731. 0.398380
\(257\) −976900. −0.922608 −0.461304 0.887242i \(-0.652618\pi\)
−0.461304 + 0.887242i \(0.652618\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.62597e6 −1.49169
\(261\) 0 0
\(262\) −1.46938e6 −1.32246
\(263\) 1.24375e6 1.10877 0.554387 0.832259i \(-0.312953\pi\)
0.554387 + 0.832259i \(0.312953\pi\)
\(264\) 0 0
\(265\) −256116. −0.224038
\(266\) 0 0
\(267\) 0 0
\(268\) 1.36393e6 1.15999
\(269\) −1.08408e6 −0.913445 −0.456722 0.889609i \(-0.650977\pi\)
−0.456722 + 0.889609i \(0.650977\pi\)
\(270\) 0 0
\(271\) 2.16627e6 1.79180 0.895900 0.444256i \(-0.146532\pi\)
0.895900 + 0.444256i \(0.146532\pi\)
\(272\) −759119. −0.622139
\(273\) 0 0
\(274\) −3.10587e6 −2.49924
\(275\) 540597. 0.431064
\(276\) 0 0
\(277\) 253859. 0.198789 0.0993946 0.995048i \(-0.468309\pi\)
0.0993946 + 0.995048i \(0.468309\pi\)
\(278\) −1.84099e6 −1.42870
\(279\) 0 0
\(280\) 0 0
\(281\) −1.14116e6 −0.862143 −0.431072 0.902318i \(-0.641864\pi\)
−0.431072 + 0.902318i \(0.641864\pi\)
\(282\) 0 0
\(283\) 609918. 0.452694 0.226347 0.974047i \(-0.427322\pi\)
0.226347 + 0.974047i \(0.427322\pi\)
\(284\) 1.11774e6 0.822327
\(285\) 0 0
\(286\) −2.86624e6 −2.07204
\(287\) 0 0
\(288\) 0 0
\(289\) −354040. −0.249349
\(290\) 2.32049e6 1.62026
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 548211. 0.357608
\(299\) 735937. 0.476061
\(300\) 0 0
\(301\) 0 0
\(302\) −1.89007e6 −1.19251
\(303\) 0 0
\(304\) 1.55040e6 0.962189
\(305\) −859259. −0.528901
\(306\) 0 0
\(307\) −293229. −0.177566 −0.0887831 0.996051i \(-0.528298\pi\)
−0.0887831 + 0.996051i \(0.528298\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −294023. −0.173771
\(311\) 2.45216e6 1.43763 0.718816 0.695200i \(-0.244684\pi\)
0.718816 + 0.695200i \(0.244684\pi\)
\(312\) 0 0
\(313\) −1.83541e6 −1.05894 −0.529471 0.848328i \(-0.677610\pi\)
−0.529471 + 0.848328i \(0.677610\pi\)
\(314\) 3.88503e6 2.22367
\(315\) 0 0
\(316\) −2.87406e6 −1.61912
\(317\) −589960. −0.329742 −0.164871 0.986315i \(-0.552721\pi\)
−0.164871 + 0.986315i \(0.552721\pi\)
\(318\) 0 0
\(319\) 2.25567e6 1.24107
\(320\) 1.64444e6 0.897726
\(321\) 0 0
\(322\) 0 0
\(323\) −2.17679e6 −1.16094
\(324\) 0 0
\(325\) −2.09993e6 −1.10280
\(326\) 769186. 0.400855
\(327\) 0 0
\(328\) −742344. −0.380996
\(329\) 0 0
\(330\) 0 0
\(331\) 177318. 0.0889577 0.0444789 0.999010i \(-0.485837\pi\)
0.0444789 + 0.999010i \(0.485837\pi\)
\(332\) −1.19357e6 −0.594296
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) −1.46205e6 −0.685908
\(341\) −285809. −0.133104
\(342\) 0 0
\(343\) 0 0
\(344\) −1.22747e6 −0.559259
\(345\) 0 0
\(346\) −3.05874e6 −1.37357
\(347\) 2.42361e6 1.08054 0.540268 0.841493i \(-0.318323\pi\)
0.540268 + 0.841493i \(0.318323\pi\)
\(348\) 0 0
