Properties

Label 448.6.a.v.1.2
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
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] = [448,6,Mod(1,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("448.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{345}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.78709\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+21.5742 q^{3} +14.7225 q^{5} -49.0000 q^{7} +222.445 q^{9} +O(q^{10})\) \(q+21.5742 q^{3} +14.7225 q^{5} -49.0000 q^{7} +222.445 q^{9} -58.5549 q^{11} -1179.39 q^{13} +317.626 q^{15} +1496.55 q^{17} -498.838 q^{19} -1057.13 q^{21} -1889.34 q^{23} -2908.25 q^{25} -443.456 q^{27} -1914.54 q^{29} +794.577 q^{31} -1263.27 q^{33} -721.404 q^{35} -2987.93 q^{37} -25444.4 q^{39} +11941.3 q^{41} -9820.19 q^{43} +3274.95 q^{45} -19636.0 q^{47} +2401.00 q^{49} +32286.9 q^{51} +19875.0 q^{53} -862.077 q^{55} -10762.0 q^{57} -35838.6 q^{59} -49975.9 q^{61} -10899.8 q^{63} -17363.6 q^{65} -48176.2 q^{67} -40761.0 q^{69} +77179.1 q^{71} -59667.3 q^{73} -62743.0 q^{75} +2869.19 q^{77} +60743.1 q^{79} -63621.3 q^{81} +46134.2 q^{83} +22033.1 q^{85} -41304.7 q^{87} +78668.7 q^{89} +57790.2 q^{91} +17142.3 q^{93} -7344.15 q^{95} -43573.3 q^{97} -13025.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 82 q^{5} - 98 q^{7} + 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 82 q^{5} - 98 q^{7} + 222 q^{9} - 340 q^{11} - 910 q^{13} + 1824 q^{15} + 3216 q^{17} + 674 q^{19} - 294 q^{21} - 1104 q^{23} + 3322 q^{25} + 3348 q^{27} - 8064 q^{29} - 6212 q^{31} + 3120 q^{33} + 4018 q^{35} + 8512 q^{37} - 29640 q^{39} - 1304 q^{41} + 10004 q^{43} + 3318 q^{45} - 12748 q^{47} + 4802 q^{49} + 5508 q^{51} + 11220 q^{53} + 26360 q^{55} - 29028 q^{57} + 12018 q^{59} - 102738 q^{61} - 10878 q^{63} - 43420 q^{65} - 24136 q^{67} - 52992 q^{69} + 89720 q^{71} - 55588 q^{73} - 159774 q^{75} + 16660 q^{77} + 48824 q^{79} - 122562 q^{81} - 35782 q^{83} - 144276 q^{85} + 54468 q^{87} - 18300 q^{89} + 44590 q^{91} + 126264 q^{93} - 120784 q^{95} - 69984 q^{97} - 12900 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 21.5742 1.38398 0.691992 0.721905i \(-0.256733\pi\)
0.691992 + 0.721905i \(0.256733\pi\)
\(4\) 0 0
\(5\) 14.7225 0.263365 0.131682 0.991292i \(-0.457962\pi\)
0.131682 + 0.991292i \(0.457962\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 222.445 0.915412
\(10\) 0 0
\(11\) −58.5549 −0.145909 −0.0729545 0.997335i \(-0.523243\pi\)
−0.0729545 + 0.997335i \(0.523243\pi\)
\(12\) 0 0
\(13\) −1179.39 −1.93553 −0.967765 0.251853i \(-0.918960\pi\)
−0.967765 + 0.251853i \(0.918960\pi\)
\(14\) 0 0
\(15\) 317.626 0.364492
\(16\) 0 0
\(17\) 1496.55 1.25594 0.627972 0.778236i \(-0.283885\pi\)
0.627972 + 0.778236i \(0.283885\pi\)
\(18\) 0 0
\(19\) −498.838 −0.317012 −0.158506 0.987358i \(-0.550668\pi\)
−0.158506 + 0.987358i \(0.550668\pi\)
\(20\) 0 0
\(21\) −1057.13 −0.523097
\(22\) 0 0
\(23\) −1889.34 −0.744716 −0.372358 0.928089i \(-0.621451\pi\)
−0.372358 + 0.928089i \(0.621451\pi\)
\(24\) 0 0
\(25\) −2908.25 −0.930639
\(26\) 0 0
\(27\) −443.456 −0.117069
\(28\) 0 0
\(29\) −1914.54 −0.422737 −0.211369 0.977406i \(-0.567792\pi\)
−0.211369 + 0.977406i \(0.567792\pi\)
\(30\) 0 0
\(31\) 794.577 0.148502 0.0742509 0.997240i \(-0.476343\pi\)
0.0742509 + 0.997240i \(0.476343\pi\)
\(32\) 0 0
\(33\) −1263.27 −0.201936
\(34\) 0 0
\(35\) −721.404 −0.0995424
\(36\) 0 0
\(37\) −2987.93 −0.358811 −0.179406 0.983775i \(-0.557417\pi\)
−0.179406 + 0.983775i \(0.557417\pi\)
\(38\) 0 0
\(39\) −25444.4 −2.67874
\(40\) 0 0
\(41\) 11941.3 1.10941 0.554704 0.832047i \(-0.312831\pi\)
0.554704 + 0.832047i \(0.312831\pi\)
\(42\) 0 0
\(43\) −9820.19 −0.809933 −0.404966 0.914332i \(-0.632717\pi\)
−0.404966 + 0.914332i \(0.632717\pi\)
\(44\) 0 0
\(45\) 3274.95 0.241087
\(46\) 0 0
\(47\) −19636.0 −1.29660 −0.648302 0.761383i \(-0.724521\pi\)
−0.648302 + 0.761383i \(0.724521\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 32286.9 1.73821
\(52\) 0 0
\(53\) 19875.0 0.971889 0.485945 0.873990i \(-0.338476\pi\)
0.485945 + 0.873990i \(0.338476\pi\)
\(54\) 0 0
\(55\) −862.077 −0.0384272
\(56\) 0 0
\(57\) −10762.0 −0.438739
\(58\) 0 0
\(59\) −35838.6 −1.34036 −0.670180 0.742199i \(-0.733783\pi\)
−0.670180 + 0.742199i \(0.733783\pi\)
\(60\) 0 0
\(61\) −49975.9 −1.71964 −0.859818 0.510601i \(-0.829423\pi\)
