Properties

Label 900.6.j.c.557.11
Level $900$
Weight $6$
Character 900.557
Analytic conductor $144.345$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,6,Mod(557,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.557");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.j (of order \(4\), degree \(2\), 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: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 557.11
Character \(\chi\) \(=\) 900.557
Dual form 900.6.j.c.593.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(123.937 + 123.937i) q^{7} +O(q^{10})\) \(q+(123.937 + 123.937i) q^{7} +296.299i q^{11} +(-16.2336 + 16.2336i) q^{13} +(547.265 - 547.265i) q^{17} +262.818i q^{19} +(2434.99 + 2434.99i) q^{23} -4327.66 q^{29} +8473.00 q^{31} +(100.890 + 100.890i) q^{37} +14606.6i q^{41} +(9268.83 - 9268.83i) q^{43} +(1485.67 - 1485.67i) q^{47} +13913.6i q^{49} +(9264.63 + 9264.63i) q^{53} -2053.40 q^{59} -10864.1 q^{61} +(-9222.90 - 9222.90i) q^{67} +10577.6i q^{71} +(-20495.6 + 20495.6i) q^{73} +(-36722.4 + 36722.4i) q^{77} -58845.3i q^{79} +(-61864.1 - 61864.1i) q^{83} +43592.7 q^{89} -4023.88 q^{91} +(75406.9 + 75406.9i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 13848 q^{31} - 28200 q^{61} + 908328 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 123.937 + 123.937i 0.955994 + 0.955994i 0.999072 0.0430780i \(-0.0137164\pi\)
−0.0430780 + 0.999072i \(0.513716\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 296.299i 0.738327i 0.929364 + 0.369164i \(0.120356\pi\)
−0.929364 + 0.369164i \(0.879644\pi\)
\(12\) 0 0
\(13\) −16.2336 + 16.2336i −0.0266414 + 0.0266414i −0.720302 0.693661i \(-0.755997\pi\)
0.693661 + 0.720302i \(0.255997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 547.265 547.265i 0.459278 0.459278i −0.439140 0.898418i \(-0.644717\pi\)
0.898418 + 0.439140i \(0.144717\pi\)
\(18\) 0 0
\(19\) 262.818i 0.167021i 0.996507 + 0.0835104i \(0.0266132\pi\)
−0.996507 + 0.0835104i \(0.973387\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2434.99 + 2434.99i 0.959793 + 0.959793i 0.999222 0.0394291i \(-0.0125539\pi\)
−0.0394291 + 0.999222i \(0.512554\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4327.66 −0.955561 −0.477781 0.878479i \(-0.658559\pi\)
−0.477781 + 0.878479i \(0.658559\pi\)
\(30\) 0 0
\(31\) 8473.00 1.58355 0.791777 0.610810i \(-0.209156\pi\)
0.791777 + 0.610810i \(0.209156\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 100.890 + 100.890i 0.0121155 + 0.0121155i 0.713139 0.701023i \(-0.247273\pi\)
−0.701023 + 0.713139i \(0.747273\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 14606.6i 1.35703i 0.734589 + 0.678513i \(0.237375\pi\)
−0.734589 + 0.678513i \(0.762625\pi\)
\(42\) 0 0
\(43\) 9268.83 9268.83i 0.764459 0.764459i −0.212666 0.977125i \(-0.568215\pi\)
0.977125 + 0.212666i \(0.0682147\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1485.67 1485.67i 0.0981017 0.0981017i −0.656353 0.754454i \(-0.727902\pi\)
0.754454 + 0.656353i \(0.227902\pi\)
\(48\) 0 0
\(49\) 13913.6i 0.827848i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9264.63 + 9264.63i 0.453042 + 0.453042i 0.896363 0.443321i \(-0.146200\pi\)
−0.443321 + 0.896363i \(0.646200\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2053.40 −0.0767970 −0.0383985 0.999263i \(-0.512226\pi\)
−0.0383985 + 0.999263i \(0.512226\pi\)
\(60\) 0 0
\(61\) −10864.1 −0.373826 −0.186913 0.982376i \(-0.559848\pi\)
−0.186913 + 0.982376i \(0.559848\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9222.90 9222.90i −0.251004 0.251004i 0.570378 0.821382i \(-0.306797\pi\)