\(349\) 2.67690e6 1.17644 0.588218 0.808702i \(-0.299830\pi\)
0.588218 + 0.808702i \(0.299830\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.42191e6 1.04184
\(353\) 950412. 0.405953 0.202976 0.979184i \(-0.434939\pi\)
0.202976 + 0.979184i \(0.434939\pi\)
\(354\) 0 0
\(355\) −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 \(-0.693453\pi\)
−0.571022 + 0.820935i \(0.693453\pi\)
\(360\) 0 0
\(361\) 1.96972e6 0.795495
\(362\) −1.14001e6 −0.457235
\(363\) 0 0
\(364\) 0 0
\(365\) −550536. −0.216299
\(366\) 0 0
\(367\) −153881. −0.0596377 −0.0298189 0.999555i \(-0.509493\pi\)
−0.0298189 + 0.999555i \(0.509493\pi\)
\(368\) −471323. −0.181426
\(369\) 0 0
\(370\) 539169. 0.204749
\(371\) 0 0
\(372\) 0 0
\(373\) −2.38381e6 −0.887156 −0.443578 0.896236i \(-0.646291\pi\)
−0.443578 + 0.896236i \(0.646291\pi\)
\(374\) −2.57729e6 −0.952763
\(375\) 0 0
\(376\) −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
\(381\) 0 0
\(382\) −1.53226e6 −0.537246
\(383\) 2.15730e6 0.751472 0.375736 0.926727i \(-0.377390\pi\)
0.375736 + 0.926727i \(0.377390\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.15904e6 2.78722
\(387\) 0 0
\(388\) 6.05669e6 2.04247
\(389\) 3.66471e6 1.22791 0.613954 0.789342i \(-0.289578\pi\)
0.613954 + 0.789342i \(0.289578\pi\)
\(390\) 0 0
\(391\) 661746. 0.218902
\(392\) 0 0
\(393\) 0 0
\(394\) 5.90440e6 1.91617
\(395\) 2.63014e6 0.848177
\(396\) 0 0
\(397\) 3.94648e6 1.25671 0.628353 0.777928i \(-0.283729\pi\)
0.628353 + 0.777928i \(0.283729\pi\)
\(398\) 3.51524e6 1.11236
\(399\) 0 0
\(400\) 1.34488e6 0.420274
\(401\) −25016.1 −0.00776887 −0.00388444 0.999992i \(-0.501236\pi\)
−0.00388444 + 0.999992i \(0.501236\pi\)
\(402\) 0 0
\(403\) 1.11021e6 0.340521
\(404\) 5.49989e6 1.67649
\(405\) 0 0
\(406\) 0 0
\(407\) 524107. 0.156832
\(408\) 0 0
\(409\) −832700. −0.246139 −0.123069 0.992398i \(-0.539274\pi\)
−0.123069 + 0.992398i \(0.539274\pi\)
\(410\) 3.64159e6 1.06987
\(411\) 0 0
\(412\) 4.56035e6 1.32360
\(413\) 0 0
\(414\) 0 0
\(415\) 1.09227e6 0.311323
\(416\) −9.40782e6 −2.66536
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 440974. 0.119124
\(425\) −1.88823e6 −0.507087
\(426\) 0 0
\(427\) 0 0
\(428\) −3.26754e6 −0.862207
\(429\) 0 0
\(430\) 6.02136e6 1.57045
\(431\) −4.07810e6 −1.05746 −0.528731 0.848790i \(-0.677332\pi\)
−0.528731 + 0.848790i \(0.677332\pi\)
\(432\) 0 0
\(433\) 1.79927e6 0.461186 0.230593 0.973050i \(-0.425933\pi\)
0.230593 + 0.973050i \(0.425933\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.78425e6 0.449511
\(437\) −1.35153e6 −0.338550
\(438\) 0 0
\(439\) −4.51827e6 −1.11895 −0.559475 0.828847i \(-0.688997\pi\)
−0.559475 + 0.828847i \(0.688997\pi\)
\(440\) 659542. 0.162409
\(441\) 0 0
\(442\) 1.00114e7 2.43746
\(443\) 2.85256e6 0.690597 0.345299 0.938493i \(-0.387778\pi\)