−0.859818 + 0.510601i \(0.829423\pi\)
\(62\) 0 0
\(63\) −10899.8 −0.345993
\(64\) 0 0
\(65\) −17363.6 −0.509750
\(66\) 0 0
\(67\) −48176.2 −1.31113 −0.655565 0.755139i \(-0.727569\pi\)
−0.655565 + 0.755139i \(0.727569\pi\)
\(68\) 0 0
\(69\) −40761.0 −1.03068
\(70\) 0 0
\(71\) 77179.1 1.81699 0.908497 0.417891i \(-0.137230\pi\)
0.908497 + 0.417891i \(0.137230\pi\)
\(72\) 0 0
\(73\) −59667.3 −1.31048 −0.655238 0.755422i \(-0.727432\pi\)
−0.655238 + 0.755422i \(0.727432\pi\)
\(74\) 0 0
\(75\) −62743.0 −1.28799
\(76\) 0 0
\(77\) 2869.19 0.0551484
\(78\) 0 0
\(79\) 60743.1 1.09504 0.547519 0.836793i \(-0.315572\pi\)
0.547519 + 0.836793i \(0.315572\pi\)
\(80\) 0 0
\(81\) −63621.3 −1.07743
\(82\) 0 0
\(83\) 46134.2 0.735068 0.367534 0.930010i \(-0.380202\pi\)
0.367534 + 0.930010i \(0.380202\pi\)
\(84\) 0 0
\(85\) 22033.1 0.330771
\(86\) 0 0
\(87\) −41304.7 −0.585061
\(88\) 0 0
\(89\) 78668.7 1.05275 0.526377 0.850251i \(-0.323550\pi\)
0.526377 + 0.850251i \(0.323550\pi\)
\(90\) 0 0
\(91\) 57790.2 0.731562
\(92\) 0 0
\(93\) 17142.3 0.205524
\(94\) 0 0
\(95\) −7344.15 −0.0834897
\(96\) 0 0
\(97\) −43573.3 −0.470209 −0.235104 0.971970i \(-0.575543\pi\)
−0.235104 + 0.971970i \(0.575543\pi\)
\(98\) 0 0
\(99\) −13025.3 −0.133567
\(100\) 0 0
\(101\) 164188. 1.60154 0.800772 0.598970i \(-0.204423\pi\)
0.800772 + 0.598970i \(0.204423\pi\)
\(102\) 0 0
\(103\) 164547. 1.52825 0.764127 0.645065i \(-0.223170\pi\)
0.764127 + 0.645065i \(0.223170\pi\)
\(104\) 0 0
\(105\) −15563.7 −0.137765
\(106\) 0 0
\(107\) 185426. 1.56571 0.782854 0.622206i \(-0.213763\pi\)
0.782854 + 0.622206i \(0.213763\pi\)
\(108\) 0 0
\(109\) −190063. −1.53225 −0.766127 0.642689i \(-0.777819\pi\)
−0.766127 + 0.642689i \(0.777819\pi\)
\(110\) 0 0
\(111\) −64462.1 −0.496589
\(112\) 0 0
\(113\) −116023. −0.854769 −0.427384 0.904070i \(-0.640565\pi\)
−0.427384 + 0.904070i \(0.640565\pi\)
\(114\) 0 0
\(115\) −27815.9 −0.196132
\(116\) 0 0
\(117\) −262350. −1.77181
\(118\) 0 0
\(119\) −73331.2 −0.474702
\(120\) 0 0
\(121\) −157622. −0.978711
\(122\) 0 0
\(123\) 257624. 1.53540
\(124\) 0 0
\(125\) −88824.6 −0.508462
\(126\) 0 0
\(127\) 71785.5 0.394936 0.197468 0.980309i \(-0.436728\pi\)
0.197468 + 0.980309i \(0.436728\pi\)
\(128\) 0 0
\(129\) −211863. −1.12093
\(130\) 0 0
\(131\) 2016.22 0.0102650 0.00513250 0.999987i \(-0.498366\pi\)
0.00513250 + 0.999987i \(0.498366\pi\)
\(132\) 0 0
\(133\) 24443.1 0.119819
\(134\) 0 0
\(135\) −6528.79 −0.0308318
\(136\) 0 0
\(137\) −243904. −1.11024 −0.555120 0.831770i \(-0.687328\pi\)
−0.555120 + 0.831770i \(0.687328\pi\)
\(138\) 0 0
\(139\) −209413. −0.919319 −0.459660 0.888095i \(-0.652029\pi\)
−0.459660 + 0.888095i \(0.652029\pi\)
\(140\) 0 0
\(141\) −423630. −1.79448
\(142\) 0 0
\(143\) 69059.3 0.282411
\(144\) 0 0
\(145\) −28186.9 −0.111334
\(146\) 0 0
\(147\) 51799.6 0.197712
\(148\) 0 0
\(149\) −22954.3 −0.0847028 −0.0423514 0.999103i \(-0.513485\pi\)
−0.0423514 + 0.999103i \(0.513485\pi\)
\(150\) 0 0
\(151\) −276737. −0.987700 −0.493850 0.869547i \(-0.664411\pi\)
−0.493850 + 0.869547i \(0.664411\pi\)
\(152\) 0 0
\(153\) 332901. 1.14971
\(154\) 0 0
\(155\) 11698.2 0.0391101
\(156\) 0 0
\(157\) −276692. −0.895874 −0.447937 0.894065i \(-0.647841\pi\)
−0.447937 + 0.894065i \(0.647841\pi\)
\(158\) 0 0
\(159\) 428786. 1.34508
\(160\) 0 0
\(161\) 92577.7 0.281476
\(162\) 0 0
\(163\) −274258. −0.808518 −0.404259 0.914645i \(-0.632471\pi\)
−0.404259 + 0.914645i \(0.632471\pi\)
\(164\) 0 0
\(165\) −18598.6 −0.0531827
\(166\) 0 0
\(167\) −215716. −0.598537 −0.299268 0.954169i \(-0.596743\pi\)
−0.299268 + 0.954169i \(0.596743\pi\)
\(168\) 0 0
\(169\) 1.01967e6 2.74628
\(170\) 0 0
\(171\) −110964. −0.290196
\(172\) 0 0
\(173\) −86252.7 −0.219108 −0.109554 0.993981i \(-0.534942\pi\)
−0.109554 + 0.993981i \(0.534942\pi\)
\(174\) 0 0
\(175\) 142504. 0.351749
\(176\) 0 0
\(177\) −773189. −1.85504
\(178\) 0 0
\(179\) 435822. 1.01666 0.508331 0.861162i \(-0.330262\pi\)
0.508331 + 0.861162i \(0.330262\pi\)
\(180\) 0 0
\(181\) −56972.2 −0.129261 −0.0646303 0.997909i \(-0.520587\pi\)
−0.0646303 + 0.997909i \(0.520587\pi\)
\(182\) 0 0
\(183\) −1.07819e6 −2.37995
\(184\) 0 0
\(185\) −43989.9 −0.0944981
\(186\) 0 0
\(187\) −87630.7 −0.183253
\(188\) 0 0
\(189\) 21729.3 0.0442478
\(190\) 0 0
\(191\) −350077. −0.694353 −0.347177 0.937800i \(-0.612860\pi\)
−0.347177 + 0.937800i \(0.612860\pi\)
