Properties

Label 320.6.c.k.129.7
Level $320$
Weight $6$
Character 320.129
Analytic conductor $51.323$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,6,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{44}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.7
Root \(4.71554i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.6.c.k.129.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.81633i q^{3} +(16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +O(q^{10})\) \(q+9.81633i q^{3} +(16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +407.674 q^{11} -465.665i q^{13} +(522.874 + 166.519i) q^{15} -284.781i q^{17} -323.952 q^{19} +2187.28 q^{21} +12.0533i q^{23} +(-2549.48 - 1807.14i) q^{25} +3824.83i q^{27} +4381.26 q^{29} -10140.5 q^{31} +4001.86i q^{33} +(-11868.7 - 3779.81i) q^{35} -6906.96i q^{37} +4571.12 q^{39} -10733.8 q^{41} -6002.36i q^{43} +(2487.51 - 7810.87i) q^{45} +25033.9i q^{47} -32842.1 q^{49} +2795.51 q^{51} -10570.8i q^{53} +(6915.56 - 21715.1i) q^{55} -3180.02i q^{57} +30921.1 q^{59} +34366.3 q^{61} -32674.4i q^{63} +(-24804.0 - 7899.27i) q^{65} +45890.9i q^{67} -118.319 q^{69} +58026.5 q^{71} -68664.7i q^{73} +(17739.5 - 25026.6i) q^{75} -90838.3i q^{77} -65066.0 q^{79} -1912.37 q^{81} -51017.2i q^{83} +(-15169.1 - 4830.87i) q^{85} +43007.9i q^{87} +59409.1 q^{89} -103760. q^{91} -99542.4i q^{93} +(-5495.34 + 17255.5i) q^{95} -73186.2i q^{97} +59781.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 60 q^{5} - 1676 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 60 q^{5} - 1676 q^{9} - 688 q^{21} + 1500 q^{25} + 21304 q^{29} - 109672 q^{41} + 75140 q^{45} - 32348 q^{49} - 64552 q^{61} - 160 q^{65} + 166352 q^{69} - 173764 q^{81} - 151360 q^{85} - 3720 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.81633i 0.629718i 0.949138 + 0.314859i \(0.101957\pi\)
−0.949138 + 0.314859i \(0.898043\pi\)
\(4\) 0 0
\(5\) 16.9634 53.2658i 0.303451 0.952847i
\(6\) 0 0
\(7\) 222.821i 1.71874i −0.511353 0.859371i \(-0.670855\pi\)
0.511353 0.859371i \(-0.329145\pi\)
\(8\) 0 0
\(9\) 146.640 0.603455
\(10\) 0 0
\(11\) 407.674 1.01585 0.507927 0.861400i \(-0.330412\pi\)
0.507927 + 0.861400i \(0.330412\pi\)
\(12\) 0 0
\(13\) 465.665i 0.764214i −0.924118 0.382107i \(-0.875199\pi\)
0.924118 0.382107i \(-0.124801\pi\)
\(14\) 0 0
\(15\) 522.874 + 166.519i 0.600025 + 0.191089i
\(16\) 0 0
\(17\) 284.781i 0.238995i −0.992834 0.119498i \(-0.961872\pi\)
0.992834 0.119498i \(-0.0381284\pi\)
\(18\) 0 0
\(19\) −323.952 −0.205872 −0.102936 0.994688i \(-0.532824\pi\)
−0.102936 + 0.994688i \(0.532824\pi\)
\(20\) 0 0
\(21\) 2187.28 1.08232
\(22\) 0 0
\(23\) 12.0533i 0.00475102i 0.999997 + 0.00237551i \(0.000756149\pi\)
−0.999997 + 0.00237551i \(0.999244\pi\)
\(24\) 0 0
\(25\) −2549.48 1807.14i −0.815835 0.578285i
\(26\) 0 0
\(27\) 3824.83i 1.00972i
\(28\) 0 0
\(29\) 4381.26 0.967395 0.483697 0.875235i \(-0.339294\pi\)
0.483697 + 0.875235i \(0.339294\pi\)
\(30\) 0 0
\(31\) −10140.5 −1.89520 −0.947599 0.319461i \(-0.896498\pi\)
−0.947599 + 0.319461i \(0.896498\pi\)
\(32\) 0 0
\(33\) 4001.86i 0.639702i
\(34\) 0 0
\(35\) −11868.7 3779.81i −1.63770 0.521554i
\(36\) 0 0
\(37\) 6906.96i 0.829435i −0.909950 0.414718i \(-0.863880\pi\)
0.909950 0.414718i \(-0.136120\pi\)
\(38\) 0 0
\(39\) 4571.12 0.481239
\(40\) 0 0
\(41\) −10733.8 −0.997231 −0.498615 0.866823i \(-0.666158\pi\)
−0.498615 + 0.866823i \(0.666158\pi\)
\(42\) 0 0
\(43\) 6002.36i 0.495052i −0.968881 0.247526i \(-0.920382\pi\)
0.968881 0.247526i \(-0.0796175\pi\)
\(44\) 0 0
\(45\) 2487.51 7810.87i 0.183119 0.575001i
\(46\) 0 0
\(47\) 25033.9i 1.65304i 0.562905 + 0.826522i \(0.309684\pi\)
−0.562905 + 0.826522i \(0.690316\pi\)
\(48\) 0 0
\(49\) −32842.1 −1.95407
\(50\) 0 0
\(51\) 2795.51 0.150499
\(52\) 0 0
\(53\) 10570.8i 0.516913i −0.966023 0.258457i \(-0.916786\pi\)
0.966023 0.258457i \(-0.0832139\pi\)
\(54\) 0 0
\(55\) 6915.56 21715.1i 0.308262 0.967954i
\(56\) 0 0
\(57\) 3180.02i 0.129641i
\(58\) 0 0
\(59\) 30921.1 1.15644 0.578222 0.815880i \(-0.303747\pi\)
0.578222 + 0.815880i \(0.303747\pi\)
\(60\) 0 0
\(61\) 34366.3 1.18252 0.591260 0.806481i \(-0.298631\pi\)
0.591260 + 0.806481i \(0.298631\pi\)
\(62\) 0 0
\(63\) 32674.4i 1.03718i
\(64\) 0 0
\(65\) −24804.0 7899.27i −0.728179 0.231902i
\(66\) 0 0
\(67\) 45890.9i 1.24894i 0.781051 + 0.624468i \(0.214684\pi\)
−0.781051 + 0.624468i \(0.785316\pi\)
\(68\) 0 0
\(69\) −118.319 −0.00299180
\(70\) 0 0
\(71\) 58026.5 1.36609 0.683047 0.730375i \(-0.260655\pi\)
0.683047 + 0.730375i \(0.260655\pi\)
\(72\) 0 0
\(73\) 68664.7i 1.50809i −0.656825 0.754043i \(-0.728101\pi\)
0.656825 0.754043i \(-0.271899\pi\)