0.345299 + 0.938493i \(0.387778\pi\)
\(444\) 0 0
\(445\) 1.29922e6 0.311016
\(446\) −7.44976e6 −1.77339
\(447\) 0 0
\(448\) 0 0
\(449\) 1.90246e6 0.445348 0.222674 0.974893i \(-0.428522\pi\)
0.222674 + 0.974893i \(0.428522\pi\)
\(450\) 0 0
\(451\) 3.53986e6 0.819491
\(452\) 13986.2 0.00321998
\(453\) 0 0
\(454\) −9.51468e6 −2.16648
\(455\) 0 0
\(456\) 0 0
\(457\) 2.64834e6 0.593176 0.296588 0.955006i \(-0.404151\pi\)
0.296588 + 0.955006i \(0.404151\pi\)
\(458\) −2.61904e6 −0.583416
\(459\) 0 0
\(460\) −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
\(465\) 0 0
\(466\) −9.59943e6 −2.04777
\(467\) −3.20935e6 −0.680966 −0.340483 0.940251i \(-0.610591\pi\)
−0.340483 + 0.940251i \(0.610591\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.50350e6 1.77563
\(471\) 0 0
\(472\) −1.29358e6 −0.267262
\(473\) 5.85315e6 1.20292
\(474\) 0 0
\(475\) 3.85647e6 0.784252
\(476\) 0 0
\(477\) 0 0
\(478\) −739102. −0.147957
\(479\) 2.31462e6 0.460936 0.230468 0.973080i \(-0.425974\pi\)
0.230468 + 0.973080i \(0.425974\pi\)
\(480\) 0 0
\(481\) −2.03587e6 −0.401225
\(482\) −4.54211e6 −0.890513
\(483\) 0 0
\(484\) −2.89885e6 −0.562486
\(485\) −5.54266e6 −1.06995
\(486\) 0 0
\(487\) −4.63735e6 −0.886028 −0.443014 0.896515i \(-0.646091\pi\)
−0.443014 + 0.896515i \(0.646091\pi\)
\(488\) 1.47945e6 0.281224
\(489\) 0 0
\(490\) 0 0
\(491\) −5.02151e6 −0.940007 −0.470003 0.882665i \(-0.655747\pi\)
−0.470003 + 0.882665i \(0.655747\pi\)
\(492\) 0 0
\(493\) −7.87872e6 −1.45995
\(494\) −2.04470e7 −3.76974
\(495\) 0 0
\(496\) −711024. −0.129772
\(497\) 0 0
\(498\) 0 0
\(499\) 3.37822e6 0.607347 0.303673 0.952776i \(-0.401787\pi\)
0.303673 + 0.952776i \(0.401787\pi\)
\(500\) 7.01582e6 1.25503
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 6.64694e6 1.14278
\(509\) 6.72466e6 1.15047 0.575236 0.817988i \(-0.304910\pi\)
0.575236 + 0.817988i \(0.304910\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 7.48361e6 1.26164
\(513\) 0 0
\(514\) −8.25112e6 −1.37754
\(515\) −4.17332e6 −0.693367
\(516\) 0 0
\(517\) 8.26595e6 1.36009
\(518\) 0 0
\(519\) 0 0
\(520\) −2.56196e6 −0.415493
\(521\) 4.42770e6 0.714635 0.357317 0.933983i \(-0.383691\pi\)
0.357317 + 0.933983i \(0.383691\pi\)
\(522\) 0 0
\(523\) −8.95911e6 −1.43222 −0.716111 0.697986i \(-0.754080\pi\)
−0.716111 + 0.697986i \(0.754080\pi\)
\(524\) −6.84372e6 −1.08884
\(525\) 0 0
\(526\) 1.05050e7 1.65551
\(527\) 998291. 0.156578
\(528\) 0 0
\(529\) −6.02548e6 −0.936165
\(530\) −2.16321e6 −0.334510
\(531\) 0 0
\(532\) 0 0
\(533\) −1.37504e7 −2.09652
\(534\) 0 0
\(535\) 2.99022e6 0.451668
\(536\) 2.14908e6 0.323103
\(537\) 0 0
\(538\) −9.15641e6 −1.36386
\(539\) 0 0
\(540\) 0 0
\(541\) −1.00467e7 −1.47581 −0.737907 0.674902i \(-0.764186\pi\)