−0.821382 + 0.570378i \(0.806797\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10577.6i 0.249024i 0.992218 + 0.124512i \(0.0397365\pi\)
−0.992218 + 0.124512i \(0.960263\pi\)
\(72\) 0 0
\(73\) −20495.6 + 20495.6i −0.450146 + 0.450146i −0.895403 0.445257i \(-0.853112\pi\)
0.445257 + 0.895403i \(0.353112\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −36722.4 + 36722.4i −0.705836 + 0.705836i
\(78\) 0 0
\(79\) 58845.3i 1.06083i −0.847740 0.530413i \(-0.822037\pi\)
0.847740 0.530413i \(-0.177963\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −61864.1 61864.1i −0.985698 0.985698i 0.0142015 0.999899i \(-0.495479\pi\)
−0.999899 + 0.0142015i \(0.995479\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 43592.7 0.583363 0.291681 0.956516i \(-0.405785\pi\)
0.291681 + 0.956516i \(0.405785\pi\)
\(90\) 0 0
\(91\) −4023.88 −0.0509380
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 75406.9 + 75406.9i 0.813733 + 0.813733i 0.985191 0.171459i \(-0.0548480\pi\)
−0.171459 + 0.985191i \(0.554848\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 136828.i 1.33466i −0.744763 0.667329i \(-0.767437\pi\)
0.744763 0.667329i \(-0.232563\pi\)
\(102\) 0 0
\(103\) −68031.2 + 68031.2i −0.631851 + 0.631851i −0.948532 0.316681i \(-0.897432\pi\)
0.316681 + 0.948532i \(0.397432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 21604.1 21604.1i 0.182422 0.182422i −0.609988 0.792410i \(-0.708826\pi\)
0.792410 + 0.609988i \(0.208826\pi\)
\(108\) 0 0
\(109\) 56892.1i 0.458655i 0.973349 + 0.229327i \(0.0736526\pi\)
−0.973349 + 0.229327i \(0.926347\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −61980.0 61980.0i −0.456621 0.456621i 0.440924 0.897544i \(-0.354651\pi\)
−0.897544 + 0.440924i \(0.854651\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 135653. 0.878134
\(120\) 0 0
\(121\) 73257.7 0.454873
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 59893.7 + 59893.7i 0.329512 + 0.329512i 0.852401 0.522889i \(-0.175146\pi\)
−0.522889 + 0.852401i \(0.675146\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 317743.i 1.61770i 0.588017 + 0.808849i \(0.299909\pi\)
−0.588017 + 0.808849i \(0.700091\pi\)
\(132\) 0 0
\(133\) −32572.8 + 32572.8i −0.159671 + 0.159671i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −283098. + 283098.i −1.28865 + 1.28865i −0.353045 + 0.935606i \(0.614854\pi\)
−0.935606 + 0.353045i \(0.885146\pi\)
\(138\) 0 0
\(139\) 215426.i 0.945716i −0.881139 0.472858i \(-0.843222\pi\)
0.881139 0.472858i \(-0.156778\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4810.01 4810.01i −0.0196701 0.0196701i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −369959. −1.36517 −0.682587 0.730804i \(-0.739145\pi\)
−0.682587 + 0.730804i \(0.739145\pi\)
\(150\) 0 0
\(151\) 280532. 1.00124 0.500621 0.865666i \(-0.333105\pi\)
0.500621 + 0.865666i \(0.333105\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 30811.3 + 30811.3i 0.0997610 + 0.0997610i 0.755226 0.655465i \(-0.227527\pi\)
−0.655465 + 0.755226i \(0.727527\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 603570.i 1.83511i
\(162\) 0 0
\(163\) 96397.3 96397.3i 0.284181 0.284181i −0.550593 0.834774i \(-0.685598\pi\)
0.834774 + 0.550593i \(0.185598\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −35684.1 + 35684.1i −0.0990109 + 0.0990109i −0.754877 0.655866i \(-0.772304\pi\)
0.655866 + 0.754877i \(0.272304\pi\)
\(168\) 0 0
\(169\) 370766.i 0.998580i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 76138.1 + 76138.1i 0.193414 + 0.193414i 0.797169 0.603756i \(-0.206330\pi\)