\(74\) 0 0
\(75\) 17739.5 25026.6i 0.364156 0.513746i
\(76\) 0 0
\(77\) 90838.3i 1.74599i
\(78\) 0 0
\(79\) −65066.0 −1.17297 −0.586484 0.809961i \(-0.699488\pi\)
−0.586484 + 0.809961i \(0.699488\pi\)
\(80\) 0 0
\(81\) −1912.37 −0.0323861
\(82\) 0 0
\(83\) 51017.2i 0.812871i −0.913679 0.406436i \(-0.866772\pi\)
0.913679 0.406436i \(-0.133228\pi\)
\(84\) 0 0
\(85\) −15169.1 4830.87i −0.227726 0.0725233i
\(86\) 0 0
\(87\) 43007.9i 0.609186i
\(88\) 0 0
\(89\) 59409.1 0.795019 0.397510 0.917598i \(-0.369874\pi\)
0.397510 + 0.917598i \(0.369874\pi\)
\(90\) 0 0
\(91\) −103760. −1.31349
\(92\) 0 0
\(93\) 99542.4i 1.19344i
\(94\) 0 0
\(95\) −5495.34 + 17255.5i −0.0624720 + 0.196164i
\(96\) 0 0
\(97\) 73186.2i 0.789768i −0.918731 0.394884i \(-0.870785\pi\)
0.918731 0.394884i \(-0.129215\pi\)
\(98\) 0 0
\(99\) 59781.2 0.613023
\(100\) 0 0
\(101\) −162003. −1.58023 −0.790116 0.612958i \(-0.789979\pi\)
−0.790116 + 0.612958i \(0.789979\pi\)
\(102\) 0 0
\(103\) 23887.3i 0.221857i −0.993828 0.110928i \(-0.964618\pi\)
0.993828 0.110928i \(-0.0353825\pi\)
\(104\) 0 0
\(105\) 37103.8 116507.i 0.328432 1.03129i
\(106\) 0 0
\(107\) 88952.0i 0.751098i 0.926803 + 0.375549i \(0.122546\pi\)
−0.926803 + 0.375549i \(0.877454\pi\)
\(108\) 0 0
\(109\) −73145.1 −0.589683 −0.294842 0.955546i \(-0.595267\pi\)
−0.294842 + 0.955546i \(0.595267\pi\)
\(110\) 0 0
\(111\) 67801.0 0.522310
\(112\) 0 0
\(113\) 162814.i 1.19949i −0.800192 0.599744i \(-0.795269\pi\)
0.800192 0.599744i \(-0.204731\pi\)
\(114\) 0 0
\(115\) 642.029 + 204.466i 0.00452700 + 0.00144170i
\(116\) 0 0
\(117\) 68284.9i 0.461169i
\(118\) 0 0
\(119\) −63455.2 −0.410771
\(120\) 0 0
\(121\) 5147.29 0.0319606
\(122\) 0 0
\(123\) 105367.i 0.627974i
\(124\) 0 0
\(125\) −139507. + 105145.i −0.798583 + 0.601884i
\(126\) 0 0
\(127\) 48080.1i 0.264518i 0.991215 + 0.132259i \(0.0422231\pi\)
−0.991215 + 0.132259i \(0.957777\pi\)
\(128\) 0 0
\(129\) 58921.1 0.311743
\(130\) 0 0
\(131\) 245815. 1.25150 0.625750 0.780024i \(-0.284793\pi\)
0.625750 + 0.780024i \(0.284793\pi\)
\(132\) 0 0
\(133\) 72183.2i 0.353840i
\(134\) 0 0
\(135\) 203733. + 64882.3i 0.962113 + 0.306402i
\(136\) 0 0
\(137\) 188160.i 0.856496i 0.903661 + 0.428248i \(0.140869\pi\)
−0.903661 + 0.428248i \(0.859131\pi\)
\(138\) 0 0
\(139\) −277146. −1.21666 −0.608332 0.793682i \(-0.708161\pi\)
−0.608332 + 0.793682i \(0.708161\pi\)
\(140\) 0 0
\(141\) −245741. −1.04095
\(142\) 0 0
\(143\) 189839.i 0.776330i
\(144\) 0 0
\(145\) 74321.2 233371.i 0.293557 0.921779i
\(146\) 0 0
\(147\) 322389.i 1.23051i
\(148\) 0 0
\(149\) −139067. −0.513167 −0.256584 0.966522i \(-0.582597\pi\)
−0.256584 + 0.966522i \(0.582597\pi\)
\(150\) 0 0
\(151\) −514519. −1.83636 −0.918182 0.396158i \(-0.870343\pi\)
−0.918182 + 0.396158i \(0.870343\pi\)
\(152\) 0 0
\(153\) 41760.2i 0.144223i
\(154\) 0 0
\(155\) −172018. + 540141.i −0.575100 + 1.80583i
\(156\) 0 0
\(157\) 578669.i 1.87362i −0.349840 0.936809i \(-0.613764\pi\)
0.349840 0.936809i \(-0.386236\pi\)
\(158\) 0 0
\(159\) 103766. 0.325509
\(160\) 0 0
\(161\) 2685.73 0.00816578
\(162\) 0 0
\(163\) 269341.i 0.794025i 0.917813 + 0.397012i \(0.129953\pi\)
−0.917813 + 0.397012i \(0.870047\pi\)
\(164\) 0 0
\(165\) 213162. + 67885.4i 0.609538 + 0.194118i
\(166\) 0 0
\(167\) 449114.i 1.24613i −0.782168 0.623067i \(-0.785886\pi\)
0.782168 0.623067i \(-0.214114\pi\)
\(168\) 0 0
\(169\) 154449. 0.415977
\(170\) 0 0
\(171\) −47504.2 −0.124234
\(172\) 0 0
\(173\) 367584.i 0.933772i 0.884317 + 0.466886i \(0.154624\pi\)
−0.884317 + 0.466886i \(0.845376\pi\)
\(174\) 0 0
\(175\) −402669. + 568078.i −0.993923 + 1.40221i
\(176\) 0 0
\(177\) 303531.i 0.728233i
\(178\) 0 0
\(179\) 55367.6 0.129159 0.0645793 0.997913i \(-0.479429\pi\)
0.0645793 + 0.997913i \(0.479429\pi\)
\(180\) 0 0
\(181\) 652653. 1.48076 0.740382 0.672186i \(-0.234645\pi\)
0.740382 + 0.672186i \(0.234645\pi\)
\(182\) 0 0
\(183\) 337351.i 0.744653i
\(184\) 0 0
\(185\) −367904. 117166.i −0.790325 0.251693i
\(186\) 0 0
\(187\) 116098.i 0.242784i
\(188\) 0 0
\(189\) 852252. 1.73546
\(190\) 0 0
\(191\) 307365. 0.609636 0.304818 0.952411i \(-0.401404\pi\)
0.304818 + 0.952411i \(0.401404\pi\)
\(192\) 0 0
\(193\) 91632.5i 0.177075i −0.996073 0.0885373i \(-0.971781\pi\)
0.996073 0.0885373i \(-0.0282193\pi\)
\(194\) 0 0
\(195\) 77541.9 243484.i 0.146033 0.458547i
\(196\) 0 0
\(197\) 256405.i 0.470718i −0.971908 0.235359i \(-0.924373\pi\)
0.971908 0.235359i \(-0.0756266\pi\)
\(198\) 0 0
\(199\) 994011. 1.77934 0.889669 0.456605i \(-0.150935\pi\)
0.889669 + 0.456605i \(0.150935\pi\)
\(200\) 0 0
\(201\) −450481. −0.786477
\(202\) 0 0
\(203\) 976235.i 1.66270i