−0.737907 + 0.674902i \(0.764186\pi\)
\(542\) 1.82968e7 2.67533
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 4.56600e6 0.643620
\(551\) 1.60913e7 2.25794
\(552\) 0 0
\(553\) 0 0
\(554\) 2.14415e6 0.296811
\(555\) 0 0
\(556\) −8.57452e6 −1.17631
\(557\) −9.06752e6 −1.23837 −0.619185 0.785245i \(-0.712537\pi\)
−0.619185 + 0.785245i \(0.712537\pi\)
\(558\) 0 0
\(559\) −2.27363e7 −3.07744
\(560\) 0 0
\(561\) 0 0
\(562\) −9.63846e6 −1.28726
\(563\) −1.05180e7 −1.39849 −0.699247 0.714880i \(-0.746481\pi\)
−0.699247 + 0.714880i \(0.746481\pi\)
\(564\) 0 0
\(565\) −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 \(-0.657231\pi\)
−0.474114 + 0.880464i \(0.657231\pi\)
\(570\) 0 0
\(571\) −6.97981e6 −0.895887 −0.447943 0.894062i \(-0.647843\pi\)
−0.447943 + 0.894062i \(0.647843\pi\)
\(572\) −1.33497e7 −1.70600
\(573\) 0 0
\(574\) 0 0
\(575\) −1.17237e6 −0.147875
\(576\) 0 0
\(577\) −5.81210e6 −0.726765 −0.363382 0.931640i \(-0.618378\pi\)
−0.363382 + 0.931640i \(0.618378\pi\)
\(578\) −2.99030e6 −0.372302
\(579\) 0 0
\(580\) 1.08078e7 1.33403
\(581\) 0 0
\(582\) 0 0
\(583\) −2.10278e6 −0.256226
\(584\) 947901. 0.115009
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 1.30385e6 0.152906
\(593\) −9.46528e6 −1.10534 −0.552671 0.833399i \(-0.686391\pi\)
−0.552671 + 0.833399i \(0.686391\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.55332e6 0.294435
\(597\) 0 0
\(598\) 6.21589e6 0.710804
\(599\) 8.52195e6 0.970448 0.485224 0.874390i \(-0.338738\pi\)
0.485224 + 0.874390i \(0.338738\pi\)
\(600\) 0 0
\(601\) −657065. −0.0742031 −0.0371016 0.999311i \(-0.511813\pi\)
−0.0371016 + 0.999311i \(0.511813\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.80310e6 −0.981846
\(605\) 2.65282e6 0.294659
\(606\) 0 0
\(607\) 5.86885e6 0.646519 0.323260 0.946310i \(-0.395221\pi\)
0.323260 + 0.946310i \(0.395221\pi\)
\(608\) 1.72773e7 1.89547
\(609\) 0 0
\(610\) −7.25750e6 −0.789700
\(611\) −3.21087e7 −3.47952
\(612\) 0 0
\(613\) 3.84402e6 0.413175 0.206588 0.978428i \(-0.433764\pi\)
0.206588 + 0.978428i \(0.433764\pi\)
\(614\) −2.47667e6 −0.265123
\(615\) 0 0
\(616\) 0 0
\(617\) −6.44660e6 −0.681739 −0.340869 0.940111i \(-0.610721\pi\)
−0.340869 + 0.940111i \(0.610721\pi\)
\(618\) 0 0
\(619\) 6.73740e6 0.706749 0.353375 0.935482i \(-0.385034\pi\)
0.353375 + 0.935482i \(0.385034\pi\)
\(620\) −1.36942e6 −0.143073
\(621\) 0 0
\(622\) 2.07115e7 2.14652
\(623\) 0 0
\(624\) 0 0
\(625\) −704759. −0.0721673
\(626\) −1.55023e7 −1.58110
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) −4.98293e6 −0.492336
\(635\) −6.08282e6 −0.598647
\(636\) 0 0
\(637\) 0 0
\(638\) 1.90518e7 1.85304
\(639\) 0 0
\(640\) 4.44978e6 0.429426
\(641\) 1.04088e6 0.100059 0.0500296 0.998748i \(-0.484068\pi\)