\(192\) 0 0
\(193\) −623227. −1.20435 −0.602175 0.798364i \(-0.705699\pi\)
−0.602175 + 0.798364i \(0.705699\pi\)
\(194\) 0 0
\(195\) −374606. −0.705486
\(196\) 0 0
\(197\) −223265. −0.409879 −0.204939 0.978775i \(-0.565700\pi\)
−0.204939 + 0.978775i \(0.565700\pi\)
\(198\) 0 0
\(199\) 756375. 1.35396 0.676978 0.736004i \(-0.263289\pi\)
0.676978 + 0.736004i \(0.263289\pi\)
\(200\) 0 0
\(201\) −1.03936e6 −1.81458
\(202\) 0 0
\(203\) 93812.7 0.159780
\(204\) 0 0
\(205\) 175806. 0.292179
\(206\) 0 0
\(207\) −420274. −0.681722
\(208\) 0 0
\(209\) 29209.4 0.0462549
\(210\) 0 0
\(211\) 30644.3 0.0473853 0.0236927 0.999719i \(-0.492458\pi\)
0.0236927 + 0.999719i \(0.492458\pi\)
\(212\) 0 0
\(213\) 1.66507e6 2.51469
\(214\) 0 0
\(215\) −144578. −0.213308
\(216\) 0 0
\(217\) −38934.3 −0.0561284
\(218\) 0 0
\(219\) −1.28727e6 −1.81368
\(220\) 0 0
\(221\) −1.76503e6 −2.43092
\(222\) 0 0
\(223\) 461375. 0.621286 0.310643 0.950527i \(-0.399456\pi\)
0.310643 + 0.950527i \(0.399456\pi\)
\(224\) 0 0
\(225\) −646925. −0.851918
\(226\) 0 0
\(227\) −346439. −0.446234 −0.223117 0.974792i \(-0.571623\pi\)
−0.223117 + 0.974792i \(0.571623\pi\)
\(228\) 0 0
\(229\) 521650. 0.657341 0.328671 0.944445i \(-0.393399\pi\)
0.328671 + 0.944445i \(0.393399\pi\)
\(230\) 0 0
\(231\) 61900.5 0.0763245
\(232\) 0 0
\(233\) 154120. 0.185981 0.0929907 0.995667i \(-0.470357\pi\)
0.0929907 + 0.995667i \(0.470357\pi\)
\(234\) 0 0
\(235\) −289091. −0.341480
\(236\) 0 0
\(237\) 1.31048e6 1.51551
\(238\) 0 0
\(239\) −478940. −0.542358 −0.271179 0.962529i \(-0.587414\pi\)
−0.271179 + 0.962529i \(0.587414\pi\)
\(240\) 0 0
\(241\) 86551.8 0.0959917 0.0479958 0.998848i \(-0.484717\pi\)
0.0479958 + 0.998848i \(0.484717\pi\)
\(242\) 0 0
\(243\) −1.26482e6 −1.37408
\(244\) 0 0
\(245\) 35348.8 0.0376235
\(246\) 0 0
\(247\) 588326. 0.613586
\(248\) 0 0
\(249\) 995307. 1.01732
\(250\) 0 0
\(251\) 1.90322e6 1.90679 0.953397 0.301719i \(-0.0975605\pi\)
0.953397 + 0.301719i \(0.0975605\pi\)
\(252\) 0 0
\(253\) 110630. 0.108661
\(254\) 0 0
\(255\) 475345. 0.457782
\(256\) 0 0
\(257\) −569922. −0.538248 −0.269124 0.963105i \(-0.586734\pi\)
−0.269124 + 0.963105i \(0.586734\pi\)
\(258\) 0 0
\(259\) 146408. 0.135618
\(260\) 0 0
\(261\) −425881. −0.386978
\(262\) 0 0
\(263\) 1.29853e6 1.15761 0.578806 0.815465i \(-0.303519\pi\)
0.578806 + 0.815465i \(0.303519\pi\)
\(264\) 0 0
\(265\) 292610. 0.255961
\(266\) 0 0
\(267\) 1.69721e6 1.45699
\(268\) 0 0
\(269\) 53721.5 0.0452655 0.0226328 0.999744i \(-0.492795\pi\)
0.0226328 + 0.999744i \(0.492795\pi\)
\(270\) 0 0
\(271\) −758053. −0.627013 −0.313506 0.949586i \(-0.601504\pi\)
−0.313506 + 0.949586i \(0.601504\pi\)
\(272\) 0 0
\(273\) 1.24678e6 1.01247
\(274\) 0 0
\(275\) 170292. 0.135789
\(276\) 0 0
\(277\) −2.07598e6 −1.62564 −0.812819 0.582516i \(-0.802068\pi\)
−0.812819 + 0.582516i \(0.802068\pi\)
\(278\) 0 0
\(279\) 176750. 0.135940
\(280\) 0 0
\(281\) 2.61195e6 1.97333 0.986664 0.162770i \(-0.0520427\pi\)
0.986664 + 0.162770i \(0.0520427\pi\)
\(282\) 0 0
\(283\) −992734. −0.736829 −0.368414 0.929662i \(-0.620099\pi\)
−0.368414 + 0.929662i \(0.620099\pi\)
\(284\) 0 0
\(285\) −158444. −0.115548
\(286\) 0 0
\(287\) −585123. −0.419317
\(288\) 0 0
\(289\) 819820. 0.577396
\(290\) 0 0
\(291\) −940057. −0.650762
\(292\) 0 0
\(293\) −371728. −0.252962 −0.126481 0.991969i \(-0.540368\pi\)
−0.126481 + 0.991969i \(0.540368\pi\)
\(294\) 0 0
\(295\) −527635. −0.353003
\(296\) 0 0
\(297\) 25966.5 0.0170814
\(298\) 0 0
\(299\) 2.22827e6 1.44142
\(300\) 0 0
\(301\) 481189. 0.306126
\(302\) 0 0
\(303\) 3.54223e6 2.21651
\(304\) 0 0
\(305\) −735772. −0.452891
\(306\) 0 0
\(307\) 2.83906e6 1.71921 0.859606 0.510958i \(-0.170709\pi\)
0.859606 + 0.510958i \(0.170709\pi\)
\(308\) 0 0
\(309\) 3.54996e6 2.11508
\(310\) 0 0
\(311\) 759362. 0.445193 0.222596 0.974911i \(-0.428547\pi\)
0.222596 + 0.974911i \(0.428547\pi\)
\(312\) 0 0
\(313\) −1.80191e6 −1.03961 −0.519806 0.854284i \(-0.673996\pi\)
−0.519806 + 0.854284i \(0.673996\pi\)
\(314\) 0 0
\(315\) −160473. −0.0911223
\(316\) 0 0
\(317\) 441899. 0.246987 0.123494 0.992345i \(-0.460590\pi\)
0.123494 + 0.992345i \(0.460590\pi\)
\(318\) 0 0
\(319\) 112106. 0.0616811
\(320\) 0 0
\(321\) 4.00041e6 2.16691
\(322\) 0 0
\(323\) −746538. −0.398149
\(324\) 0 0
\(325\) 3.42997e6 1.80128
\(326\) 0 0
\(327\) −4.10045e6 −2.12061
\(328\) 0 0