\(204\) 0 0
\(205\) −182083. + 571746.i −0.302611 + 0.950208i
\(206\) 0 0
\(207\) 1767.49i 0.00286703i
\(208\) 0 0
\(209\) −132067. −0.209136
\(210\) 0 0
\(211\) −81503.0 −0.126028 −0.0630140 0.998013i \(-0.520071\pi\)
−0.0630140 + 0.998013i \(0.520071\pi\)
\(212\) 0 0
\(213\) 569607.i 0.860253i
\(214\) 0 0
\(215\) −319720. 101821.i −0.471709 0.150224i
\(216\) 0 0
\(217\) 2.25951e6i 3.25736i
\(218\) 0 0
\(219\) 674035. 0.949669
\(220\) 0 0
\(221\) −132612. −0.182643
\(222\) 0 0
\(223\) 944128.i 1.27136i −0.771953 0.635680i \(-0.780720\pi\)
0.771953 0.635680i \(-0.219280\pi\)
\(224\) 0 0
\(225\) −373855. 264999.i −0.492320 0.348969i
\(226\) 0 0
\(227\) 225503.i 0.290462i 0.989398 + 0.145231i \(0.0463925\pi\)
−0.989398 + 0.145231i \(0.953608\pi\)
\(228\) 0 0
\(229\) 1.08157e6 1.36290 0.681450 0.731865i \(-0.261350\pi\)
0.681450 + 0.731865i \(0.261350\pi\)
\(230\) 0 0
\(231\) 891699. 1.09948
\(232\) 0 0
\(233\) 768283.i 0.927110i 0.886068 + 0.463555i \(0.153426\pi\)
−0.886068 + 0.463555i \(0.846574\pi\)
\(234\) 0 0
\(235\) 1.33345e6 + 424661.i 1.57510 + 0.501618i
\(236\) 0 0
\(237\) 638709.i 0.738639i
\(238\) 0 0
\(239\) −683119. −0.773573 −0.386787 0.922169i \(-0.626415\pi\)
−0.386787 + 0.922169i \(0.626415\pi\)
\(240\) 0 0
\(241\) −304420. −0.337622 −0.168811 0.985648i \(-0.553993\pi\)
−0.168811 + 0.985648i \(0.553993\pi\)
\(242\) 0 0
\(243\) 910662.i 0.989330i
\(244\) 0 0
\(245\) −557115. + 1.74936e6i −0.592966 + 1.86193i
\(246\) 0 0
\(247\) 150853.i 0.157330i
\(248\) 0 0
\(249\) 500802. 0.511880
\(250\) 0 0
\(251\) −630802. −0.631988 −0.315994 0.948761i \(-0.602338\pi\)
−0.315994 + 0.948761i \(0.602338\pi\)
\(252\) 0 0
\(253\) 4913.83i 0.00482635i
\(254\) 0 0
\(255\) 47421.4 148905.i 0.0456692 0.143403i
\(256\) 0 0
\(257\) 847796.i 0.800679i 0.916367 + 0.400340i \(0.131108\pi\)
−0.916367 + 0.400340i \(0.868892\pi\)
\(258\) 0 0
\(259\) −1.53901e6 −1.42559
\(260\) 0 0
\(261\) 642466. 0.583780
\(262\) 0 0
\(263\) 709631.i 0.632620i 0.948656 + 0.316310i \(0.102444\pi\)
−0.948656 + 0.316310i \(0.897556\pi\)
\(264\) 0 0
\(265\) −563061. 179317.i −0.492539 0.156858i
\(266\) 0 0
\(267\) 583179.i 0.500638i
\(268\) 0 0
\(269\) −424943. −0.358055 −0.179028 0.983844i \(-0.557295\pi\)
−0.179028 + 0.983844i \(0.557295\pi\)
\(270\) 0 0
\(271\) −833724. −0.689603 −0.344801 0.938676i \(-0.612054\pi\)
−0.344801 + 0.938676i \(0.612054\pi\)
\(272\) 0 0
\(273\) 1.01854e6i 0.827126i
\(274\) 0 0
\(275\) −1.03936e6 736725.i −0.828769 0.587454i
\(276\) 0 0
\(277\) 424122.i 0.332117i 0.986116 + 0.166058i \(0.0531040\pi\)
−0.986116 + 0.166058i \(0.946896\pi\)
\(278\) 0 0
\(279\) −1.48700e6 −1.14367
\(280\) 0 0
\(281\) 1.72721e6 1.30491 0.652453 0.757830i \(-0.273740\pi\)
0.652453 + 0.757830i \(0.273740\pi\)
\(282\) 0 0
\(283\) 823843.i 0.611475i −0.952116 0.305737i \(-0.901097\pi\)
0.952116 0.305737i \(-0.0989029\pi\)
\(284\) 0 0
\(285\) −169386. 53944.0i −0.123528 0.0393397i
\(286\) 0 0
\(287\) 2.39172e6i 1.71398i
\(288\) 0 0
\(289\) 1.33876e6 0.942881
\(290\) 0 0
\(291\) 718419. 0.497331
\(292\) 0 0
\(293\) 2.60970e6i 1.77591i 0.459930 + 0.887955i \(0.347875\pi\)
−0.459930 + 0.887955i \(0.652125\pi\)
\(294\) 0 0
\(295\) 524528. 1.64703e6i 0.350924 1.10191i
\(296\) 0 0
\(297\) 1.55929e6i 1.02573i
\(298\) 0 0
\(299\) 5612.80 0.00363080
\(300\) 0 0
\(301\) −1.33745e6 −0.850866
\(302\) 0 0
\(303\) 1.59028e6i 0.995100i
\(304\) 0 0
\(305\) 582971. 1.83055e6i 0.358837 1.12676i
\(306\) 0 0
\(307\) 747194.i 0.452467i 0.974073 + 0.226234i \(0.0726413\pi\)
−0.974073 + 0.226234i \(0.927359\pi\)
\(308\) 0 0
\(309\) 234485. 0.139707
\(310\) 0 0
\(311\) −1.52486e6 −0.893984 −0.446992 0.894538i \(-0.647505\pi\)
−0.446992 + 0.894538i \(0.647505\pi\)
\(312\) 0 0
\(313\) 285672.i 0.164819i 0.996599 + 0.0824094i \(0.0262615\pi\)
−0.996599 + 0.0824094i \(0.973738\pi\)
\(314\) 0 0
\(315\) −1.74043e6 554270.i −0.988278 0.314735i
\(316\) 0 0
\(317\) 3.06761e6i 1.71456i −0.514854 0.857278i \(-0.672154\pi\)
0.514854 0.857278i \(-0.327846\pi\)
\(318\) 0 0
\(319\) 1.78613e6 0.982733
\(320\) 0 0
\(321\) −873182. −0.472980
\(322\) 0 0
\(323\) 92255.4i 0.0492023i
\(324\) 0 0
\(325\) −841522. + 1.18720e6i −0.441933 + 0.623472i
\(326\) 0 0
\(327\) 718016.i 0.371334i
\(328\) 0 0
\(329\) 5.57808e6 2.84116
\(330\) 0 0
\(331\) 2.05250e6 1.02971 0.514854 0.857278i \(-0.327846\pi\)
0.514854 + 0.857278i \(0.327846\pi\)
\(332\) 0 0
\(333\) 1.01283e6i 0.500527i
\(334\) 0 0
\(335\) 2.44442e6 + 778468.i 1.19004 + 0.378991i
\(336\) 0 0
\(337\) 271975.i 0.130453i 0.997870 + 0.0652265i \(0.0207770\pi\)
−0.997870 + 0.0652265i \(0.979223\pi\)
\(338\) 0 0
\(339\) 1.59824e6 0.755339
