Properties

Label 432.7.e.k.161.3
Level $432$
Weight $7$
Character 432.161
Analytic conductor $99.383$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,7,Mod(161,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 432.e (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: \(99.3833641238\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 75x^{4} - 90x^{3} + 1861x^{2} + 7864x + 10098 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(7.70376 - 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 432.161
Dual form 432.7.e.k.161.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-62.7833i q^{5} +496.882 q^{7} +O(q^{10})\) \(q-62.7833i q^{5} +496.882 q^{7} -1437.06i q^{11} +3905.03 q^{13} +8385.53i q^{17} +3625.19 q^{19} +24070.3i q^{23} +11683.3 q^{25} -38898.5i q^{29} +10728.8 q^{31} -31195.9i q^{35} -45162.5 q^{37} +35120.1i q^{41} +21584.6 q^{43} +103509. i q^{47} +129242. q^{49} -70152.9i q^{53} -90223.6 q^{55} +185774. i q^{59} -301079. q^{61} -245171. i q^{65} +130275. q^{67} +464741. i q^{71} +274352. q^{73} -714051. i q^{77} -705830. q^{79} +233654. i q^{83} +526471. q^{85} -1.16112e6i q^{89} +1.94034e6 q^{91} -227602. i q^{95} +129972. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 114 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 114 q^{7} + 4026 q^{13} - 1854 q^{19} + 7686 q^{25} - 708 q^{31} - 66198 q^{37} + 116172 q^{43} + 237072 q^{49} + 158544 q^{55} + 170154 q^{61} + 84162 q^{67} + 748986 q^{73} + 560130 q^{79} + 1073040 q^{85} + 2795982 q^{91} + 1967418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 62.7833i − 0.502266i −0.967953 0.251133i \(-0.919197\pi\)
0.967953 0.251133i \(-0.0808032\pi\)
\(6\) 0 0
\(7\) 496.882 1.44863 0.724317 0.689467i \(-0.242155\pi\)
0.724317 + 0.689467i \(0.242155\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1437.06i − 1.07969i −0.841765 0.539844i \(-0.818483\pi\)
0.841765 0.539844i \(-0.181517\pi\)
\(12\) 0 0
\(13\) 3905.03 1.77744 0.888719 0.458452i \(-0.151596\pi\)
0.888719 + 0.458452i \(0.151596\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8385.53i 1.70681i 0.521252 + 0.853403i \(0.325465\pi\)
−0.521252 + 0.853403i \(0.674535\pi\)
\(18\) 0 0
\(19\) 3625.19 0.528531 0.264265 0.964450i \(-0.414871\pi\)
0.264265 + 0.964450i \(0.414871\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 24070.3i 1.97833i 0.146820 + 0.989163i \(0.453096\pi\)
−0.146820 + 0.989163i \(0.546904\pi\)
\(24\) 0 0
\(25\) 11683.3 0.747729
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 38898.5i − 1.59492i −0.603372 0.797460i \(-0.706177\pi\)
0.603372 0.797460i \(-0.293823\pi\)
\(30\) 0 0
\(31\) 10728.8 0.360136 0.180068 0.983654i \(-0.442368\pi\)
0.180068 + 0.983654i \(0.442368\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 31195.9i − 0.727600i
\(36\) 0 0
\(37\) −45162.5 −0.891606 −0.445803 0.895131i \(-0.647082\pi\)
−0.445803 + 0.895131i \(0.647082\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 35120.1i 0.509571i 0.966998 + 0.254785i \(0.0820048\pi\)
−0.966998 + 0.254785i \(0.917995\pi\)
\(42\) 0 0
\(43\) 21584.6 0.271481 0.135740 0.990744i \(-0.456659\pi\)
0.135740 + 0.990744i \(0.456659\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 103509.i 0.996974i 0.866897 + 0.498487i \(0.166111\pi\)
−0.866897 + 0.498487i \(0.833889\pi\)
\(48\) 0 0
\(49\) 129242. 1.09854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 70152.9i − 0.471214i −0.971848 0.235607i \(-0.924292\pi\)
0.971848 0.235607i \(-0.0757078\pi\)
\(54\) 0 0
\(55\) −90223.6 −0.542291
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 185774.i 0.904541i 0.891881 + 0.452271i \(0.149386\pi\)
−0.891881 + 0.452271i \(0.850614\pi\)
\(60\) 0 0
\(61\) −301079. −1.32645 −0.663226 0.748419i \(-0.730813\pi\)
−0.663226 + 0.748419i \(0.730813\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 245171.i − 0.892748i
\(66\) 0 0
\(67\) 130275. 0.433148 0.216574 0.976266i \(-0.430512\pi\)
0.216574 + 0.976266i \(0.430512\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 464741.i 1.29848i 0.760583 + 0.649241i \(0.224913\pi\)
−0.760583 + 0.649241i \(0.775087\pi\)