\(329\) 962162. 0.490070
\(330\) 0 0
\(331\) 3.15779e6 1.58421 0.792107 0.610383i \(-0.208984\pi\)
0.792107 + 0.610383i \(0.208984\pi\)
\(332\) 0 0
\(333\) −664650. −0.328460
\(334\) 0 0
\(335\) −709275. −0.345305
\(336\) 0 0
\(337\) −1.66612e6 −0.799158 −0.399579 0.916699i \(-0.630844\pi\)
−0.399579 + 0.916699i \(0.630844\pi\)
\(338\) 0 0
\(339\) −2.50310e6 −1.18299
\(340\) 0 0
\(341\) −46526.4 −0.0216677
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) 0 0
\(345\) −600104. −0.271443
\(346\) 0 0
\(347\) 3.41953e6 1.52455 0.762277 0.647251i \(-0.224081\pi\)
0.762277 + 0.647251i \(0.224081\pi\)
\(348\) 0 0
\(349\) 3.18396e6 1.39928 0.699638 0.714497i \(-0.253345\pi\)
0.699638 + 0.714497i \(0.253345\pi\)
\(350\) 0 0
\(351\) 523009. 0.226590
\(352\) 0 0
\(353\) 3.02506e6 1.29210 0.646051 0.763294i \(-0.276419\pi\)
0.646051 + 0.763294i \(0.276419\pi\)
\(354\) 0 0
\(355\) 1.13627e6 0.478532
\(356\) 0 0
\(357\) −1.58206e6 −0.656980
\(358\) 0 0
\(359\) −1.12353e6 −0.460098 −0.230049 0.973179i \(-0.573889\pi\)
−0.230049 + 0.973179i \(0.573889\pi\)
\(360\) 0 0
\(361\) −2.22726e6 −0.899504
\(362\) 0 0
\(363\) −3.40057e6 −1.35452
\(364\) 0 0
\(365\) −878453. −0.345133
\(366\) 0 0
\(367\) 3.39115e6 1.31426 0.657131 0.753777i \(-0.271770\pi\)
0.657131 + 0.753777i \(0.271770\pi\)
\(368\) 0 0
\(369\) 2.65628e6 1.01557
\(370\) 0 0
\(371\) −973873. −0.367340
\(372\) 0 0
\(373\) −1.33823e6 −0.498033 −0.249017 0.968499i \(-0.580107\pi\)
−0.249017 + 0.968499i \(0.580107\pi\)
\(374\) 0 0
\(375\) −1.91632e6 −0.703703
\(376\) 0 0
\(377\) 2.25800e6 0.818221
\(378\) 0 0
\(379\) −3.11403e6 −1.11359 −0.556795 0.830650i \(-0.687969\pi\)
−0.556795 + 0.830650i \(0.687969\pi\)
\(380\) 0 0
\(381\) 1.54871e6 0.546586
\(382\) 0 0
\(383\) −334533. −0.116531 −0.0582656 0.998301i \(-0.518557\pi\)
−0.0582656 + 0.998301i \(0.518557\pi\)
\(384\) 0 0
\(385\) 42241.8 0.0145241
\(386\) 0 0
\(387\) −2.18445e6 −0.741422
\(388\) 0 0
\(389\) −1.59871e6 −0.535667 −0.267834 0.963465i \(-0.586308\pi\)
−0.267834 + 0.963465i \(0.586308\pi\)
\(390\) 0 0
\(391\) −2.82750e6 −0.935322
\(392\) 0 0
\(393\) 43498.2 0.0142066
\(394\) 0 0
\(395\) 894292. 0.288394
\(396\) 0 0
\(397\) 2.20789e6 0.703074 0.351537 0.936174i \(-0.385659\pi\)
0.351537 + 0.936174i \(0.385659\pi\)
\(398\) 0 0
\(399\) 527339. 0.165828
\(400\) 0 0
\(401\) 1.89272e6 0.587793 0.293897 0.955837i \(-0.405048\pi\)
0.293897 + 0.955837i \(0.405048\pi\)
\(402\) 0 0
\(403\) −937118. −0.287430
\(404\) 0 0
\(405\) −936667. −0.283758
\(406\) 0 0
\(407\) 174958. 0.0523537
\(408\) 0 0
\(409\) 1.14418e6 0.338210 0.169105 0.985598i \(-0.445912\pi\)
0.169105 + 0.985598i \(0.445912\pi\)
\(410\) 0 0
\(411\) −5.26202e6 −1.53655
\(412\) 0 0
\(413\) 1.75609e6 0.506608
\(414\) 0 0
\(415\) 679212. 0.193591
\(416\) 0 0
\(417\) −4.51791e6 −1.27232
\(418\) 0 0
\(419\) −5.17298e6 −1.43948 −0.719740 0.694244i \(-0.755739\pi\)
−0.719740 + 0.694244i \(0.755739\pi\)
\(420\) 0 0
\(421\) 1.40960e6 0.387606 0.193803 0.981040i \(-0.437918\pi\)
0.193803 + 0.981040i \(0.437918\pi\)
\(422\) 0 0
\(423\) −4.36792e6 −1.18693
\(424\) 0 0
\(425\) −4.35235e6 −1.16883
\(426\) 0 0
\(427\) 2.44882e6 0.649961
\(428\) 0 0
\(429\) 1.48990e6 0.390853
\(430\) 0 0
\(431\) 6.17076e6 1.60009 0.800047 0.599938i \(-0.204808\pi\)
0.800047 + 0.599938i \(0.204808\pi\)
\(432\) 0 0
\(433\) 387046. 0.0992071 0.0496036 0.998769i \(-0.484204\pi\)
0.0496036 + 0.998769i \(0.484204\pi\)
\(434\) 0 0
\(435\) −608110. −0.154084
\(436\) 0 0
\(437\) 942475. 0.236084
\(438\) 0 0
\(439\) 3.32974e6 0.824611 0.412305 0.911046i \(-0.364724\pi\)
0.412305 + 0.911046i \(0.364724\pi\)
\(440\) 0 0
\(441\) 534091. 0.130773
\(442\) 0 0
\(443\) 1.44003e6 0.348627 0.174314 0.984690i \(-0.444229\pi\)
0.174314 + 0.984690i \(0.444229\pi\)
\(444\) 0 0
\(445\) 1.15820e6 0.277258
\(446\) 0 0
\(447\) −495219. −0.117227
\(448\) 0 0
\(449\) 4.28977e6 1.00420 0.502098 0.864811i \(-0.332562\pi\)
0.502098 + 0.864811i \(0.332562\pi\)
\(450\) 0 0
\(451\) −699222. −0.161873
\(452\) 0 0
\(453\) −5.97037e6 −1.36696
\(454\) 0 0
\(455\) 850819. 0.192667
\(456\) 0 0
\(457\) −5.27889e6 −1.18237 −0.591183 0.806537i \(-0.701339\pi\)
−0.591183 + 0.806537i \(0.701339\pi\)
\(458\) 0 0
\(459\) −663656. −0.147032
\(460\) 0 0
\(461\) 3.72322e6 0.815955 0.407977 0.912992i \(-0.366234\pi\)
0.407977 + 0.912992i \(0.366234\pi\)
\(462\) 0 0