−0.603756 + 0.797169i \(0.706330\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 411064. 0.958907 0.479454 0.877567i \(-0.340835\pi\)
0.479454 + 0.877567i \(0.340835\pi\)
\(180\) 0 0
\(181\) 272889. 0.619141 0.309571 0.950876i \(-0.399815\pi\)
0.309571 + 0.950876i \(0.399815\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 162154. + 162154.i 0.339098 + 0.339098i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 868065.i 1.72175i 0.508820 + 0.860873i \(0.330082\pi\)
−0.508820 + 0.860873i \(0.669918\pi\)
\(192\) 0 0
\(193\) −23668.8 + 23668.8i −0.0457387 + 0.0457387i −0.729606 0.683868i \(-0.760297\pi\)
0.683868 + 0.729606i \(0.260297\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −629819. + 629819.i −1.15625 + 1.15625i −0.170969 + 0.985276i \(0.554690\pi\)
−0.985276 + 0.170969i \(0.945310\pi\)
\(198\) 0 0
\(199\) 120544.i 0.215780i 0.994163 + 0.107890i \(0.0344094\pi\)
−0.994163 + 0.107890i \(0.965591\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −536357. 536357.i −0.913511 0.913511i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −77872.7 −0.123316
\(210\) 0 0
\(211\) 1.16641e6 1.80362 0.901809 0.432135i \(-0.142240\pi\)
0.901809 + 0.432135i \(0.142240\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.05012e6 + 1.05012e6i 1.51387 + 1.51387i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 17768.2i 0.0244716i
\(222\) 0 0
\(223\) −794666. + 794666.i −1.07010 + 1.07010i −0.0727445 + 0.997351i \(0.523176\pi\)
−0.997351 + 0.0727445i \(0.976824\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −614503. + 614503.i −0.791515 + 0.791515i −0.981740 0.190225i \(-0.939078\pi\)
0.190225 + 0.981740i \(0.439078\pi\)
\(228\) 0 0
\(229\) 1.41946e6i 1.78869i −0.447377 0.894346i \(-0.647642\pi\)
0.447377 0.894346i \(-0.352358\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −580216. 580216.i −0.700165 0.700165i 0.264281 0.964446i \(-0.414865\pi\)
−0.964446 + 0.264281i \(0.914865\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 438358. 0.496402 0.248201 0.968709i \(-0.420161\pi\)
0.248201 + 0.968709i \(0.420161\pi\)
\(240\) 0 0
\(241\) −61779.3 −0.0685173 −0.0342586 0.999413i \(-0.510907\pi\)
−0.0342586 + 0.999413i \(0.510907\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4266.48 4266.48i −0.00444967 0.00444967i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 152110.i 0.152396i 0.997093 + 0.0761978i \(0.0242780\pi\)
−0.997093 + 0.0761978i \(0.975722\pi\)
\(252\) 0 0
\(253\) −721486. + 721486.i −0.708642 + 0.708642i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −794182. + 794182.i −0.750045 + 0.750045i −0.974487 0.224442i \(-0.927944\pi\)
0.224442 + 0.974487i \(0.427944\pi\)
\(258\) 0 0
\(259\) 25007.9i 0.0231648i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −846425. 846425.i −0.754570 0.754570i 0.220759 0.975328i \(-0.429147\pi\)
−0.975328 + 0.220759i \(0.929147\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.38502e6 1.16701 0.583507 0.812108i \(-0.301680\pi\)
0.583507 + 0.812108i \(0.301680\pi\)
\(270\) 0 0
\(271\) −450412. −0.372552 −0.186276 0.982497i \(-0.559642\pi\)
−0.186276 + 0.982497i \(0.559642\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.25916e6 + 1.25916e6i 0.986012 + 0.986012i 0.999904 0.0138917i \(-0.00442202\pi\)
−0.0138917 + 0.999904i \(0.504422\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.96636e6i 1.48558i 0.669522 + 0.742792i \(0.266499\pi\)
−0.669522 + 0.742792i \(0.733501\pi\)
\(282\) 0 0
\(283\) −963548. + 963548.i −0.715167 + 0.715167i −0.967611 0.252445i \(-0.918765\pi\)