\(340\) 0 0
\(341\) −4.13402e6 −1.92525
\(342\) 0 0
\(343\) 3.57296e6i 1.63981i
\(344\) 0 0
\(345\) −2007.10 + 6302.37i −0.000907866 + 0.00285073i
\(346\) 0 0
\(347\) 1.43593e6i 0.640192i 0.947385 + 0.320096i \(0.103715\pi\)
−0.947385 + 0.320096i \(0.896285\pi\)
\(348\) 0 0
\(349\) 2.95015e6 1.29653 0.648263 0.761417i \(-0.275496\pi\)
0.648263 + 0.761417i \(0.275496\pi\)
\(350\) 0 0
\(351\) 1.78109e6 0.771645
\(352\) 0 0
\(353\) 3.45504e6i 1.47576i −0.674931 0.737881i \(-0.735827\pi\)
0.674931 0.737881i \(-0.264173\pi\)
\(354\) 0 0
\(355\) 984328. 3.09082e6i 0.414543 1.30168i
\(356\) 0 0
\(357\) 622897.i 0.258670i
\(358\) 0 0
\(359\) −2.38425e6 −0.976372 −0.488186 0.872740i \(-0.662341\pi\)
−0.488186 + 0.872740i \(0.662341\pi\)
\(360\) 0 0
\(361\) −2.37115e6 −0.957617
\(362\) 0 0
\(363\) 50527.4i 0.0201262i
\(364\) 0 0
\(365\) −3.65748e6 1.16479e6i −1.43698 0.457631i
\(366\) 0 0
\(367\) 631924.i 0.244906i −0.992474 0.122453i \(-0.960924\pi\)
0.992474 0.122453i \(-0.0390761\pi\)
\(368\) 0 0
\(369\) −1.57401e6 −0.601784
\(370\) 0 0
\(371\) −2.35539e6 −0.888440
\(372\) 0 0
\(373\) 2.09617e6i 0.780107i 0.920792 + 0.390053i \(0.127543\pi\)
−0.920792 + 0.390053i \(0.872457\pi\)
\(374\) 0 0
\(375\) −1.03214e6 1.36944e6i −0.379017 0.502882i
\(376\) 0 0
\(377\) 2.04020e6i 0.739297i
\(378\) 0 0
\(379\) 15578.4 0.00557088 0.00278544 0.999996i \(-0.499113\pi\)
0.00278544 + 0.999996i \(0.499113\pi\)
\(380\) 0 0
\(381\) −471970. −0.166572
\(382\) 0 0
\(383\) 2.67131e6i 0.930524i −0.885173 0.465262i \(-0.845960\pi\)
0.885173 0.465262i \(-0.154040\pi\)
\(384\) 0 0
\(385\) −4.83857e6 1.54093e6i −1.66366 0.529823i
\(386\) 0 0
\(387\) 880183.i 0.298742i
\(388\) 0 0
\(389\) 2.15353e6 0.721568 0.360784 0.932649i \(-0.382509\pi\)
0.360784 + 0.932649i \(0.382509\pi\)
\(390\) 0 0
\(391\) 3432.56 0.00113547
\(392\) 0 0
\(393\) 2.41300e6i 0.788092i
\(394\) 0 0
\(395\) −1.10374e6 + 3.46579e6i −0.355939 + 1.11766i
\(396\) 0 0
\(397\) 1.79394e6i 0.571256i −0.958341 0.285628i \(-0.907798\pi\)
0.958341 0.285628i \(-0.0922021\pi\)
\(398\) 0 0
\(399\) −708574. −0.222819
\(400\) 0 0
\(401\) −1.26450e6 −0.392696 −0.196348 0.980534i \(-0.562908\pi\)
−0.196348 + 0.980534i \(0.562908\pi\)
\(402\) 0 0
\(403\) 4.72207e6i 1.44834i
\(404\) 0 0
\(405\) −32440.3 + 101864.i −0.00982760 + 0.0308590i
\(406\) 0 0
\(407\) 2.81579e6i 0.842586i
\(408\) 0 0
\(409\) −476802. −0.140939 −0.0704693 0.997514i \(-0.522450\pi\)
−0.0704693 + 0.997514i \(0.522450\pi\)
\(410\) 0 0
\(411\) −1.84704e6 −0.539351
\(412\) 0 0
\(413\) 6.88986e6i 1.98763i
\(414\) 0 0
\(415\) −2.71747e6 865428.i −0.774542 0.246667i
\(416\) 0 0
\(417\) 2.72055e6i 0.766156i
\(418\) 0 0
\(419\) 4.75187e6 1.32230 0.661149 0.750255i \(-0.270069\pi\)
0.661149 + 0.750255i \(0.270069\pi\)
\(420\) 0 0
\(421\) 2.82485e6 0.776765 0.388383 0.921498i \(-0.373034\pi\)
0.388383 + 0.921498i \(0.373034\pi\)
\(422\) 0 0
\(423\) 3.67097e6i 0.997538i
\(424\) 0 0
\(425\) −514640. + 726045.i −0.138207 + 0.194980i
\(426\) 0 0
\(427\) 7.65753e6i 2.03245i
\(428\) 0 0
\(429\) 1.86353e6 0.488869
\(430\) 0 0
\(431\) 2.09727e6 0.543827 0.271914 0.962322i \(-0.412344\pi\)
0.271914 + 0.962322i \(0.412344\pi\)
\(432\) 0 0
\(433\) 2.22584e6i 0.570524i −0.958450 0.285262i \(-0.907919\pi\)
0.958450 0.285262i \(-0.0920806\pi\)
\(434\) 0 0
\(435\) 2.29085e6 + 729561.i 0.580461 + 0.184858i
\(436\) 0 0
\(437\) 3904.69i 0.000978100i
\(438\) 0 0
\(439\) 4.60019e6 1.13924 0.569618 0.821909i \(-0.307091\pi\)
0.569618 + 0.821909i \(0.307091\pi\)
\(440\) 0 0
\(441\) −4.81596e6 −1.17920
\(442\) 0 0
\(443\) 95739.0i 0.0231782i −0.999933 0.0115891i \(-0.996311\pi\)
0.999933 0.0115891i \(-0.00368901\pi\)
\(444\) 0 0
\(445\) 1.00778e6 3.16447e6i 0.241250 0.757532i
\(446\) 0 0
\(447\) 1.36513e6i 0.323150i
\(448\) 0 0
\(449\) 6.03855e6 1.41357 0.706784 0.707430i \(-0.250145\pi\)
0.706784 + 0.707430i \(0.250145\pi\)
\(450\) 0 0
\(451\) −4.37591e6 −1.01304
\(452\) 0 0
\(453\) 5.05069e6i 1.15639i
\(454\) 0 0
\(455\) −1.76012e6 + 5.52684e6i −0.398579 + 1.25155i
\(456\) 0 0
\(457\) 2.79696e6i 0.626464i 0.949677 + 0.313232i \(0.101412\pi\)
−0.949677 + 0.313232i \(0.898588\pi\)
\(458\) 0 0
\(459\) 1.08924e6 0.241319
\(460\) 0 0
\(461\) −1.65757e6 −0.363261 −0.181631 0.983367i \(-0.558138\pi\)
−0.181631 + 0.983367i \(0.558138\pi\)
\(462\) 0 0
\(463\) 6.11162e6i 1.32496i 0.749078 + 0.662481i \(0.230497\pi\)
−0.749078 + 0.662481i \(0.769503\pi\)
\(464\) 0 0
\(465\) −5.30220e6 1.68858e6i −1.13717 0.362151i
\(466\) 0 0
\(467\) 3.00858e6i 0.638366i −0.947693 0.319183i \(-0.896591\pi\)
0.947693 0.319183i \(-0.103409\pi\)
\(468\) 0 0
\(469\) 1.02255e7 2.14660
\(470\) 0 0