\(463\) 1.04809e6 0.227220 0.113610 0.993525i \(-0.463759\pi\)
0.113610 + 0.993525i \(0.463759\pi\)
\(464\) 0 0
\(465\) 252379. 0.0541278
\(466\) 0 0
\(467\) −6.04244e6 −1.28210 −0.641048 0.767501i \(-0.721500\pi\)
−0.641048 + 0.767501i \(0.721500\pi\)
\(468\) 0 0
\(469\) 2.36063e6 0.495560
\(470\) 0 0
\(471\) −5.96940e6 −1.23988
\(472\) 0 0
\(473\) 575021. 0.118176
\(474\) 0 0
\(475\) 1.45074e6 0.295024
\(476\) 0 0
\(477\) 4.42109e6 0.889679
\(478\) 0 0
\(479\) −2.06868e6 −0.411959 −0.205980 0.978556i \(-0.566038\pi\)
−0.205980 + 0.978556i \(0.566038\pi\)
\(480\) 0 0
\(481\) 3.52394e6 0.694490
\(482\) 0 0
\(483\) 1.99729e6 0.389559
\(484\) 0 0
\(485\) −641509. −0.123836
\(486\) 0 0
\(487\) −2.82966e6 −0.540645 −0.270323 0.962770i \(-0.587130\pi\)
−0.270323 + 0.962770i \(0.587130\pi\)
\(488\) 0 0
\(489\) −5.91688e6 −1.11898
\(490\) 0 0
\(491\) 4.44554e6 0.832186 0.416093 0.909322i \(-0.363399\pi\)
0.416093 + 0.909322i \(0.363399\pi\)
\(492\) 0 0
\(493\) −2.86522e6 −0.530934
\(494\) 0 0
\(495\) −191765. −0.0351767
\(496\) 0 0
\(497\) −3.78177e6 −0.686759
\(498\) 0 0
\(499\) 1.34623e6 0.242029 0.121015 0.992651i \(-0.461385\pi\)
0.121015 + 0.992651i \(0.461385\pi\)
\(500\) 0 0
\(501\) −4.65389e6 −0.828365
\(502\) 0 0
\(503\) −4.88909e6 −0.861604 −0.430802 0.902446i \(-0.641769\pi\)
−0.430802 + 0.902446i \(0.641769\pi\)
\(504\) 0 0
\(505\) 2.41727e6 0.421790
\(506\) 0 0
\(507\) 2.19986e7 3.80081
\(508\) 0 0
\(509\) −6.55518e6 −1.12148 −0.560738 0.827993i \(-0.689483\pi\)
−0.560738 + 0.827993i \(0.689483\pi\)
\(510\) 0 0
\(511\) 2.92370e6 0.495313
\(512\) 0 0
\(513\) 221213. 0.0371122
\(514\) 0 0
\(515\) 2.42254e6 0.402488
\(516\) 0 0
\(517\) 1.14978e6 0.189186
\(518\) 0 0
\(519\) −1.86083e6 −0.303242
\(520\) 0 0
\(521\) −4.77290e6 −0.770349 −0.385175 0.922844i \(-0.625859\pi\)
−0.385175 + 0.922844i \(0.625859\pi\)
\(522\) 0 0
\(523\) 828135. 0.132387 0.0661937 0.997807i \(-0.478914\pi\)
0.0661937 + 0.997807i \(0.478914\pi\)
\(524\) 0 0
\(525\) 3.07441e6 0.486814
\(526\) 0 0
\(527\) 1.18913e6 0.186510
\(528\) 0 0
\(529\) −2.86673e6 −0.445398
\(530\) 0 0
\(531\) −7.97212e6 −1.22698
\(532\) 0 0
\(533\) −1.40835e7 −2.14730
\(534\) 0 0
\(535\) 2.72994e6 0.412352
\(536\) 0 0
\(537\) 9.40251e6 1.40705
\(538\) 0 0
\(539\) −140590. −0.0208441
\(540\) 0 0
\(541\) −6.93356e6 −1.01851 −0.509253 0.860617i \(-0.670078\pi\)
−0.509253 + 0.860617i \(0.670078\pi\)
\(542\) 0 0
\(543\) −1.22913e6 −0.178895
\(544\) 0 0
\(545\) −2.79820e6 −0.403541
\(546\) 0 0
\(547\) −559077. −0.0798920 −0.0399460 0.999202i \(-0.512719\pi\)
−0.0399460 + 0.999202i \(0.512719\pi\)
\(548\) 0 0
\(549\) −1.11169e7 −1.57417
\(550\) 0 0
\(551\) 955047. 0.134013
\(552\) 0 0
\(553\) −2.97641e6 −0.413885
\(554\) 0 0
\(555\) −949045. −0.130784
\(556\) 0 0
\(557\) 3.90444e6 0.533237 0.266619 0.963802i \(-0.414094\pi\)
0.266619 + 0.963802i \(0.414094\pi\)
\(558\) 0 0
\(559\) 1.15819e7 1.56765
\(560\) 0 0
\(561\) −1.89056e6 −0.253620
\(562\) 0 0
\(563\) 4.02307e6 0.534918 0.267459 0.963569i \(-0.413816\pi\)
0.267459 + 0.963569i \(0.413816\pi\)
\(564\) 0 0
\(565\) −1.70815e6 −0.225116
\(566\) 0 0
\(567\) 3.11745e6 0.407231
\(568\) 0 0
\(569\) −2.35801e6 −0.305327 −0.152663 0.988278i \(-0.548785\pi\)
−0.152663 + 0.988278i \(0.548785\pi\)
\(570\) 0 0
\(571\) 7.76733e6 0.996969 0.498484 0.866899i \(-0.333890\pi\)
0.498484 + 0.866899i \(0.333890\pi\)
\(572\) 0 0
\(573\) −7.55263e6 −0.960974
\(574\) 0 0
\(575\) 5.49467e6 0.693062
\(576\) 0 0
\(577\) 1.38363e7 1.73013 0.865066 0.501657i \(-0.167276\pi\)
0.865066 + 0.501657i \(0.167276\pi\)
\(578\) 0 0
\(579\) −1.34456e7 −1.66680
\(580\) 0 0
\(581\) −2.26057e6 −0.277830
\(582\) 0 0
\(583\) −1.16378e6 −0.141807
\(584\) 0 0
\(585\) −3.86246e6 −0.466631
\(586\) 0 0
\(587\) −2.13747e6 −0.256038 −0.128019 0.991772i \(-0.540862\pi\)
−0.128019 + 0.991772i \(0.540862\pi\)
\(588\) 0 0
\(589\) −396365. −0.0470768
\(590\) 0 0
\(591\) −4.81676e6 −0.567265
\(592\) 0 0
\(593\) −1.11381e7 −1.30069 −0.650345 0.759639i \(-0.725376\pi\)
−0.650345 + 0.759639i \(0.725376\pi\)
\(594\) 0 0
\(595\) −1.07962e6 −0.125020
\(596\) 0 0
\(597\) 1.63182e7 1.87385
\(598\) 0 0
\(599\) 1.10044e7 1.25314 0.626572 0.779364i \(-0.284457\pi\)
0.626572 + 0.779364i \(0.284457\pi\)
\(600\) 0 0
\(601\) 1.28046e6 0.144604 0.0723019 0.997383i \(-0.476965\pi\)
0.0723019 + 0.997383i \(0.476965\pi\)
\(602\) 0 0