\(72\) 0 0
\(73\) 274352. 0.705245 0.352623 0.935766i \(-0.385290\pi\)
0.352623 + 0.935766i \(0.385290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 714051.i − 1.56407i
\(78\) 0 0
\(79\) −705830. −1.43159 −0.715795 0.698311i \(-0.753935\pi\)
−0.715795 + 0.698311i \(0.753935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 233654.i 0.408639i 0.978904 + 0.204319i \(0.0654981\pi\)
−0.978904 + 0.204319i \(0.934502\pi\)
\(84\) 0 0
\(85\) 526471. 0.857271
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.16112e6i − 1.64706i −0.567275 0.823529i \(-0.692002\pi\)
0.567275 0.823529i \(-0.307998\pi\)
\(90\) 0 0
\(91\) 1.94034e6 2.57486
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 227602.i − 0.265463i
\(96\) 0 0
\(97\) 129972. 0.142408 0.0712038 0.997462i \(-0.477316\pi\)
0.0712038 + 0.997462i \(0.477316\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 44362.7i − 0.0430580i −0.999768 0.0215290i \(-0.993147\pi\)
0.999768 0.0215290i \(-0.00685342\pi\)
\(102\) 0 0
\(103\) 1.21733e6 1.11403 0.557013 0.830504i \(-0.311947\pi\)
0.557013 + 0.830504i \(0.311947\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 194040.i 0.158394i 0.996859 + 0.0791972i \(0.0252357\pi\)
−0.996859 + 0.0791972i \(0.974764\pi\)
\(108\) 0 0
\(109\) 2.22984e6 1.72184 0.860921 0.508738i \(-0.169888\pi\)
0.860921 + 0.508738i \(0.169888\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.17695e6i 0.815684i 0.913052 + 0.407842i \(0.133719\pi\)
−0.913052 + 0.407842i \(0.866281\pi\)
\(114\) 0 0
\(115\) 1.51121e6 0.993647
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.16662e6i 2.47254i
\(120\) 0 0
\(121\) −293593. −0.165725
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.71450e6i − 0.877825i
\(126\) 0 0
\(127\) −2.57158e6 −1.25542 −0.627710 0.778448i \(-0.716007\pi\)
−0.627710 + 0.778448i \(0.716007\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.08627e6i 0.928020i 0.885830 + 0.464010i \(0.153590\pi\)
−0.885830 + 0.464010i \(0.846410\pi\)
\(132\) 0 0
\(133\) 1.80129e6 0.765648
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.52222e6i − 0.980892i −0.871471 0.490446i \(-0.836834\pi\)
0.871471 0.490446i \(-0.163166\pi\)
\(138\) 0 0
\(139\) −558638. −0.208011 −0.104006 0.994577i \(-0.533166\pi\)
−0.104006 + 0.994577i \(0.533166\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 5.61178e6i − 1.91908i
\(144\) 0 0
\(145\) −2.44218e6 −0.801074
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 951789.i 0.287728i 0.989597 + 0.143864i \(0.0459528\pi\)
−0.989597 + 0.143864i \(0.954047\pi\)
\(150\) 0 0
\(151\) 3.52962e6 1.02517 0.512586 0.858636i \(-0.328687\pi\)
0.512586 + 0.858636i \(0.328687\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 673590.i − 0.180884i
\(156\) 0 0
\(157\) 678623. 0.175360 0.0876798 0.996149i \(-0.472055\pi\)
0.0876798 + 0.996149i \(0.472055\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.19601e7i 2.86587i
\(162\) 0 0
\(163\) 2.74597e6 0.634064 0.317032 0.948415i \(-0.397314\pi\)
0.317032 + 0.948415i \(0.397314\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.51214e6i − 1.39822i −0.715016 0.699108i \(-0.753580\pi\)
0.715016 0.699108i \(-0.246420\pi\)
\(168\) 0 0
\(169\) 1.04225e7 2.15929
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3.02492e6i − 0.584220i −0.956385 0.292110i \(-0.905643\pi\)
0.956385 0.292110i \(-0.0943573\pi\)
\(174\) 0 0
\(175\) 5.80520e6 1.08319
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.43892e6i 0.599602i 0.954002 + 0.299801i \(0.0969202\pi\)
−0.954002 + 0.299801i \(0.903080\pi\)
\(180\) 0 0
\(181\) 4.76769e6 0.804030 0.402015 0.915633i \(-0.368310\pi\)
0.402015 + 0.915633i \(0.368310\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.83545e6i 0.447824i
\(186\) 0 0
\(187\) 1.20506e7 1.84282
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 1.43616e6i − 0.206112i −0.994676 0.103056i \(-0.967138\pi\)
0.994676 0.103056i \(-0.0328621\pi\)
\(192\) 0 0
\(193\) 6.88018e6 0.957036 0.478518 0.878078i \(-0.341174\pi\)
0.478518 + 0.878078i \(0.341174\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4.46415e6i − 0.583902i −0.956433 0.291951i \(-0.905696\pi\)