\(471\) 5.68041e6 1.17985
\(472\) 0 0
\(473\) 2.44701e6i 0.502901i
\(474\) 0 0
\(475\) 825910. + 585427.i 0.167957 + 0.119052i
\(476\) 0 0
\(477\) 1.55010e6i 0.311934i
\(478\) 0 0
\(479\) 233940. 0.0465872 0.0232936 0.999729i \(-0.492585\pi\)
0.0232936 + 0.999729i \(0.492585\pi\)
\(480\) 0 0
\(481\) −3.21633e6 −0.633866
\(482\) 0 0
\(483\) 26364.0i 0.00514214i
\(484\) 0 0
\(485\) −3.89832e6 1.24149e6i −0.752528 0.239656i
\(486\) 0 0
\(487\) 4.85784e6i 0.928156i −0.885794 0.464078i \(-0.846386\pi\)
0.885794 0.464078i \(-0.153614\pi\)
\(488\) 0 0
\(489\) −2.64394e6 −0.500011
\(490\) 0 0
\(491\) 7.32174e6 1.37060 0.685300 0.728261i \(-0.259671\pi\)
0.685300 + 0.728261i \(0.259671\pi\)
\(492\) 0 0
\(493\) 1.24770e6i 0.231203i
\(494\) 0 0
\(495\) 1.01410e6 3.18429e6i 0.186023 0.584117i
\(496\) 0 0
\(497\) 1.29295e7i 2.34796i
\(498\) 0 0
\(499\) −8.86841e6 −1.59439 −0.797195 0.603722i \(-0.793684\pi\)
−0.797195 + 0.603722i \(0.793684\pi\)
\(500\) 0 0
\(501\) 4.40865e6 0.784713
\(502\) 0 0
\(503\) 7.27944e6i 1.28286i −0.767183 0.641428i \(-0.778342\pi\)
0.767183 0.641428i \(-0.221658\pi\)
\(504\) 0 0
\(505\) −2.74813e6 + 8.62923e6i −0.479523 + 1.50572i
\(506\) 0 0
\(507\) 1.51613e6i 0.261948i
\(508\) 0 0
\(509\) −598060. −0.102318 −0.0511588 0.998691i \(-0.516291\pi\)
−0.0511588 + 0.998691i \(0.516291\pi\)
\(510\) 0 0
\(511\) −1.52999e7 −2.59201
\(512\) 0 0
\(513\) 1.23906e6i 0.207874i
\(514\) 0 0
\(515\) −1.27237e6 405210.i −0.211396 0.0673228i
\(516\) 0 0
\(517\) 1.02057e7i 1.67925i
\(518\) 0 0
\(519\) −3.60832e6 −0.588013
\(520\) 0 0
\(521\) 4.12658e6 0.666034 0.333017 0.942921i \(-0.391933\pi\)
0.333017 + 0.942921i \(0.391933\pi\)
\(522\) 0 0
\(523\) 5.94827e6i 0.950903i 0.879742 + 0.475451i \(0.157715\pi\)
−0.879742 + 0.475451i \(0.842285\pi\)
\(524\) 0 0
\(525\) −5.57644e6 3.95273e6i −0.882996 0.625891i
\(526\) 0 0
\(527\) 2.88782e6i 0.452943i
\(528\) 0 0
\(529\) 6.43620e6 0.999977
\(530\) 0 0
\(531\) 4.53426e6 0.697862
\(532\) 0 0
\(533\) 4.99837e6i 0.762097i
\(534\) 0 0
\(535\) 4.73810e6 + 1.50893e6i 0.715681 + 0.227921i
\(536\) 0 0
\(537\) 543506.i 0.0813334i
\(538\) 0 0
\(539\) −1.33889e7 −1.98505
\(540\) 0 0
\(541\) −8.27787e6 −1.21598 −0.607989 0.793946i \(-0.708023\pi\)
−0.607989 + 0.793946i \(0.708023\pi\)
\(542\) 0 0
\(543\) 6.40666e6i 0.932464i
\(544\) 0 0
\(545\) −1.24079e6 + 3.89613e6i −0.178940 + 0.561878i
\(546\) 0 0
\(547\) 1.18115e7i 1.68786i 0.536453 + 0.843930i \(0.319764\pi\)
−0.536453 + 0.843930i \(0.680236\pi\)
\(548\) 0 0
\(549\) 5.03946e6 0.713598
\(550\) 0 0
\(551\) −1.41932e6 −0.199159
\(552\) 0 0
\(553\) 1.44981e7i 2.01603i
\(554\) 0 0
\(555\) 1.15014e6 3.61147e6i 0.158496 0.497682i
\(556\) 0 0
\(557\) 7.81776e6i 1.06769i 0.845583 + 0.533844i \(0.179253\pi\)
−0.845583 + 0.533844i \(0.820747\pi\)
\(558\) 0 0
\(559\) −2.79508e6 −0.378325
\(560\) 0 0
\(561\) 1.13966e6 0.152886
\(562\) 0 0
\(563\) 1.16975e7i 1.55533i −0.628677 0.777666i \(-0.716403\pi\)
0.628677 0.777666i \(-0.283597\pi\)
\(564\) 0 0
\(565\) −8.67242e6 2.76189e6i −1.14293 0.363986i
\(566\) 0 0
\(567\) 426115.i 0.0556634i
\(568\) 0 0
\(569\) 1.48869e7 1.92763 0.963813 0.266578i \(-0.0858931\pi\)
0.963813 + 0.266578i \(0.0858931\pi\)
\(570\) 0 0
\(571\) −995656. −0.127797 −0.0638983 0.997956i \(-0.520353\pi\)
−0.0638983 + 0.997956i \(0.520353\pi\)
\(572\) 0 0
\(573\) 3.01719e6i 0.383899i
\(574\) 0 0
\(575\) 21782.0 30729.7i 0.00274745 0.00387605i
\(576\) 0 0
\(577\) 1.00279e6i 0.125392i −0.998033 0.0626961i \(-0.980030\pi\)
0.998033 0.0626961i \(-0.0199699\pi\)
\(578\) 0 0
\(579\) 899495. 0.111507
\(580\) 0 0
\(581\) −1.13677e7 −1.39712
\(582\) 0 0
\(583\) 4.30944e6i 0.525109i
\(584\) 0 0
\(585\) −3.63725e6 1.15835e6i −0.439423 0.139942i
\(586\) 0 0
\(587\) 3.54906e6i 0.425127i −0.977147 0.212563i \(-0.931819\pi\)
0.977147 0.212563i \(-0.0681812\pi\)
\(588\) 0 0
\(589\) 3.28503e6 0.390168
\(590\) 0 0
\(591\) 2.51696e6 0.296420
\(592\) 0 0
\(593\) 4.20405e6i 0.490944i 0.969404 + 0.245472i \(0.0789429\pi\)
−0.969404 + 0.245472i \(0.921057\pi\)
\(594\) 0 0
\(595\) −1.07642e6 + 3.37999e6i −0.124649 + 0.391402i
\(596\) 0 0
\(597\) 9.75754e6i 1.12048i
\(598\) 0 0
\(599\) 1.21200e7 1.38017 0.690087 0.723726i \(-0.257572\pi\)
0.690087 + 0.723726i \(0.257572\pi\)
\(600\) 0 0
\(601\) 1.15778e6 0.130749 0.0653745 0.997861i \(-0.479176\pi\)
0.0653745 + 0.997861i \(0.479176\pi\)
\(602\) 0 0
\(603\) 6.72943e6i 0.753677i
\(604\) 0 0
\(605\) 87315.7 274174.i 0.00969848 0.0304536i
\(606\) 0 0
\(607\) 6.62196e6i 0.729482i 0.931109 + 0.364741i \(0.118842\pi\)
−0.931109 + 0.364741i \(0.881158\pi\)
\(608\) 0 0
\(609\) 9.58305e6 1.04703
\(610\) 0 0