\(603\) −1.07166e7 −1.20022
\(604\) 0 0
\(605\) −2.32060e6 −0.257758
\(606\) 0 0
\(607\) −2.08327e6 −0.229496 −0.114748 0.993395i \(-0.536606\pi\)
−0.114748 + 0.993395i \(0.536606\pi\)
\(608\) 0 0
\(609\) 2.02393e6 0.221132
\(610\) 0 0
\(611\) 2.31585e7 2.50962
\(612\) 0 0
\(613\) 1.37116e7 1.47380 0.736899 0.676003i \(-0.236289\pi\)
0.736899 + 0.676003i \(0.236289\pi\)
\(614\) 0 0
\(615\) 3.79287e6 0.404371
\(616\) 0 0
\(617\) −1.65172e7 −1.74672 −0.873361 0.487073i \(-0.838064\pi\)
−0.873361 + 0.487073i \(0.838064\pi\)
\(618\) 0 0
\(619\) −3.51321e6 −0.368534 −0.184267 0.982876i \(-0.558991\pi\)
−0.184267 + 0.982876i \(0.558991\pi\)
\(620\) 0 0
\(621\) 837839. 0.0871830
\(622\) 0 0
\(623\) −3.85477e6 −0.397904
\(624\) 0 0
\(625\) 7.78055e6 0.796728
\(626\) 0 0
\(627\) 630169. 0.0640160
\(628\) 0 0
\(629\) −4.47160e6 −0.450647
\(630\) 0 0
\(631\) −8.54135e6 −0.853991 −0.426995 0.904254i \(-0.640428\pi\)
−0.426995 + 0.904254i \(0.640428\pi\)
\(632\) 0 0
\(633\) 661126. 0.0655805
\(634\) 0 0
\(635\) 1.05686e6 0.104012
\(636\) 0 0
\(637\) −2.83172e6 −0.276504
\(638\) 0 0
\(639\) 1.71681e7 1.66330
\(640\) 0 0
\(641\) −9.95519e6 −0.956983 −0.478492 0.878092i \(-0.658816\pi\)
−0.478492 + 0.878092i \(0.658816\pi\)
\(642\) 0 0
\(643\) 3.98313e6 0.379924 0.189962 0.981791i \(-0.439163\pi\)
0.189962 + 0.981791i \(0.439163\pi\)
\(644\) 0 0
\(645\) −3.11915e6 −0.295214
\(646\) 0 0
\(647\) −3.84121e6 −0.360750 −0.180375 0.983598i \(-0.557731\pi\)
−0.180375 + 0.983598i \(0.557731\pi\)
\(648\) 0 0
\(649\) 2.09853e6 0.195570
\(650\) 0 0
\(651\) −839975. −0.0776808
\(652\) 0 0
\(653\) −1.84286e7 −1.69126 −0.845628 0.533772i \(-0.820774\pi\)
−0.845628 + 0.533772i \(0.820774\pi\)
\(654\) 0 0
\(655\) 29683.8 0.00270344
\(656\) 0 0
\(657\) −1.32727e7 −1.19963
\(658\) 0 0
\(659\) −2.15854e7 −1.93619 −0.968093 0.250590i \(-0.919376\pi\)
−0.968093 + 0.250590i \(0.919376\pi\)
\(660\) 0 0
\(661\) −1.44072e7 −1.28256 −0.641278 0.767308i \(-0.721596\pi\)
−0.641278 + 0.767308i \(0.721596\pi\)
\(662\) 0 0
\(663\) −3.80790e7 −3.36435
\(664\) 0 0
\(665\) 359864. 0.0315561
\(666\) 0 0
\(667\) 3.61723e6 0.314819
\(668\) 0 0
\(669\) 9.95378e6 0.859850
\(670\) 0 0
\(671\) 2.92634e6 0.250910
\(672\) 0 0
\(673\) −5.55230e6 −0.472536 −0.236268 0.971688i \(-0.575924\pi\)
−0.236268 + 0.971688i \(0.575924\pi\)
\(674\) 0 0
\(675\) 1.28968e6 0.108949
\(676\) 0 0
\(677\) 9.92009e6 0.831848 0.415924 0.909399i \(-0.363458\pi\)
0.415924 + 0.909399i \(0.363458\pi\)
\(678\) 0 0
\(679\) 2.13509e6 0.177722
\(680\) 0 0
\(681\) −7.47414e6 −0.617581
\(682\) 0 0
\(683\) 1.01291e7 0.830846 0.415423 0.909628i \(-0.363634\pi\)
0.415423 + 0.909628i \(0.363634\pi\)
\(684\) 0 0
\(685\) −3.59088e6 −0.292398
\(686\) 0 0
\(687\) 1.12542e7 0.909750
\(688\) 0 0
\(689\) −2.34404e7 −1.88112
\(690\) 0 0
\(691\) −1.98747e7 −1.58345 −0.791727 0.610875i \(-0.790818\pi\)
−0.791727 + 0.610875i \(0.790818\pi\)
\(692\) 0 0
\(693\) 638238. 0.0504835
\(694\) 0 0
\(695\) −3.08309e6 −0.242116
\(696\) 0 0
\(697\) 1.78708e7 1.39336
\(698\) 0 0
\(699\) 3.32502e6 0.257395
\(700\) 0 0
\(701\) 2.13147e7 1.63826 0.819132 0.573605i \(-0.194455\pi\)
0.819132 + 0.573605i \(0.194455\pi\)
\(702\) 0 0
\(703\) 1.49049e6 0.113747
\(704\) 0 0
\(705\) −6.23690e6 −0.472602
\(706\) 0 0
\(707\) −8.04523e6 −0.605327
\(708\) 0 0
\(709\) −9.83200e6 −0.734559 −0.367279 0.930111i \(-0.619711\pi\)
−0.367279 + 0.930111i \(0.619711\pi\)
\(710\) 0 0
\(711\) 1.35120e7 1.00241
\(712\) 0 0
\(713\) −1.50123e6 −0.110592
\(714\) 0 0
\(715\) 1.01673e6 0.0743771
\(716\) 0 0
\(717\) −1.03327e7 −0.750615
\(718\) 0 0
\(719\) −1.80674e7 −1.30338 −0.651692 0.758484i \(-0.725940\pi\)
−0.651692 + 0.758484i \(0.725940\pi\)
\(720\) 0 0
\(721\) −8.06278e6 −0.577626
\(722\) 0 0
\(723\) 1.86728e6 0.132851
\(724\) 0 0
\(725\) 5.56797e6 0.393416
\(726\) 0 0
\(727\) −1.82003e7 −1.27715 −0.638576 0.769559i \(-0.720476\pi\)
−0.638576 + 0.769559i \(0.720476\pi\)
\(728\) 0 0
\(729\) −1.18274e7 −0.824274
\(730\) 0 0
\(731\) −1.46965e7 −1.01723
\(732\) 0 0
\(733\) −6.70629e6 −0.461023 −0.230512 0.973070i \(-0.574040\pi\)
−0.230512 + 0.973070i \(0.574040\pi\)
\(734\) 0 0
\(735\) 762621. 0.0520703
\(736\) 0 0
\(737\) 2.82095e6 0.191305
\(738\) 0 0
\(739\) −2.16502e7 −1.45831 −0.729156 0.684347i \(-0.760087\pi\)
−0.729156 + 0.684347i \(0.760087\pi\)
\(740\) 0 0
\(741\) 1.26926e7 0.849193