0.956433 0.291951i \(-0.0943045\pi\)
\(198\) 0 0
\(199\) 1.21686e7 1.54412 0.772058 0.635552i \(-0.219227\pi\)
0.772058 + 0.635552i \(0.219227\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.93280e7i − 2.31046i
\(204\) 0 0
\(205\) 2.20496e6 0.255940
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 5.20964e6i − 0.570648i
\(210\) 0 0
\(211\) −1.68577e7 −1.79453 −0.897265 0.441491i \(-0.854450\pi\)
−0.897265 + 0.441491i \(0.854450\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 1.35515e6i − 0.136356i
\(216\) 0 0
\(217\) 5.33095e6 0.521706
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.27458e7i 3.03374i
\(222\) 0 0
\(223\) 1.09013e6 0.0983019 0.0491510 0.998791i \(-0.484348\pi\)
0.0491510 + 0.998791i \(0.484348\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.12524e6i 0.523656i 0.965115 + 0.261828i \(0.0843253\pi\)
−0.965115 + 0.261828i \(0.915675\pi\)
\(228\) 0 0
\(229\) −9.73155e6 −0.810356 −0.405178 0.914238i \(-0.632790\pi\)
−0.405178 + 0.914238i \(0.632790\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.98923e6i 0.473482i 0.971573 + 0.236741i \(0.0760793\pi\)
−0.971573 + 0.236741i \(0.923921\pi\)
\(234\) 0 0
\(235\) 6.49862e6 0.500746
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.14769e6i 0.523567i 0.965127 + 0.261783i \(0.0843106\pi\)
−0.965127 + 0.261783i \(0.915689\pi\)
\(240\) 0 0
\(241\) −2.98986e6 −0.213599 −0.106800 0.994281i \(-0.534060\pi\)
−0.106800 + 0.994281i \(0.534060\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 8.11427e6i − 0.551761i
\(246\) 0 0
\(247\) 1.41565e7 0.939431
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.85894e7i − 1.80794i −0.427598 0.903969i \(-0.640640\pi\)
0.427598 0.903969i \(-0.359360\pi\)
\(252\) 0 0
\(253\) 3.45906e7 2.13597
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.45391e6i − 0.556945i −0.960444 0.278472i \(-0.910172\pi\)
0.960444 0.278472i \(-0.0898281\pi\)
\(258\) 0 0
\(259\) −2.24404e7 −1.29161
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.26436e7i 0.695030i 0.937674 + 0.347515i \(0.112974\pi\)
−0.937674 + 0.347515i \(0.887026\pi\)
\(264\) 0 0
\(265\) −4.40443e6 −0.236675
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.91952e7i 0.986133i 0.869992 + 0.493067i \(0.164124\pi\)
−0.869992 + 0.493067i \(0.835876\pi\)
\(270\) 0 0
\(271\) 3.10242e6 0.155881 0.0779404 0.996958i \(-0.475166\pi\)
0.0779404 + 0.996958i \(0.475166\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.67896e7i − 0.807313i
\(276\) 0 0
\(277\) −1.97207e7 −0.927861 −0.463930 0.885872i \(-0.653561\pi\)
−0.463930 + 0.885872i \(0.653561\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 3.26616e7i − 1.47203i −0.676963 0.736017i \(-0.736704\pi\)
0.676963 0.736017i \(-0.263296\pi\)
\(282\) 0 0
\(283\) 1.16099e7 0.512233 0.256117 0.966646i \(-0.417557\pi\)
0.256117 + 0.966646i \(0.417557\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.74506e7i 0.738182i
\(288\) 0 0
\(289\) −4.61796e7 −1.91318
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 3.03466e7i − 1.20645i −0.797573 0.603223i \(-0.793883\pi\)
0.797573 0.603223i \(-0.206117\pi\)
\(294\) 0 0
\(295\) 1.16635e7 0.454321
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.39953e7i 3.51635i
\(300\) 0 0
\(301\) 1.07250e7 0.393276
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.89027e7i 0.666232i
\(306\) 0 0
\(307\) −4.99035e6 −0.172471 −0.0862355 0.996275i \(-0.527484\pi\)
−0.0862355 + 0.996275i \(0.527484\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 1.75448e7i − 0.583266i −0.956530 0.291633i \(-0.905801\pi\)
0.956530 0.291633i \(-0.0941985\pi\)
\(312\) 0 0
\(313\) 1.42107e7 0.463429 0.231714 0.972784i \(-0.425567\pi\)
0.231714 + 0.972784i \(0.425567\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.79914e7i − 1.19264i −0.802749 0.596318i \(-0.796630\pi\)
0.802749 0.596318i \(-0.203370\pi\)
\(318\) 0 0
\(319\) −5.58996e7 −1.72202
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.03992e7i 0.902099i
\(324\) 0 0
\(325\) 4.56235e7 1.32904
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.14316e7i 1.44425i