\(611\) 1.16574e7 1.26328
\(612\) 0 0
\(613\) 8.15388e6i 0.876421i 0.898872 + 0.438211i \(0.144388\pi\)
−0.898872 + 0.438211i \(0.855612\pi\)
\(614\) 0 0
\(615\) −5.61245e6 1.78739e6i −0.598363 0.190559i
\(616\) 0 0
\(617\) 8.07144e6i 0.853568i −0.904353 0.426784i \(-0.859646\pi\)
0.904353 0.426784i \(-0.140354\pi\)
\(618\) 0 0
\(619\) 1.78058e7 1.86782 0.933910 0.357509i \(-0.116374\pi\)
0.933910 + 0.357509i \(0.116374\pi\)
\(620\) 0 0
\(621\) −46101.9 −0.00479722
\(622\) 0 0
\(623\) 1.32376e7i 1.36643i
\(624\) 0 0
\(625\) 3.23411e6 + 9.21455e6i 0.331173 + 0.943570i
\(626\) 0 0
\(627\) 1.29641e6i 0.131696i
\(628\) 0 0
\(629\) −1.96697e6 −0.198231
\(630\) 0 0
\(631\) 5.93229e6 0.593129 0.296564 0.955013i \(-0.404159\pi\)
0.296564 + 0.955013i \(0.404159\pi\)
\(632\) 0 0
\(633\) 800060.i 0.0793621i
\(634\) 0 0
\(635\) 2.56102e6 + 815604.i 0.252046 + 0.0802684i
\(636\) 0 0
\(637\) 1.52934e7i 1.49333i
\(638\) 0 0
\(639\) 8.50898e6 0.824376
\(640\) 0 0
\(641\) 584130. 0.0561519 0.0280760 0.999606i \(-0.491062\pi\)
0.0280760 + 0.999606i \(0.491062\pi\)
\(642\) 0 0
\(643\) 1.65155e7i 1.57530i −0.616123 0.787650i \(-0.711298\pi\)
0.616123 0.787650i \(-0.288702\pi\)
\(644\) 0 0
\(645\) 999504. 3.13848e6i 0.0945988 0.297043i
\(646\) 0 0
\(647\) 1.35366e7i 1.27130i 0.771976 + 0.635652i \(0.219269\pi\)
−0.771976 + 0.635652i \(0.780731\pi\)
\(648\) 0 0
\(649\) 1.26057e7 1.17478
\(650\) 0 0
\(651\) −2.21801e7 −2.05122
\(652\) 0 0
\(653\) 8.32959e6i 0.764435i −0.924072 0.382218i \(-0.875160\pi\)
0.924072 0.382218i \(-0.124840\pi\)
\(654\) 0 0
\(655\) 4.16987e6 1.30935e7i 0.379769 1.19249i
\(656\) 0 0
\(657\) 1.00690e7i 0.910063i
\(658\) 0 0
\(659\) 6.89776e6 0.618720 0.309360 0.950945i \(-0.399885\pi\)
0.309360 + 0.950945i \(0.399885\pi\)
\(660\) 0 0
\(661\) 1.00381e7 0.893606 0.446803 0.894632i \(-0.352562\pi\)
0.446803 + 0.894632i \(0.352562\pi\)
\(662\) 0 0
\(663\) 1.30177e6i 0.115014i
\(664\) 0 0
\(665\) 3.84489e6 + 1.22448e6i 0.337156 + 0.107373i
\(666\) 0 0
\(667\) 52808.7i 0.00459611i
\(668\) 0 0
\(669\) 9.26787e6 0.800598
\(670\) 0 0
\(671\) 1.40103e7 1.20127
\(672\) 0 0
\(673\) 1.76094e7i 1.49867i 0.662190 + 0.749336i \(0.269627\pi\)
−0.662190 + 0.749336i \(0.730373\pi\)
\(674\) 0 0
\(675\) 6.91201e6 9.75134e6i 0.583909 0.823768i
\(676\) 0 0
\(677\) 1.55786e7i 1.30634i 0.757209 + 0.653172i \(0.226562\pi\)
−0.757209 + 0.653172i \(0.773438\pi\)
\(678\) 0 0
\(679\) −1.63074e7 −1.35741
\(680\) 0 0
\(681\) −2.21362e6 −0.182909
\(682\) 0 0
\(683\) 2.77212e6i 0.227384i −0.993516 0.113692i \(-0.963732\pi\)
0.993516 0.113692i \(-0.0362677\pi\)
\(684\) 0 0
\(685\) 1.00225e7 + 3.19184e6i 0.816109 + 0.259905i
\(686\) 0 0
\(687\) 1.06170e7i 0.858242i
\(688\) 0 0
\(689\) −4.92244e6 −0.395032
\(690\) 0 0
\(691\) −5.49060e6 −0.437446 −0.218723 0.975787i \(-0.570189\pi\)
−0.218723 + 0.975787i \(0.570189\pi\)
\(692\) 0 0
\(693\) 1.33205e7i 1.05363i
\(694\) 0 0
\(695\) −4.70134e6 + 1.47624e7i −0.369198 + 1.15930i
\(696\) 0 0
\(697\) 3.05680e6i 0.238333i
\(698\) 0 0
\(699\) −7.54172e6 −0.583818
\(700\) 0 0
\(701\) −1.01658e7 −0.781351 −0.390676 0.920528i \(-0.627759\pi\)
−0.390676 + 0.920528i \(0.627759\pi\)
\(702\) 0 0
\(703\) 2.23752e6i 0.170757i
\(704\) 0 0
\(705\) −4.16862e6 + 1.30896e7i −0.315878 + 0.991867i
\(706\) 0 0
\(707\) 3.60977e7i 2.71601i
\(708\) 0 0
\(709\) 1.01024e7 0.754761 0.377380 0.926058i \(-0.376825\pi\)
0.377380 + 0.926058i \(0.376825\pi\)
\(710\) 0 0
\(711\) −9.54125e6 −0.707834
\(712\) 0 0
\(713\) 122227.i 0.00900413i
\(714\) 0 0
\(715\) −1.01119e7 3.22033e6i −0.739724 0.235578i
\(716\) 0 0
\(717\) 6.70572e6i 0.487133i
\(718\) 0 0
\(719\) −1.48567e7 −1.07176 −0.535882 0.844293i \(-0.680021\pi\)
−0.535882 + 0.844293i \(0.680021\pi\)
\(720\) 0 0
\(721\) −5.32258e6 −0.381315
\(722\) 0 0
\(723\) 2.98829e6i 0.212607i
\(724\) 0 0
\(725\) −1.11699e7 7.91755e6i −0.789234 0.559430i
\(726\) 0 0
\(727\) 1.24685e7i 0.874939i −0.899233 0.437469i \(-0.855875\pi\)
0.899233 0.437469i \(-0.144125\pi\)
\(728\) 0 0
\(729\) −9.40406e6 −0.655385
\(730\) 0 0
\(731\) −1.70936e6 −0.118315
\(732\) 0 0
\(733\) 2.01823e7i 1.38743i 0.720251 + 0.693713i \(0.244027\pi\)
−0.720251 + 0.693713i \(0.755973\pi\)
\(734\) 0 0
\(735\) −1.71723e7 5.46883e6i −1.17249 0.373401i
\(736\) 0 0
\(737\) 1.87086e7i 1.26874i
\(738\) 0 0
\(739\) 1.18948e7 0.801209 0.400605 0.916251i \(-0.368800\pi\)
0.400605 + 0.916251i \(0.368800\pi\)
\(740\) 0 0
\(741\) −1.48082e6 −0.0990735
\(742\) 0 0
\(743\) 1.70571e7i 1.13353i −0.823879 0.566766i \(-0.808195\pi\)
0.823879 0.566766i \(-0.191805\pi\)
\(744\) 0 0
\(745\) −2.35906e6 + 7.40752e6i −0.155721 + 0.488970i