\(742\) 0 0
\(743\) −1.40437e7 −0.933276 −0.466638 0.884448i \(-0.654535\pi\)
−0.466638 + 0.884448i \(0.654535\pi\)
\(744\) 0 0
\(745\) −337945. −0.0223077
\(746\) 0 0
\(747\) 1.02623e7 0.672890
\(748\) 0 0
\(749\) −9.08586e6 −0.591782
\(750\) 0 0
\(751\) 8.91381e6 0.576718 0.288359 0.957522i \(-0.406890\pi\)
0.288359 + 0.957522i \(0.406890\pi\)
\(752\) 0 0
\(753\) 4.10603e7 2.63897
\(754\) 0 0
\(755\) −4.07427e6 −0.260125
\(756\) 0 0
\(757\) 1.04419e7 0.662276 0.331138 0.943582i \(-0.392567\pi\)
0.331138 + 0.943582i \(0.392567\pi\)
\(758\) 0 0
\(759\) 2.38676e6 0.150385
\(760\) 0 0
\(761\) 2.96214e7 1.85414 0.927072 0.374883i \(-0.122317\pi\)
0.927072 + 0.374883i \(0.122317\pi\)
\(762\) 0 0
\(763\) 9.31307e6 0.579138
\(764\) 0 0
\(765\) 4.90115e6 0.302792
\(766\) 0 0
\(767\) 4.22678e7 2.59431
\(768\) 0 0
\(769\) −4.24042e6 −0.258579 −0.129289 0.991607i \(-0.541270\pi\)
−0.129289 + 0.991607i \(0.541270\pi\)
\(770\) 0 0
\(771\) −1.22956e7 −0.744927
\(772\) 0 0
\(773\) 1.37072e7 0.825086 0.412543 0.910938i \(-0.364641\pi\)
0.412543 + 0.910938i \(0.364641\pi\)
\(774\) 0 0
\(775\) −2.31083e6 −0.138202
\(776\) 0 0
\(777\) 3.15864e6 0.187693
\(778\) 0 0
\(779\) −5.95677e6 −0.351696
\(780\) 0 0
\(781\) −4.51922e6 −0.265116
\(782\) 0 0
\(783\) 849016. 0.0494893
\(784\) 0 0
\(785\) −4.07360e6 −0.235942
\(786\) 0 0
\(787\) −1.02427e7 −0.589491 −0.294745 0.955576i \(-0.595235\pi\)
−0.294745 + 0.955576i \(0.595235\pi\)
\(788\) 0 0
\(789\) 2.80148e7 1.60212
\(790\) 0 0
\(791\) 5.68513e6 0.323072
\(792\) 0 0
\(793\) 5.89413e7 3.32841
\(794\) 0 0
\(795\) 6.31281e6 0.354246
\(796\) 0 0
\(797\) −3.00087e7 −1.67340 −0.836702 0.547658i \(-0.815520\pi\)
−0.836702 + 0.547658i \(0.815520\pi\)
\(798\) 0 0
\(799\) −2.93863e7 −1.62846
\(800\) 0 0
\(801\) 1.74995e7 0.963704
\(802\) 0 0
\(803\) 3.49381e6 0.191210
\(804\) 0 0
\(805\) 1.36298e6 0.0741309
\(806\) 0 0
\(807\) 1.15900e6 0.0626468
\(808\) 0 0
\(809\) 922081. 0.0495333 0.0247667 0.999693i \(-0.492116\pi\)
0.0247667 + 0.999693i \(0.492116\pi\)
\(810\) 0 0
\(811\) 2.49323e7 1.33110 0.665549 0.746355i \(-0.268198\pi\)
0.665549 + 0.746355i \(0.268198\pi\)
\(812\) 0 0
\(813\) −1.63544e7 −0.867776
\(814\) 0 0
\(815\) −4.03776e6 −0.212935
\(816\) 0 0
\(817\) 4.89868e6 0.256758
\(818\) 0 0
\(819\) 1.28552e7 0.669680
\(820\) 0 0
\(821\) 8.06228e6 0.417446 0.208723 0.977975i \(-0.433069\pi\)
0.208723 + 0.977975i \(0.433069\pi\)
\(822\) 0 0
\(823\) −5.68168e6 −0.292400 −0.146200 0.989255i \(-0.546704\pi\)
−0.146200 + 0.989255i \(0.546704\pi\)
\(824\) 0 0
\(825\) 3.67392e6 0.187929
\(826\) 0 0
\(827\) 976611. 0.0496544 0.0248272 0.999692i \(-0.492096\pi\)
0.0248272 + 0.999692i \(0.492096\pi\)
\(828\) 0 0
\(829\) 4.94857e6 0.250088 0.125044 0.992151i \(-0.460093\pi\)
0.125044 + 0.992151i \(0.460093\pi\)
\(830\) 0 0
\(831\) −4.47876e7 −2.24986
\(832\) 0 0
\(833\) 3.59323e6 0.179421
\(834\) 0 0
\(835\) −3.17588e6 −0.157633
\(836\) 0 0
\(837\) −352360. −0.0173849
\(838\) 0 0
\(839\) −2.03716e7 −0.999125 −0.499562 0.866278i \(-0.666506\pi\)
−0.499562 + 0.866278i \(0.666506\pi\)
\(840\) 0 0
\(841\) −1.68457e7 −0.821293
\(842\) 0 0
\(843\) 5.63507e7 2.73105
\(844\) 0 0
\(845\) 1.50122e7 0.723273
\(846\) 0 0
\(847\) 7.72349e6 0.369918
\(848\) 0 0
\(849\) −2.14174e7 −1.01976
\(850\) 0 0
\(851\) 5.64521e6 0.267212
\(852\) 0 0
\(853\) −1.41579e7 −0.666234 −0.333117 0.942885i \(-0.608100\pi\)
−0.333117 + 0.942885i \(0.608100\pi\)
\(854\) 0 0
\(855\) −1.63367e6 −0.0764274
\(856\) 0 0
\(857\) −1.24397e7 −0.578575 −0.289287 0.957242i \(-0.593418\pi\)
−0.289287 + 0.957242i \(0.593418\pi\)
\(858\) 0 0
\(859\) −1.72105e7 −0.795810 −0.397905 0.917427i \(-0.630263\pi\)
−0.397905 + 0.917427i \(0.630263\pi\)
\(860\) 0 0
\(861\) −1.26236e7 −0.580328
\(862\) 0 0
\(863\) 2.04374e7 0.934113 0.467057 0.884227i \(-0.345314\pi\)
0.467057 + 0.884227i \(0.345314\pi\)
\(864\) 0 0
\(865\) −1.26986e6 −0.0577052
\(866\) 0 0
\(867\) 1.76869e7 0.799107
\(868\) 0 0
\(869\) −3.55681e6 −0.159776
\(870\) 0 0
\(871\) 5.68187e7 2.53773
\(872\) 0 0
\(873\) −9.69266e6 −0.430435
\(874\) 0 0
\(875\) 4.35241e6 0.192181
\(876\) 0 0
\(877\) 1.72398e7 0.756890 0.378445 0.925624i \(-0.376459\pi\)
0.378445 + 0.925624i \(0.376459\pi\)
\(878\) 0 0
\(879\) −8.01972e6 −0.350096
\(880\) 0 0
\(881\) −3.86589e7 −1.67807 −0.839034 0.544080i \(-0.816879\pi\)