0.0500296 + 0.998748i \(0.484068\pi\)
\(642\) 0 0
\(643\) −9.08713e6 −0.866761 −0.433381 0.901211i \(-0.642679\pi\)
−0.433381 + 0.901211i \(0.642679\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.83857e7 −1.73340
\(647\) −2.20711e6 −0.207283 −0.103641 0.994615i \(-0.533049\pi\)
−0.103641 + 0.994615i \(0.533049\pi\)
\(648\) 0 0
\(649\) 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 \(-0.181069\pi\)
0.842524 + 0.538658i \(0.181069\pi\)
\(654\) 0 0
\(655\) 6.26289e6 0.570390
\(656\) 8.80632e6 0.798978
\(657\) 0 0
\(658\) 0 0
\(659\) −6.21208e6 −0.557216 −0.278608 0.960405i \(-0.589873\pi\)
−0.278608 + 0.960405i \(0.589873\pi\)
\(660\) 0 0
\(661\) 1.54230e7 1.37298 0.686491 0.727138i \(-0.259150\pi\)
0.686491 + 0.727138i \(0.259150\pi\)
\(662\) 1.49767e6 0.132822
\(663\) 0 0
\(664\) −1.88065e6 −0.165534
\(665\) 0 0
\(666\) 0 0
\(667\) −4.89176e6 −0.425746
\(668\) −1.23827e7 −1.07368
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 3.72500e7 3.13516
\(677\) −1.36173e7 −1.14188 −0.570940 0.820992i \(-0.693421\pi\)
−0.570940 + 0.820992i \(0.693421\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.30369e6 −0.191052
\(681\) 0 0
\(682\) −2.41401e6 −0.198736
\(683\) −2.39985e6 −0.196849 −0.0984245 0.995145i \(-0.531380\pi\)
−0.0984245 + 0.995145i \(0.531380\pi\)
\(684\) 0 0
\(685\) 1.32380e7 1.07795
\(686\) 0 0
\(687\) 0 0
\(688\) 1.45612e7 1.17281
\(689\) 8.16816e6 0.655505
\(690\) 0 0
\(691\) −1.40365e6 −0.111831 −0.0559156 0.998435i \(-0.517808\pi\)
−0.0559156 + 0.998435i \(0.517808\pi\)
\(692\) −1.42462e7 −1.13093
\(693\) 0 0
\(694\) 2.04703e7 1.61334
\(695\) 7.84680e6 0.616212
\(696\) 0 0
\(697\) −1.23642e7 −0.964018
\(698\) 2.26097e7 1.75653
\(699\) 0 0
\(700\) 0 0
\(701\) 5.78991e6 0.445017 0.222509 0.974931i \(-0.428575\pi\)
0.222509 + 0.974931i \(0.428575\pi\)
\(702\) 0 0
\(703\) 3.73884e6 0.285330
\(704\) 1.35013e7 1.02670
\(705\) 0 0
\(706\) 8.02739e6 0.606126
\(707\) 0 0
\(708\) 0 0
\(709\) −1.13143e7 −0.845304 −0.422652 0.906292i \(-0.638901\pi\)
−0.422652 + 0.906292i \(0.638901\pi\)
\(710\) −8.63944e6 −0.643191
\(711\) 0 0
\(712\) −2.23697e6 −0.165371
\(713\) 619821. 0.0456607
\(714\) 0 0
\(715\) 1.22167e7 0.893692
\(716\) 6.84085e6 0.498686
\(717\) 0 0
\(718\) −2.35549e7 −1.70518
\(719\) 2.73780e7 1.97506 0.987529 0.157437i \(-0.0503231\pi\)
0.987529 + 0.157437i \(0.0503231\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.66367e7 1.18775
\(723\) 0 0
\(724\) −5.30967e6 −0.376462
\(725\) 1.39582e7 0.986241
\(726\) 0 0
\(727\) −9.86471e6 −0.692226 −0.346113 0.938193i \(-0.612499\pi\)
−0.346113 + 0.938193i \(0.612499\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.64995e6 −0.322954
\(731\) −2.04442e7 −1.41507
\(732\) 0 0