\(330\) 0 0
\(331\) −3.85294e6 −0.106245 −0.0531224 0.998588i \(-0.516917\pi\)
−0.0531224 + 0.998588i \(0.516917\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 8.17909e6i − 0.217556i
\(336\) 0 0
\(337\) −4.67260e7 −1.22087 −0.610434 0.792067i \(-0.709005\pi\)
−0.610434 + 0.792067i \(0.709005\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 1.54180e7i − 0.388835i
\(342\) 0 0
\(343\) 5.76057e6 0.142752
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 8.68841e6i − 0.207947i −0.994580 0.103973i \(-0.966844\pi\)
0.994580 0.103973i \(-0.0331557\pi\)
\(348\) 0 0
\(349\) 3.56685e7 0.839089 0.419545 0.907735i \(-0.362190\pi\)
0.419545 + 0.907735i \(0.362190\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.34071e7i − 1.44150i −0.693196 0.720749i \(-0.743798\pi\)
0.693196 0.720749i \(-0.256202\pi\)
\(354\) 0 0
\(355\) 2.91780e7 0.652184
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 4.47602e7i − 0.967405i −0.875233 0.483702i \(-0.839292\pi\)
0.875233 0.483702i \(-0.160708\pi\)
\(360\) 0 0
\(361\) −3.39039e7 −0.720655
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.72247e7i − 0.354221i
\(366\) 0 0
\(367\) −1.51940e7 −0.307378 −0.153689 0.988119i \(-0.549115\pi\)
−0.153689 + 0.988119i \(0.549115\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 3.48577e7i − 0.682616i
\(372\) 0 0
\(373\) −9.43540e7 −1.81817 −0.909084 0.416614i \(-0.863217\pi\)
−0.909084 + 0.416614i \(0.863217\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.51900e8i − 2.83487i
\(378\) 0 0
\(379\) −8.08369e6 −0.148488 −0.0742441 0.997240i \(-0.523654\pi\)
−0.0742441 + 0.997240i \(0.523654\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.63217e7i 1.71446i 0.514932 + 0.857231i \(0.327817\pi\)
−0.514932 + 0.857231i \(0.672183\pi\)
\(384\) 0 0
\(385\) −4.48305e7 −0.785581
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.43211e7i 0.413176i 0.978428 + 0.206588i \(0.0662360\pi\)
−0.978428 + 0.206588i \(0.933764\pi\)
\(390\) 0 0
\(391\) −2.01842e8 −3.37662
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.43143e7i 0.719039i
\(396\) 0 0
\(397\) 4.70196e7 0.751463 0.375732 0.926729i \(-0.377391\pi\)
0.375732 + 0.926729i \(0.377391\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.79234e6i 0.105338i 0.998612 + 0.0526692i \(0.0167729\pi\)
−0.998612 + 0.0526692i \(0.983227\pi\)
\(402\) 0 0
\(403\) 4.18964e7 0.640120
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.49015e7i 0.962656i
\(408\) 0 0
\(409\) −8.43151e7 −1.23235 −0.616177 0.787608i \(-0.711319\pi\)
−0.616177 + 0.787608i \(0.711319\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.23076e7i 1.31035i
\(414\) 0 0
\(415\) 1.46696e7 0.205245
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 1.72937e7i − 0.235097i −0.993067 0.117548i \(-0.962496\pi\)
0.993067 0.117548i \(-0.0375035\pi\)
\(420\) 0 0
\(421\) −1.25301e8 −1.67922 −0.839612 0.543187i \(-0.817217\pi\)
−0.839612 + 0.543187i \(0.817217\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.79704e7i 1.27623i
\(426\) 0 0
\(427\) −1.49601e8 −1.92154
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 6.28715e7i − 0.785275i −0.919693 0.392638i \(-0.871563\pi\)
0.919693 0.392638i \(-0.128437\pi\)
\(432\) 0 0
\(433\) −5.66021e7 −0.697219 −0.348609 0.937268i \(-0.613346\pi\)
−0.348609 + 0.937268i \(0.613346\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.72595e7i 1.04561i
\(438\) 0 0
\(439\) −3.82688e7 −0.452326 −0.226163 0.974089i \(-0.572618\pi\)
−0.226163 + 0.974089i \(0.572618\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 8.79866e7i − 1.01206i −0.862517 0.506029i \(-0.831113\pi\)
0.862517 0.506029i \(-0.168887\pi\)
\(444\) 0 0
\(445\) −7.28992e7 −0.827262
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.97838e7i 0.991879i 0.868357 + 0.495940i \(0.165176\pi\)
−0.868357 + 0.495940i \(0.834824\pi\)
\(450\) 0 0
\(451\) 5.04699e7 0.550177
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.21821e8i − 1.29327i
\(456\) 0 0
\(457\) 1.76314e8 1.84731 0.923653 0.383229i \(-0.125188\pi\)