\(746\) 0 0
\(747\) 7.48115e6i 0.490532i
\(748\) 0 0
\(749\) 1.98204e7 1.29094
\(750\) 0 0
\(751\) 1.69025e7 1.09358 0.546792 0.837268i \(-0.315849\pi\)
0.546792 + 0.837268i \(0.315849\pi\)
\(752\) 0 0
\(753\) 6.19216e6i 0.397974i
\(754\) 0 0
\(755\) −8.72801e6 + 2.74062e7i −0.557247 + 1.74977i
\(756\) 0 0
\(757\) 2.47458e6i 0.156950i 0.996916 + 0.0784750i \(0.0250051\pi\)
−0.996916 + 0.0784750i \(0.974995\pi\)
\(758\) 0 0
\(759\) −48235.7 −0.00303924
\(760\) 0 0
\(761\) −2.38180e7 −1.49088 −0.745442 0.666570i \(-0.767762\pi\)
−0.745442 + 0.666570i \(0.767762\pi\)
\(762\) 0 0
\(763\) 1.62982e7i 1.01351i
\(764\) 0 0
\(765\) −2.22439e6 708397.i −0.137422 0.0437646i
\(766\) 0 0
\(767\) 1.43989e7i 0.883770i
\(768\) 0 0
\(769\) −6.78796e6 −0.413927 −0.206963 0.978349i \(-0.566358\pi\)
−0.206963 + 0.978349i \(0.566358\pi\)
\(770\) 0 0
\(771\) −8.32224e6 −0.504202
\(772\) 0 0
\(773\) 1.43751e7i 0.865291i −0.901564 0.432646i \(-0.857580\pi\)
0.901564 0.432646i \(-0.142420\pi\)
\(774\) 0 0
\(775\) 2.58530e7 + 1.83253e7i 1.54617 + 1.09597i
\(776\) 0 0
\(777\) 1.51075e7i 0.897716i
\(778\) 0 0
\(779\) 3.47725e6 0.205301
\(780\) 0 0
\(781\) 2.36559e7 1.38775
\(782\) 0 0
\(783\) 1.67576e7i 0.976802i
\(784\) 0 0
\(785\) −3.08233e7 9.81622e6i −1.78527 0.568552i
\(786\) 0 0
\(787\) 2.78490e7i 1.60277i −0.598146 0.801387i \(-0.704096\pi\)
0.598146 0.801387i \(-0.295904\pi\)
\(788\) 0 0
\(789\) −6.96597e6 −0.398372
\(790\) 0 0
\(791\) −3.62784e7 −2.06161
\(792\) 0 0
\(793\) 1.60032e7i 0.903697i
\(794\) 0 0
\(795\) 1.76023e6 5.52719e6i 0.0987762 0.310161i
\(796\) 0 0
\(797\) 3.40150e6i 0.189681i −0.995492 0.0948406i \(-0.969766\pi\)
0.995492 0.0948406i \(-0.0302341\pi\)
\(798\) 0 0
\(799\) 7.12919e6 0.395069
\(800\) 0 0
\(801\) 8.71173e6 0.479759
\(802\) 0 0
\(803\) 2.79928e7i 1.53200i
\(804\) 0 0
\(805\) 45559.2 143057.i 0.00247792 0.00778074i
\(806\) 0 0
\(807\) 4.17138e6i 0.225474i
\(808\) 0 0
\(809\) −1.06065e7 −0.569773 −0.284887 0.958561i \(-0.591956\pi\)
−0.284887 + 0.958561i \(0.591956\pi\)
\(810\) 0 0
\(811\) 3.49545e7 1.86617 0.933084 0.359658i \(-0.117107\pi\)
0.933084 + 0.359658i \(0.117107\pi\)
\(812\) 0 0
\(813\) 8.18411e6i 0.434255i
\(814\) 0 0
\(815\) 1.43467e7 + 4.56895e6i 0.756584 + 0.240948i
\(816\) 0 0
\(817\) 1.94447e6i 0.101917i
\(818\) 0 0
\(819\) −1.52153e7 −0.792630
\(820\) 0 0
\(821\) 1.06256e7 0.550170 0.275085 0.961420i \(-0.411294\pi\)
0.275085 + 0.961420i \(0.411294\pi\)
\(822\) 0 0
\(823\) 2.25239e6i 0.115916i 0.998319 + 0.0579582i \(0.0184590\pi\)
−0.998319 + 0.0579582i \(0.981541\pi\)
\(824\) 0 0
\(825\) 7.23193e6 1.02027e7i 0.369930 0.521891i
\(826\) 0 0
\(827\) 2.38273e7i 1.21147i −0.795668 0.605733i \(-0.792880\pi\)
0.795668 0.605733i \(-0.207120\pi\)
\(828\) 0 0
\(829\) −885468. −0.0447493 −0.0223747 0.999750i \(-0.507123\pi\)
−0.0223747 + 0.999750i \(0.507123\pi\)
\(830\) 0 0
\(831\) −4.16332e6 −0.209140
\(832\) 0 0
\(833\) 9.35281e6i 0.467014i
\(834\) 0 0
\(835\) −2.39224e7 7.61851e6i −1.18738 0.378141i
\(836\) 0 0
\(837\) 3.87857e7i 1.91363i
\(838\) 0 0
\(839\) 2.09943e7 1.02966 0.514832 0.857291i \(-0.327854\pi\)
0.514832 + 0.857291i \(0.327854\pi\)
\(840\) 0 0
\(841\) −1.31573e6 −0.0641470
\(842\) 0 0
\(843\) 1.69548e7i 0.821722i
\(844\) 0 0
\(845\) 2.61999e6 8.22687e6i 0.126229 0.396363i
\(846\) 0 0
\(847\) 1.14692e6i 0.0549320i
\(848\) 0 0
\(849\) 8.08712e6 0.385056
\(850\) 0 0
\(851\) 83251.8 0.00394067
\(852\) 0 0
\(853\) 342279.i 0.0161067i −0.999968 0.00805336i \(-0.997437\pi\)
0.999968 0.00805336i \(-0.00256349\pi\)
\(854\) 0 0
\(855\) −805834. + 2.53035e6i −0.0376991 + 0.118376i
\(856\) 0 0
\(857\) 2.13031e7i 0.990811i −0.868662 0.495405i \(-0.835020\pi\)
0.868662 0.495405i \(-0.164980\pi\)
\(858\) 0 0
\(859\) 1.20032e7 0.555028 0.277514 0.960722i \(-0.410490\pi\)
0.277514 + 0.960722i \(0.410490\pi\)
\(860\) 0 0
\(861\) −2.34779e7 −1.07932
\(862\) 0 0
\(863\) 1.11350e7i 0.508937i 0.967081 + 0.254469i \(0.0819005\pi\)
−0.967081 + 0.254469i \(0.918099\pi\)
\(864\) 0 0
\(865\) 1.95796e7 + 6.23548e6i 0.889742 + 0.283354i
\(866\) 0 0
\(867\) 1.31417e7i 0.593749i
\(868\) 0 0
\(869\) −2.65257e7 −1.19157
\(870\) 0 0
\(871\) 2.13698e7 0.954454
\(872\) 0 0
\(873\) 1.07320e7i 0.476590i
\(874\) 0 0
\(875\) 2.34285e7 + 3.10850e7i 1.03448 + 1.37256i
\(876\) 0 0
\(877\) 3.41443e6i 0.149906i 0.997187 + 0.0749530i \(0.0238807\pi\)
−0.997187 + 0.0749530i \(0.976119\pi\)
\(878\) 0 0
\(879\) −2.56177e7 −1.11832
\(880\) 0 0
\(881\) 1.89727e7 0.823548 0.411774 0.911286i \(-0.364909\pi\)
0.411774 + 0.911286i \(0.364909\pi\)
\(882\) 0 0
\(883\) 3.46909e7i 1.49732i 0.662957 + 0.748658i \(0.269301\pi\)