\(733\) 3.87876e6 0.266645 0.133322 0.991073i \(-0.457435\pi\)
0.133322 + 0.991073i \(0.457435\pi\)
\(734\) −1.29972e6 −0.0890448
\(735\) 0 0
\(736\) −5.25229e6 −0.357400
\(737\) −1.02479e7 −0.694967
\(738\) 0 0
\(739\) −7.95498e6 −0.535831 −0.267916 0.963442i \(-0.586335\pi\)
−0.267916 + 0.963442i \(0.586335\pi\)
\(740\) 2.51121e6 0.168579
\(741\) 0 0
\(742\) 0 0
\(743\) −1.65977e7 −1.10300 −0.551500 0.834175i \(-0.685944\pi\)
−0.551500 + 0.834175i \(0.685944\pi\)
\(744\) 0 0
\(745\) −2.33662e6 −0.154240
\(746\) −2.01342e7 −1.32461
\(747\) 0 0
\(748\) −1.20039e7 −0.784453
\(749\) 0 0
\(750\) 0 0
\(751\) 1.51072e7 0.977426 0.488713 0.872445i \(-0.337467\pi\)
0.488713 + 0.872445i \(0.337467\pi\)
\(752\) 2.05637e7 1.32604
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 4.70498e6 0.295477
\(761\) −2.54270e7 −1.59160 −0.795799 0.605561i \(-0.792949\pi\)
−0.795799 + 0.605561i \(0.792949\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −7.13656e6 −0.442340
\(765\) 0 0
\(766\) 1.82210e7 1.12202
\(767\) −2.39609e7 −1.47067
\(768\) 0 0
\(769\) 1.53909e7 0.938532 0.469266 0.883057i \(-0.344519\pi\)
0.469266 + 0.883057i \(0.344519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.80011e7 2.29484
\(773\) −905393. −0.0544990 −0.0272495 0.999629i \(-0.508675\pi\)
−0.0272495 + 0.999629i \(0.508675\pi\)
\(774\) 0 0
\(775\) −1.76860e6 −0.105773
\(776\) 9.54323e6 0.568907
\(777\) 0 0
\(778\) 3.09530e7 1.83338
\(779\) 2.52523e7 1.49093
\(780\) 0 0
\(781\) −8.39810e6 −0.492667
\(782\) 5.58926e6 0.326841
\(783\) 0 0
\(784\) 0 0
\(785\) −1.65590e7 −0.959093
\(786\) 0 0
\(787\) −2.79334e7 −1.60763 −0.803817 0.594877i \(-0.797201\pi\)
−0.803817 + 0.594877i \(0.797201\pi\)
\(788\) 2.75000e7 1.57767
\(789\) 0 0
\(790\) 2.22148e7 1.26641
\(791\) 0 0
\(792\) 0 0
\(793\) 2.74039e7 1.54749
\(794\) 3.33328e7 1.87638
\(795\) 0 0
\(796\) 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
\(801\) 0 0
\(802\) −211291. −0.0115997
\(803\) −4.52005e6 −0.247374
\(804\) 0 0
\(805\) 0 0
\(806\) 9.37710e6 0.508430
\(807\) 0 0
\(808\) 8.66590e6 0.466966
\(809\) −2.44194e7 −1.31179 −0.655893 0.754854i \(-0.727708\pi\)
−0.655893 + 0.754854i \(0.727708\pi\)
\(810\) 0 0
\(811\) 4.46711e6 0.238492 0.119246 0.992865i \(-0.461952\pi\)
0.119246 + 0.992865i \(0.461952\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.42673e6 0.234165
\(815\) −3.27847e6 −0.172893
\(816\) 0 0
\(817\) 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 \(-0.613391\pi\)
−0.348743 + 0.937219i \(0.613391\pi\)
\(822\) 0 0
\(823\) −4.03958e6 −0.207891 −0.103946 0.994583i \(-0.533147\pi\)
−0.103946 + 0.994583i \(0.533147\pi\)
\(824\) 7.18552e6 0.368672
\(825\) 0 0
\(826\) 0 0
\(827\) −1.24927e7 −0.635175 −0.317588 0.948229i \(-0.602873\pi\)