0.923653 + 0.383229i \(0.125188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.28602e7i − 0.539543i −0.962924 0.269771i \(-0.913052\pi\)
0.962924 0.269771i \(-0.0869481\pi\)
\(462\) 0 0
\(463\) 6.74136e6 0.0679211 0.0339605 0.999423i \(-0.489188\pi\)
0.0339605 + 0.999423i \(0.489188\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.62620e7i 0.159670i 0.996808 + 0.0798351i \(0.0254394\pi\)
−0.996808 + 0.0798351i \(0.974561\pi\)
\(468\) 0 0
\(469\) 6.47312e7 0.627473
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 3.10185e7i − 0.293114i
\(474\) 0 0
\(475\) 4.23541e7 0.395198
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 6.65819e6i − 0.0605829i −0.999541 0.0302914i \(-0.990356\pi\)
0.999541 0.0302914i \(-0.00964354\pi\)
\(480\) 0 0
\(481\) −1.76361e8 −1.58478
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 8.16004e6i − 0.0715265i
\(486\) 0 0
\(487\) 8.41405e7 0.728481 0.364241 0.931305i \(-0.381329\pi\)
0.364241 + 0.931305i \(0.381329\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.06143e7i − 0.174150i −0.996202 0.0870751i \(-0.972248\pi\)
0.996202 0.0870751i \(-0.0277520\pi\)
\(492\) 0 0
\(493\) 3.26185e8 2.72222
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.30921e8i 1.88103i
\(498\) 0 0
\(499\) −6.76628e7 −0.544563 −0.272282 0.962218i \(-0.587778\pi\)
−0.272282 + 0.962218i \(0.587778\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 8.12807e7i − 0.638680i −0.947640 0.319340i \(-0.896539\pi\)
0.947640 0.319340i \(-0.103461\pi\)
\(504\) 0 0
\(505\) −2.78523e6 −0.0216266
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 1.46456e8i − 1.11059i −0.831655 0.555293i \(-0.812606\pi\)
0.831655 0.555293i \(-0.187394\pi\)
\(510\) 0 0
\(511\) 1.36321e8 1.02164
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 7.64278e7i − 0.559538i
\(516\) 0 0
\(517\) 1.48749e8 1.07642
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.00866e8i 0.713232i 0.934251 + 0.356616i \(0.116069\pi\)
−0.934251 + 0.356616i \(0.883931\pi\)
\(522\) 0 0
\(523\) −1.23966e8 −0.866555 −0.433277 0.901261i \(-0.642643\pi\)
−0.433277 + 0.901261i \(0.642643\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.99669e7i 0.614682i
\(528\) 0 0
\(529\) −4.31343e8 −2.91378
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.37145e8i 0.905731i
\(534\) 0 0
\(535\) 1.21825e7 0.0795562
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.85730e8i − 1.18608i
\(540\) 0 0
\(541\) 1.57932e8 0.997419 0.498709 0.866769i \(-0.333807\pi\)
0.498709 + 0.866769i \(0.333807\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1.39996e8i − 0.864823i
\(546\) 0 0
\(547\) −1.50060e7 −0.0916862 −0.0458431 0.998949i \(-0.514597\pi\)
−0.0458431 + 0.998949i \(0.514597\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1.41015e8i − 0.842964i
\(552\) 0 0
\(553\) −3.50714e8 −2.07385
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 9.89257e7i − 0.572458i −0.958161 0.286229i \(-0.907598\pi\)
0.958161 0.286229i \(-0.0924018\pi\)
\(558\) 0 0
\(559\) 8.42886e7 0.482540
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.24717e7i 0.462146i 0.972936 + 0.231073i \(0.0742237\pi\)
−0.972936 + 0.231073i \(0.925776\pi\)
\(564\) 0 0
\(565\) 7.38927e7 0.409691
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.93991e8i 1.05304i 0.850162 + 0.526521i \(0.176504\pi\)
−0.850162 + 0.526521i \(0.823496\pi\)
\(570\) 0 0
\(571\) −1.87396e8 −1.00659 −0.503293 0.864116i \(-0.667878\pi\)
−0.503293 + 0.864116i \(0.667878\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.81220e8i 1.47925i
\(576\) 0 0
\(577\) 1.29755e8 0.675457 0.337729 0.941244i \(-0.390341\pi\)
0.337729 + 0.941244i \(0.390341\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.16099e8i 0.591968i
\(582\) 0 0
\(583\) −1.00814e8 −0.508764
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 3.63761e8i − 1.79846i −0.437473 0.899231i \(-0.644127\pi\)
0.437473 0.899231i \(-0.355873\pi\)
\(588\) 0 0
\(589\) 3.88940e7 0.190343
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 6.88503e7i − 0.330173i −0.986279 0.165086i \(-0.947210\pi\)
0.986279 0.165086i \(-0.0527903\pi\)
\(594\) 0 0
\(595\) 2.61594e8 1.24187