−0.662957 + 0.748658i \(0.730699\pi\)
\(884\) 0 0
\(885\) 1.61678e7 + 5.14894e6i 0.693895 + 0.220983i
\(886\) 0 0
\(887\) 4.95749e6i 0.211569i −0.994389 0.105785i \(-0.966265\pi\)
0.994389 0.105785i \(-0.0337354\pi\)
\(888\) 0 0
\(889\) 1.07132e7 0.454639
\(890\) 0 0
\(891\) −779623. −0.0328996
\(892\) 0 0
\(893\) 8.10979e6i 0.340315i
\(894\) 0 0
\(895\) 939225. 2.94920e6i 0.0391933 0.123068i
\(896\) 0 0
\(897\) 55097.1i 0.00228638i
\(898\) 0 0
\(899\) −4.44281e7 −1.83341
\(900\) 0 0
\(901\) −3.01036e6 −0.123540
\(902\) 0 0
\(903\) 1.31288e7i 0.535806i
\(904\) 0 0
\(905\) 1.10712e7 3.47641e7i 0.449340 1.41094i
\(906\) 0 0
\(907\) 2.33839e7i 0.943841i −0.881641 0.471921i \(-0.843561\pi\)
0.881641 0.471921i \(-0.156439\pi\)
\(908\) 0 0
\(909\) −2.37561e7 −0.953599
\(910\) 0 0
\(911\) −2.03543e7 −0.812569 −0.406285 0.913747i \(-0.633176\pi\)
−0.406285 + 0.913747i \(0.633176\pi\)
\(912\) 0 0
\(913\) 2.07984e7i 0.825759i
\(914\) 0 0
\(915\) 1.79693e7 + 5.72263e6i 0.709541 + 0.225966i
\(916\) 0 0
\(917\) 5.47728e7i 2.15101i
\(918\) 0 0
\(919\) −2.90338e7 −1.13401 −0.567003 0.823716i \(-0.691897\pi\)
−0.567003 + 0.823716i \(0.691897\pi\)
\(920\) 0 0
\(921\) −7.33470e6 −0.284927
\(922\) 0 0
\(923\) 2.70209e7i 1.04399i
\(924\) 0 0
\(925\) −1.24818e7 + 1.76092e7i −0.479650 + 0.676682i
\(926\) 0 0
\(927\) 3.50282e6i 0.133881i
\(928\) 0 0
\(929\) −3.90013e7 −1.48266 −0.741328 0.671143i \(-0.765804\pi\)
−0.741328 + 0.671143i \(0.765804\pi\)
\(930\) 0 0
\(931\) 1.06393e7 0.402288
\(932\) 0 0
\(933\) 1.49686e7i 0.562958i
\(934\) 0 0
\(935\) −6.18405e6 1.96942e6i −0.231336 0.0736732i
\(936\) 0 0
\(937\) 2.32825e7i 0.866324i 0.901316 + 0.433162i \(0.142602\pi\)
−0.901316 + 0.433162i \(0.857398\pi\)
\(938\) 0 0
\(939\) −2.80425e6 −0.103789
\(940\) 0 0
\(941\) 3.90746e6 0.143853 0.0719267 0.997410i \(-0.477085\pi\)
0.0719267 + 0.997410i \(0.477085\pi\)
\(942\) 0 0
\(943\) 129378.i 0.00473786i
\(944\) 0 0
\(945\) 1.44571e7 4.53959e7i 0.526626 1.65362i
\(946\) 0 0
\(947\) 1.52048e7i 0.550943i −0.961309 0.275471i \(-0.911166\pi\)
0.961309 0.275471i \(-0.0888339\pi\)
\(948\) 0 0
\(949\) −3.19747e7 −1.15250
\(950\) 0 0
\(951\) 3.01127e7 1.07969
\(952\) 0 0
\(953\) 3.60102e7i 1.28438i −0.766545 0.642190i \(-0.778026\pi\)
0.766545 0.642190i \(-0.221974\pi\)
\(954\) 0 0
\(955\) 5.21396e6 1.63720e7i 0.184995 0.580890i
\(956\) 0 0
\(957\) 1.75332e7i 0.618844i
\(958\) 0 0
\(959\) 4.19259e7 1.47210
\(960\) 0 0
\(961\) 7.42004e7 2.59178
\(962\) 0 0
\(963\) 1.30439e7i 0.453254i
\(964\) 0 0
\(965\) −4.88088e6 1.55440e6i −0.168725 0.0537335i
\(966\) 0 0
\(967\) 2.39347e7i 0.823119i −0.911383 0.411559i \(-0.864984\pi\)
0.911383 0.411559i \(-0.135016\pi\)
\(968\) 0 0
\(969\) −905609. −0.0309836
\(970\) 0 0
\(971\) −2.87801e7 −0.979589 −0.489794 0.871838i \(-0.662928\pi\)
−0.489794 + 0.871838i \(0.662928\pi\)
\(972\) 0 0
\(973\) 6.17538e7i 2.09113i
\(974\) 0 0
\(975\) −1.16540e7 8.26065e6i −0.392612 0.278293i
\(976\) 0 0
\(977\) 1.96671e7i 0.659181i 0.944124 + 0.329590i \(0.106911\pi\)
−0.944124 + 0.329590i \(0.893089\pi\)
\(978\) 0 0
\(979\) 2.42195e7 0.807624
\(980\) 0 0
\(981\) −1.07260e7 −0.355848
\(982\) 0 0
\(983\) 4.03974e7i 1.33343i 0.745314 + 0.666713i \(0.232300\pi\)
−0.745314 + 0.666713i \(0.767700\pi\)
\(984\) 0 0
\(985\) −1.36576e7 4.34951e6i −0.448522 0.142840i
\(986\) 0 0
\(987\) 5.47563e7i 1.78913i
\(988\) 0 0
\(989\) 72348.3 0.00235200
\(990\) 0 0
\(991\) 1.08921e7 0.352313 0.176156 0.984362i \(-0.443634\pi\)
0.176156 + 0.984362i \(0.443634\pi\)
\(992\) 0 0
\(993\) 2.01481e7i 0.648425i
\(994\) 0 0
\(995\) 1.68618e7 5.29468e7i 0.539943 1.69544i
\(996\) 0 0
\(997\) 3.25183e7i 1.03607i −0.855359 0.518035i \(-0.826664\pi\)
0.855359 0.518035i \(-0.173336\pi\)
\(998\) 0 0
\(999\) 2.64180e7 0.837501
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 320.6.c.k.129.7 12
4.3 odd 2 inner 320.6.c.k.129.5 12
5.4 even 2 inner 320.6.c.k.129.6 12
8.3 odd 2 160.6.c.c.129.8 yes 12
8.5 even 2 160.6.c.c.129.6 yes 12
20.19 odd 2 inner 320.6.c.k.129.8 12
40.3 even 4 800.6.a.ba.1.3 6
40.13 odd 4 800.6.a.ba.1.4 6
40.19 odd 2 160.6.c.c.129.5 12
40.27 even 4 800.6.a.z.1.4 6
40.29 even 2 160.6.c.c.129.7 yes 12
40.37 odd 4 800.6.a.z.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.c.c.129.5 12 40.19 odd 2
160.6.c.c.129.6 yes 12 8.5 even 2
160.6.c.c.129.7 yes 12 40.29 even 2
160.6.c.c.129.8 yes 12 8.3 odd 2
320.6.c.k.129.5 12 4.3 odd 2 inner
320.6.c.k.129.6 12 5.4 even 2 inner
320.6.c.k.129.7 12 1.1 even 1 trivial
320.6.c.k.129.8 12 20.19 odd 2 inner
800.6.a.z.1.3 6 40.37 odd 4
800.6.a.z.1.4 6 40.27 even 4
800.6.a.ba.1.3 6 40.3 even 4
800.6.a.ba.1.4 6 40.13 odd 4