−0.317588 + 0.948229i \(0.602873\pi\)
\(828\) 0 0
\(829\) −1.45980e7 −0.737749 −0.368874 0.929479i \(-0.620257\pi\)
−0.368874 + 0.929479i \(0.620257\pi\)
\(830\) 9.22557e6 0.464834
\(831\) 0 0
\(832\) −5.24453e7 −2.62663
\(833\) 0 0
\(834\) 0 0
\(835\) 1.13318e7 0.562449
\(836\) 2.45163e7 1.21322
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) −1.60168e7 −0.773964
\(845\) −3.40886e7 −1.64236
\(846\) 0 0
\(847\) 0 0
\(848\) −5.23121e6 −0.249812
\(849\) 0 0
\(850\) −1.59484e7 −0.757129
\(851\) −1.13661e6 −0.0538005
\(852\) 0 0
\(853\) −8.68253e6 −0.408577 −0.204289 0.978911i \(-0.565488\pi\)
−0.204289 + 0.978911i \(0.565488\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.14850e6 −0.240158
\(857\) −1.04988e7 −0.488302 −0.244151 0.969737i \(-0.578509\pi\)
−0.244151 + 0.969737i \(0.578509\pi\)
\(858\) 0 0
\(859\) 9.03780e6 0.417907 0.208954 0.977926i \(-0.432994\pi\)
0.208954 + 0.977926i \(0.432994\pi\)
\(860\) 2.80448e7 1.29302
\(861\) 0 0
\(862\) −3.44445e7 −1.57889
\(863\) 1.59858e7 0.730645 0.365322 0.930881i \(-0.380959\pi\)
0.365322 + 0.930881i \(0.380959\pi\)
\(864\) 0 0
\(865\) 1.30371e7 0.592437
\(866\) 1.51970e7 0.688594
\(867\) 0 0
\(868\) 0 0
\(869\) 2.15942e7 0.970035
\(870\) 0 0
\(871\) 3.98073e7 1.77794
\(872\) 2.81136e6 0.125206
\(873\) 0 0
\(874\) −1.14153e7 −0.505487
\(875\) 0 0
\(876\) 0 0
\(877\) 2.67453e7 1.17422 0.587109 0.809508i \(-0.300266\pi\)
0.587109 + 0.809508i \(0.300266\pi\)
\(878\) −3.81623e7 −1.67070
\(879\) 0 0
\(880\) −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
\(885\) 0 0
\(886\) 2.40933e7 1.03113
\(887\) −8.71625e6 −0.371981 −0.185990 0.982552i \(-0.559549\pi\)
−0.185990 + 0.982552i \(0.559549\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.09735e7 0.464377
\(891\) 0 0
\(892\) −3.46976e7 −1.46011
\(893\) 5.89669e7 2.47446
\(894\) 0 0
\(895\) −6.26026e6 −0.261237
\(896\) 0 0
\(897\) 0 0
\(898\) 1.60686e7 0.664946
\(899\) −7.37956e6 −0.304531
\(900\) 0 0
\(901\) 7.34472e6 0.301414
\(902\) 2.98984e7 1.22358
\(903\) 0 0
\(904\) 22037.4 0.000896888 0
\(905\) 4.85904e6 0.197210
\(906\) 0 0
\(907\) −2.36255e7 −0.953594 −0.476797 0.879014i \(-0.658202\pi\)
−0.476797 + 0.879014i \(0.658202\pi\)
\(908\) −4.43151e7 −1.78376
\(909\) 0 0
\(910\) 0 0
\(911\) 1.69360e7 0.676105 0.338053 0.941127i \(-0.390232\pi\)
0.338053 + 0.941127i \(0.390232\pi\)
\(912\) 0 0
\(913\) 8.96785e6 0.356050
\(914\) 2.23685e7 0.885668
\(915\) 0 0
\(916\) −1.21983e7 −0.480353
\(917\) 0 0
\(918\) 0 0
\(919\) −3.13804e7 −1.22566 −0.612829 0.790216i \(-0.709968\pi\)
−0.612829 + 0.790216i \(0.709968\pi\)
\(920\) −1.43032e6 −0.0557138
\(921\) 0 0
\(922\) 9.20896e6 0.356766
\(923\) 3.26220e7 1.26040
\(924\) 0 0
\(925\) 3.24320e6 0.124629