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.17543e8i 0.546911i 0.961885 + 0.273455i \(0.0881666\pi\)
−0.961885 + 0.273455i \(0.911833\pi\)
\(600\) 0 0
\(601\) −1.38103e7 −0.0636179 −0.0318090 0.999494i \(-0.510127\pi\)
−0.0318090 + 0.999494i \(0.510127\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.84327e7i 0.0832383i
\(606\) 0 0
\(607\) 2.93369e8 1.31174 0.655871 0.754873i \(-0.272302\pi\)
0.655871 + 0.754873i \(0.272302\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.04205e8i 1.77206i
\(612\) 0 0
\(613\) 2.86835e8 1.24523 0.622617 0.782526i \(-0.286069\pi\)
0.622617 + 0.782526i \(0.286069\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.64357e7i 0.325417i 0.986674 + 0.162709i \(0.0520230\pi\)
−0.986674 + 0.162709i \(0.947977\pi\)
\(618\) 0 0
\(619\) −3.68179e7 −0.155234 −0.0776170 0.996983i \(-0.524731\pi\)
−0.0776170 + 0.996983i \(0.524731\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 5.76942e8i − 2.38598i
\(624\) 0 0
\(625\) 7.49088e7 0.306826
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 3.78712e8i − 1.52180i
\(630\) 0 0
\(631\) 1.91018e8 0.760301 0.380150 0.924925i \(-0.375872\pi\)
0.380150 + 0.924925i \(0.375872\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.61452e8i 0.630555i
\(636\) 0 0
\(637\) 5.04696e8 1.95259
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 5.16010e8i − 1.95922i −0.200899 0.979612i \(-0.564386\pi\)
0.200899 0.979612i \(-0.435614\pi\)
\(642\) 0 0
\(643\) −3.35849e8 −1.26332 −0.631658 0.775248i \(-0.717625\pi\)
−0.631658 + 0.775248i \(0.717625\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.13350e8i 1.15695i 0.815698 + 0.578477i \(0.196353\pi\)
−0.815698 + 0.578477i \(0.803647\pi\)
\(648\) 0 0
\(649\) 2.66969e8 0.976622
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 3.59248e8i − 1.29019i −0.764102 0.645096i \(-0.776817\pi\)
0.764102 0.645096i \(-0.223183\pi\)
\(654\) 0 0
\(655\) 1.30983e8 0.466113
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.36947e8i 1.87618i 0.346389 + 0.938091i \(0.387408\pi\)
−0.346389 + 0.938091i \(0.612592\pi\)
\(660\) 0 0
\(661\) 2.98758e8 1.03446 0.517231 0.855846i \(-0.326963\pi\)
0.517231 + 0.855846i \(0.326963\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1.13091e8i − 0.384559i
\(666\) 0 0
\(667\) 9.36298e8 3.15527
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.32670e8i 1.43215i
\(672\) 0 0
\(673\) −3.23451e8 −1.06112 −0.530558 0.847649i \(-0.678018\pi\)
−0.530558 + 0.847649i \(0.678018\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.09203e8i 1.31878i 0.751800 + 0.659391i \(0.229186\pi\)
−0.751800 + 0.659391i \(0.770814\pi\)
\(678\) 0 0
\(679\) 6.45805e7 0.206297
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.86785e8i 0.586244i 0.956075 + 0.293122i \(0.0946943\pi\)
−0.956075 + 0.293122i \(0.905306\pi\)
\(684\) 0 0
\(685\) −1.58353e8 −0.492669
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 2.73949e8i − 0.837553i
\(690\) 0 0
\(691\) 3.75447e8 1.13793 0.568964 0.822362i \(-0.307344\pi\)
0.568964 + 0.822362i \(0.307344\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.50732e7i 0.104477i
\(696\) 0 0
\(697\) −2.94501e8 −0.869738
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.11037e8i 1.48354i 0.670657 + 0.741768i \(0.266012\pi\)
−0.670657 + 0.741768i \(0.733988\pi\)
\(702\) 0 0
\(703\) −1.63723e8 −0.471242
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.20430e7i − 0.0623753i
\(708\) 0 0
\(709\) 4.42507e8 1.24160 0.620800 0.783969i \(-0.286808\pi\)
0.620800 + 0.783969i \(0.286808\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.58246e8i 0.712467i
\(714\) 0 0
\(715\) −3.52326e8 −0.963889
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 7.20610e8i − 1.93871i −0.245659 0.969356i \(-0.579004\pi\)
0.245659 0.969356i \(-0.420996\pi\)
\(720\) 0 0
\(721\) 6.04867e8 1.61382
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 4.54461e8i − 1.19257i
\(726\) 0 0
\(727\) 2.46407e8 0.641282 0.320641 0.947201i \(-0.396102\pi\)
0.320641 + 0.947201i \(0.396102\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.80998e8i 0.463364i
\(732\) 0 0