−0.839034 + 0.544080i \(0.816879\pi\)
\(882\) 0 0
\(883\) 3.29454e7 1.42198 0.710990 0.703203i \(-0.248247\pi\)
0.710990 + 0.703203i \(0.248247\pi\)
\(884\) 0 0
\(885\) −1.13833e7 −0.488551
\(886\) 0 0
\(887\) 3.61545e7 1.54295 0.771477 0.636258i \(-0.219518\pi\)
0.771477 + 0.636258i \(0.219518\pi\)
\(888\) 0 0
\(889\) −3.51749e6 −0.149272
\(890\) 0 0
\(891\) 3.72534e6 0.157207
\(892\) 0 0
\(893\) 9.79516e6 0.411039
\(894\) 0 0
\(895\) 6.41641e6 0.267753
\(896\) 0 0
\(897\) 4.80732e7 1.99490
\(898\) 0 0
\(899\) −1.52125e6 −0.0627772
\(900\) 0 0
\(901\) 2.97440e7 1.22064
\(902\) 0 0
\(903\) 1.03813e7 0.423673
\(904\) 0 0
\(905\) −838774. −0.0340427
\(906\) 0 0
\(907\) 4.15620e7 1.67756 0.838781 0.544469i \(-0.183269\pi\)
0.838781 + 0.544469i \(0.183269\pi\)
\(908\) 0 0
\(909\) 3.65229e7 1.46607
\(910\) 0 0
\(911\) −9.55917e6 −0.381614 −0.190807 0.981628i \(-0.561110\pi\)
−0.190807 + 0.981628i \(0.561110\pi\)
\(912\) 0 0
\(913\) −2.70138e6 −0.107253
\(914\) 0 0
\(915\) −1.58737e7 −0.626794
\(916\) 0 0
\(917\) −98794.6 −0.00387980
\(918\) 0 0
\(919\) −1.20453e7 −0.470469 −0.235234 0.971939i \(-0.575586\pi\)
−0.235234 + 0.971939i \(0.575586\pi\)
\(920\) 0 0
\(921\) 6.12505e7 2.37936
\(922\) 0 0
\(923\) −9.10244e7 −3.51685
\(924\) 0 0
\(925\) 8.68963e6 0.333924
\(926\) 0 0
\(927\) 3.66026e7 1.39898
\(928\) 0 0
\(929\) −2.42707e7 −0.922661 −0.461331 0.887228i \(-0.652628\pi\)
−0.461331 + 0.887228i \(0.652628\pi\)
\(930\) 0 0
\(931\) −1.19771e6 −0.0452874
\(932\) 0 0
\(933\) 1.63826e7 0.616140
\(934\) 0 0
\(935\) −1.29015e6 −0.0482625
\(936\) 0 0
\(937\) 2.02014e7 0.751679 0.375840 0.926685i \(-0.377354\pi\)
0.375840 + 0.926685i \(0.377354\pi\)
\(938\) 0 0
\(939\) −3.88747e7 −1.43881
\(940\) 0 0
\(941\) 6.68084e6 0.245956 0.122978 0.992409i \(-0.460756\pi\)
0.122978 + 0.992409i \(0.460756\pi\)
\(942\) 0 0
\(943\) −2.25612e7 −0.826195
\(944\) 0 0
\(945\) 319911. 0.0116533
\(946\) 0 0
\(947\) 3.19373e7 1.15724 0.578619 0.815598i \(-0.303592\pi\)
0.578619 + 0.815598i \(0.303592\pi\)
\(948\) 0 0
\(949\) 7.03712e7 2.53647
\(950\) 0 0
\(951\) 9.53361e6 0.341827
\(952\) 0 0
\(953\) 2.26242e7 0.806938 0.403469 0.914993i \(-0.367804\pi\)
0.403469 + 0.914993i \(0.367804\pi\)
\(954\) 0 0
\(955\) −5.15402e6 −0.182868
\(956\) 0 0
\(957\) 2.41859e6 0.0853657
\(958\) 0 0
\(959\) 1.19513e7 0.419631
\(960\) 0 0
\(961\) −2.79978e7 −0.977947
\(962\) 0 0
\(963\) 4.12470e7 1.43327
\(964\) 0 0
\(965\) −9.17547e6 −0.317183
\(966\) 0 0
\(967\) −2.38201e7 −0.819178 −0.409589 0.912270i \(-0.634328\pi\)
−0.409589 + 0.912270i \(0.634328\pi\)
\(968\) 0 0
\(969\) −1.61059e7 −0.551032
\(970\) 0 0
\(971\) −1.87074e7 −0.636746 −0.318373 0.947965i \(-0.603136\pi\)
−0.318373 + 0.947965i \(0.603136\pi\)
\(972\) 0 0
\(973\) 1.02612e7 0.347470
\(974\) 0 0
\(975\) 7.39987e7 2.49294
\(976\) 0 0
\(977\) 1.50783e7 0.505376 0.252688 0.967548i \(-0.418685\pi\)
0.252688 + 0.967548i \(0.418685\pi\)
\(978\) 0 0
\(979\) −4.60644e6 −0.153606
\(980\) 0 0
\(981\) −4.22785e7 −1.40264
\(982\) 0 0
\(983\) 2.38312e6 0.0786614 0.0393307 0.999226i \(-0.487477\pi\)
0.0393307 + 0.999226i \(0.487477\pi\)
\(984\) 0 0
\(985\) −3.28703e6 −0.107948
\(986\) 0 0
\(987\) 2.07579e7 0.678250
\(988\) 0 0
\(989\) 1.85537e7 0.603170
\(990\) 0 0
\(991\) −4.86331e7 −1.57307 −0.786535 0.617546i \(-0.788127\pi\)
−0.786535 + 0.617546i \(0.788127\pi\)
\(992\) 0 0
\(993\) 6.81268e7 2.19253
\(994\) 0 0
\(995\) 1.11357e7 0.356584
\(996\) 0 0
\(997\) −1.73445e7 −0.552617 −0.276309 0.961069i \(-0.589111\pi\)
−0.276309 + 0.961069i \(0.589111\pi\)
\(998\) 0 0
\(999\) 1.32501e6 0.0420056
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 448.6.a.v.1.2 2
4.3 odd 2 448.6.a.t.1.1 2
8.3 odd 2 112.6.a.i.1.2 2
8.5 even 2 56.6.a.e.1.1 2
24.5 odd 2 504.6.a.i.1.2 2
24.11 even 2 1008.6.a.bd.1.2 2
56.5 odd 6 392.6.i.i.361.1 4
56.13 odd 2 392.6.a.d.1.2 2
56.27 even 2 784.6.a.u.1.1 2
56.37 even 6 392.6.i.j.361.2 4
56.45 odd 6 392.6.i.i.177.1 4
56.53 even 6 392.6.i.j.177.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.e.1.1 2 8.5 even 2
112.6.a.i.1.2 2 8.3 odd 2
392.6.a.d.1.2 2 56.13 odd 2
392.6.i.i.177.1 4 56.45 odd 6
392.6.i.i.361.1 4 56.5 odd 6
392.6.i.j.177.2 4 56.53 even 6
392.6.i.j.361.2 4 56.37 even 6
448.6.a.t.1.1 2 4.3 odd 2
448.6.a.v.1.2 2 1.1 even 1 trivial
504.6.a.i.1.2 2 24.5 odd 2
784.6.a.u.1.1 2 56.27 even 2
1008.6.a.bd.1.2 2 24.11 even 2