\(926\) −2.11875e7 −0.811992
\(927\) 0 0
\(928\) 6.25336e7 2.38365
\(929\) 1.43089e7 0.543959 0.271980 0.962303i \(-0.412322\pi\)
0.271980 + 0.962303i \(0.412322\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −4.47098e7 −1.68602
\(933\) 0 0
\(934\) −2.71069e7 −1.01675
\(935\) 1.09851e7 0.410937
\(936\) 0 0
\(937\) 2.81206e7 1.04635 0.523173 0.852227i \(-0.324748\pi\)
0.523173 + 0.852227i \(0.324748\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3.96054e7 1.46196
\(941\) 3.29569e7 1.21331 0.606656 0.794964i \(-0.292510\pi\)
0.606656 + 0.794964i \(0.292510\pi\)
\(942\) 0 0
\(943\) −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 \(-0.595409\pi\)
−0.295268 + 0.955415i \(0.595409\pi\)
\(948\) 0 0
\(949\) 1.75579e7 0.632861
\(950\) 3.25726e7 1.17096
\(951\) 0 0
\(952\) 0 0
\(953\) 3.03230e7 1.08153 0.540767 0.841172i \(-0.318134\pi\)
0.540767 + 0.841172i \(0.318134\pi\)
\(954\) 0 0
\(955\) 6.53088e6 0.231720
\(956\) −3.44240e6 −0.121820
\(957\) 0 0
\(958\) 1.95498e7 0.688221
\(959\) 0 0
\(960\) 0 0
\(961\) −2.76941e7 −0.967339
\(962\) −1.71954e7 −0.599067
\(963\) 0 0
\(964\) −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
\(969\) 0 0
\(970\) −4.68145e7 −1.59754
\(971\) −1.65608e7 −0.563679 −0.281840 0.959462i \(-0.590945\pi\)
−0.281840 + 0.959462i \(0.590945\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −3.91681e7 −1.32292
\(975\) 0 0
\(976\) −1.75505e7 −0.589747
\(977\) −7.56862e6 −0.253676 −0.126838 0.991923i \(-0.540483\pi\)
−0.126838 + 0.991923i \(0.540483\pi\)
\(978\) 0 0
\(979\) 1.06670e7 0.355700
\(980\) 0 0
\(981\) 0 0
\(982\) −4.24128e7 −1.40352
\(983\) 3.32042e7 1.09600 0.547998 0.836480i \(-0.315390\pi\)
0.547998 + 0.836480i \(0.315390\pi\)
\(984\) 0 0
\(985\) −2.51661e7 −0.826466
\(986\) −6.65454e7 −2.17985
\(987\) 0 0
\(988\) −9.52327e7 −3.10380
\(989\) −1.26935e7 −0.412657
\(990\) 0 0
\(991\) 3.48003e7 1.12564 0.562819 0.826580i \(-0.309717\pi\)
0.562819 + 0.826580i \(0.309717\pi\)
\(992\) −7.92345e6 −0.255644
\(993\) 0 0
\(994\) 0 0
\(995\) −1.49829e7 −0.479774
\(996\) 0 0
\(997\) 9.40852e6 0.299767 0.149883 0.988704i \(-0.452110\pi\)
0.149883 + 0.988704i \(0.452110\pi\)
\(998\) 2.85332e7 0.906826
\(999\) 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 441.6.a.q.1.2 2
3.2 odd 2 147.6.a.h.1.1 2
7.6 odd 2 441.6.a.r.1.2 2
21.2 odd 6 147.6.e.n.67.2 4
21.5 even 6 147.6.e.m.67.2 4
21.11 odd 6 147.6.e.n.79.2 4
21.17 even 6 147.6.e.m.79.2 4
21.20 even 2 147.6.a.j.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.h.1.1 2 3.2 odd 2
147.6.a.j.1.1 yes 2 21.20 even 2
147.6.e.m.67.2 4 21.5 even 6
147.6.e.m.79.2 4 21.17 even 6
147.6.e.n.67.2 4 21.2 odd 6
147.6.e.n.79.2 4 21.11 odd 6
441.6.a.q.1.2 2 1.1 even 1 trivial
441.6.a.r.1.2 2 7.6 odd 2