\(733\) −1.34987e8 −0.342751 −0.171375 0.985206i \(-0.554821\pi\)
−0.171375 + 0.985206i \(0.554821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.87213e8i − 0.467665i
\(738\) 0 0
\(739\) −5.93121e8 −1.46964 −0.734819 0.678264i \(-0.762733\pi\)
−0.734819 + 0.678264i \(0.762733\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.95231e6i 0.0120737i 0.999982 + 0.00603686i \(0.00192160\pi\)
−0.999982 + 0.00603686i \(0.998078\pi\)
\(744\) 0 0
\(745\) 5.97564e7 0.144516
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.64149e7i 0.229456i
\(750\) 0 0
\(751\) −8.35754e7 −0.197314 −0.0986572 0.995121i \(-0.531455\pi\)
−0.0986572 + 0.995121i \(0.531455\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2.21601e8i − 0.514910i
\(756\) 0 0
\(757\) 8.99345e7 0.207319 0.103659 0.994613i \(-0.466945\pi\)
0.103659 + 0.994613i \(0.466945\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 3.68572e8i − 0.836312i −0.908375 0.418156i \(-0.862677\pi\)
0.908375 0.418156i \(-0.137323\pi\)
\(762\) 0 0
\(763\) 1.10796e9 2.49432
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.25453e8i 1.60777i
\(768\) 0 0
\(769\) −5.28138e8 −1.16136 −0.580682 0.814130i \(-0.697214\pi\)
−0.580682 + 0.814130i \(0.697214\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 3.87278e8i − 0.838465i −0.907879 0.419232i \(-0.862299\pi\)
0.907879 0.419232i \(-0.137701\pi\)
\(774\) 0 0
\(775\) 1.25348e8 0.269284
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.27317e8i 0.269324i
\(780\) 0 0
\(781\) 6.67862e8 1.40195
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 4.26062e7i − 0.0880772i
\(786\) 0 0
\(787\) 6.36033e8 1.30483 0.652417 0.757860i \(-0.273755\pi\)
0.652417 + 0.757860i \(0.273755\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.84804e8i 1.18163i
\(792\) 0 0
\(793\) −1.17572e9 −2.35769
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.61669e8i 0.911918i 0.890001 + 0.455959i \(0.150704\pi\)
−0.890001 + 0.455959i \(0.849296\pi\)
\(798\) 0 0
\(799\) −8.67976e8 −1.70164
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 3.94262e8i − 0.761444i
\(804\) 0 0
\(805\) 7.50894e8 1.43943
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.25161e8i 0.425252i 0.977134 + 0.212626i \(0.0682016\pi\)
−0.977134 + 0.212626i \(0.931798\pi\)
\(810\) 0 0
\(811\) −3.06936e8 −0.575420 −0.287710 0.957718i \(-0.592894\pi\)
−0.287710 + 0.957718i \(0.592894\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.72401e8i − 0.318469i
\(816\) 0 0
\(817\) 7.82484e7 0.143486
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 2.73840e8i − 0.494843i −0.968908 0.247422i \(-0.920417\pi\)
0.968908 0.247422i \(-0.0795832\pi\)
\(822\) 0 0
\(823\) −1.03168e9 −1.85073 −0.925366 0.379074i \(-0.876243\pi\)
−0.925366 + 0.379074i \(0.876243\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 5.97417e8i − 1.05624i −0.849171 0.528118i \(-0.822898\pi\)
0.849171 0.528118i \(-0.177102\pi\)
\(828\) 0 0
\(829\) 2.40482e8 0.422103 0.211051 0.977475i \(-0.432311\pi\)
0.211051 + 0.977475i \(0.432311\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.08377e9i 1.87500i
\(834\) 0 0
\(835\) −4.08854e8 −0.702277
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 6.75223e8i − 1.14330i −0.820497 0.571651i \(-0.806303\pi\)
0.820497 0.571651i \(-0.193697\pi\)
\(840\) 0 0
\(841\) −9.18270e8 −1.54377
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 6.54357e8i − 1.08454i
\(846\) 0 0
\(847\) −1.45881e8 −0.240076
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.08708e9i − 1.76389i
\(852\) 0 0
\(853\) 2.33689e8 0.376522 0.188261 0.982119i \(-0.439715\pi\)
0.188261 + 0.982119i \(0.439715\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.09818e8i − 0.651102i −0.945525 0.325551i \(-0.894450\pi\)
0.945525 0.325551i \(-0.105550\pi\)
\(858\) 0 0
\(859\) 4.80708e8 0.758406 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.08940e8i 0.169495i 0.996402 + 0.0847475i \(0.0270083\pi\)
−0.996402 + 0.0847475i \(0.972992\pi\)
\(864\) 0 0
\(865\) −1.89915e8 −0.293434
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.01432e9i 1.54567i
\(870\) 0 0
\(871\) 5.08728e8 0.769894
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 8.51905e8i − 1.27165i
\(876\) 0 0
\(877\) 1.29815e7 0.0192454 0.00962271 0.999954i \(-0.496937\pi\)
0.00962271 + 0.999954i \(0.496937\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.01837e8i 0.587655i 0.955859 + 0.293827i \(0.0949290\pi\)
−0.955859 + 0.293827i \(0.905071\pi\)
\(882\) 0 0
\(883\) −3.93666e8 −0.571803 −0.285901 0.958259i \(-0.592293\pi\)
−0.285901 + 0.958259i \(0.592293\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.78240e8i − 1.25847i −0.777216 0.629234i \(-0.783369\pi\)
0.777216 0.629234i \(-0.216631\pi\)
\(888\) 0 0
\(889\) −1.27777e9 −1.81864
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.75239e8i 0.526931i
\(894\) 0 0
\(895\) 2.15907e8 0.301160
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 4.17335e8i − 0.574388i
\(900\) 0 0
\(901\) 5.88269e8 0.804270
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2.99331e8i − 0.403837i
\(906\) 0 0
\(907\) −5.02203e8 −0.673066 −0.336533 0.941672i \(-0.609254\pi\)
−0.336533 + 0.941672i \(0.609254\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.26644e9i 1.67506i 0.546393 + 0.837529i \(0.316001\pi\)
−0.546393 + 0.837529i \(0.683999\pi\)
\(912\) 0 0
\(913\) 3.35776e8 0.441202
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.03663e9i 1.34436i
\(918\) 0 0
\(919\) 1.07637e9 1.38681 0.693405 0.720548i \(-0.256110\pi\)
0.693405 + 0.720548i \(0.256110\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.81483e9i 2.30797i
\(924\) 0 0
\(925\) −5.27646e8 −0.666680
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.92619e8i 1.11332i 0.830741 + 0.556658i \(0.187917\pi\)
−0.830741 + 0.556658i \(0.812083\pi\)
\(930\) 0 0
\(931\) 4.68529e8 0.580614
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 7.56573e8i − 0.925585i
\(936\) 0 0
\(937\) 1.04330e8 0.126821 0.0634105 0.997988i \(-0.479802\pi\)
0.0634105 + 0.997988i \(0.479802\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.03692e9i 1.24445i 0.782839 + 0.622224i \(0.213771\pi\)
−0.782839 + 0.622224i \(0.786229\pi\)
\(942\) 0 0
\(943\) −8.45352e8 −1.00810
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.27799e9i 1.50479i 0.658710 + 0.752397i \(0.271102\pi\)
−0.658710 + 0.752397i \(0.728898\pi\)
\(948\) 0 0
\(949\) 1.07135e9 1.25353
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.38519e8i 0.622189i 0.950379 + 0.311095i \(0.100696\pi\)
−0.950379 + 0.311095i \(0.899304\pi\)
\(954\) 0 0
\(955\) −9.01670e7 −0.103523
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1.25325e9i − 1.42095i
\(960\) 0 0
\(961\) −7.72396e8 −0.870302
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 4.31961e8i − 0.480687i
\(966\) 0 0
\(967\) 2.17113e8 0.240108 0.120054 0.992767i \(-0.461693\pi\)
0.120054 + 0.992767i \(0.461693\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1.72634e8i − 0.188568i −0.995545 0.0942841i \(-0.969944\pi\)
0.995545 0.0942841i \(-0.0300562\pi\)
\(972\) 0 0
\(973\) −2.77577e8 −0.301332
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.23627e8i − 0.775946i −0.921671 0.387973i \(-0.873175\pi\)
0.921671 0.387973i \(-0.126825\pi\)
\(978\) 0 0
\(979\) −1.66861e9 −1.77831
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 7.66788e7i − 0.0807263i −0.999185 0.0403631i \(-0.987149\pi\)
0.999185 0.0403631i \(-0.0128515\pi\)
\(984\) 0 0
\(985\) −2.80274e8 −0.293274
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.19548e8i 0.537077i
\(990\) 0 0
\(991\) −4.77700e8 −0.490834 −0.245417 0.969418i \(-0.578925\pi\)
−0.245417 + 0.969418i \(0.578925\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 7.63983e8i − 0.775558i
\(996\) 0 0
\(997\) −1.42542e9 −1.43833 −0.719165 0.694839i \(-0.755475\pi\)
−0.719165 + 0.694839i \(0.755475\pi\)
\(998\) 0 0
\(999\) 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 432.7.e.k.161.3 6
3.2 odd 2 inner 432.7.e.k.161.4 6
4.3 odd 2 216.7.e.b.161.3 6
12.11 even 2 216.7.e.b.161.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.7.e.b.161.3 6 4.3 odd 2
216.7.e.b.161.4 yes 6 12.11 even 2
432.7.e.k.161.3 6 1.1 even 1 trivial
432.7.e.k.161.4 6 3.2 odd 2 inner