0.252445 + 0.967611i \(0.418765\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.81029e6 + 1.81029e6i −1.29731 + 1.29731i
\(288\) 0 0
\(289\) 820858.i 0.578127i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.78311e6 1.78311e6i −1.21342 1.21342i −0.969895 0.243522i \(-0.921697\pi\)
−0.243522 0.969895i \(-0.578303\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −79057.4 −0.0511405
\(300\) 0 0
\(301\) 2.29750e6 1.46164
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.26197e6 + 1.26197e6i 0.764193 + 0.764193i 0.977077 0.212885i \(-0.0682859\pi\)
−0.212885 + 0.977077i \(0.568286\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.26700e6i 1.32908i 0.747253 + 0.664540i \(0.231372\pi\)
−0.747253 + 0.664540i \(0.768628\pi\)
\(312\) 0 0
\(313\) 1.03818e6 1.03818e6i 0.598979 0.598979i −0.341062 0.940041i \(-0.610787\pi\)
0.940041 + 0.341062i \(0.110787\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −895082. + 895082.i −0.500282 + 0.500282i −0.911526 0.411244i \(-0.865095\pi\)
0.411244 + 0.911526i \(0.365095\pi\)
\(318\) 0 0
\(319\) 1.28228e6i 0.705517i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 143831. + 143831.i 0.0767090 + 0.0767090i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 368257. 0.187569
\(330\) 0 0
\(331\) 1.59270e6 0.799031 0.399515 0.916727i \(-0.369178\pi\)
0.399515 + 0.916727i \(0.369178\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.14157e6 2.14157e6i −1.02721 1.02721i −0.999619 0.0275860i \(-0.991218\pi\)
−0.0275860 0.999619i \(-0.508782\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.51054e6i 1.16918i
\(342\) 0 0
\(343\) 358594. 358594.i 0.164576 0.164576i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.35898e6 + 1.35898e6i −0.605883 + 0.605883i −0.941868 0.335984i \(-0.890931\pi\)
0.335984 + 0.941868i \(0.390931\pi\)
\(348\) 0 0
\(349\) 331480.i 0.145678i −0.997344 0.0728390i \(-0.976794\pi\)
0.997344 0.0728390i \(-0.0232059\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.53861e6 + 1.53861e6i 0.657192 + 0.657192i 0.954715 0.297523i \(-0.0961604\pi\)
−0.297523 + 0.954715i \(0.596160\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.46509e6 −1.41899 −0.709493 0.704712i \(-0.751076\pi\)
−0.709493 + 0.704712i \(0.751076\pi\)
\(360\) 0 0
\(361\) 2.40703e6 0.972104
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.95418e6 + 2.95418e6i 1.14491 + 1.14491i 0.987539 + 0.157373i \(0.0503026\pi\)
0.157373 + 0.987539i \(0.449697\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.29646e6i 0.866211i
\(372\) 0 0
\(373\) −1.99621e6 + 1.99621e6i −0.742906 + 0.742906i −0.973136 0.230231i \(-0.926052\pi\)
0.230231 + 0.973136i \(0.426052\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 70253.6 70253.6i 0.0254575 0.0254575i
\(378\) 0 0
\(379\) 3.85257e6i 1.37769i 0.724907 + 0.688847i \(0.241883\pi\)
−0.724907 + 0.688847i \(0.758117\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −114173. 114173.i −0.0397709 0.0397709i 0.686942 0.726713i \(-0.258953\pi\)
−0.726713 + 0.686942i \(0.758953\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.51434e6 1.17752 0.588762 0.808307i \(-0.299616\pi\)
0.588762 + 0.808307i \(0.299616\pi\)
\(390\) 0 0
\(391\) 2.66517e6 0.881624
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.18815e6 + 3.18815e6i 1.01523 + 1.01523i 0.999882 + 0.0153441i \(0.00488438\pi\)
0.0153441 + 0.999882i \(0.495116\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.76155e6i 1.16817i 0.811692 + 0.584086i \(0.198547\pi\)
−0.811692 + 0.584086i \(0.801453\pi\)
\(402\) 0 0
\(403\) −137547. + 137547.i −0.0421881 + 0.0421881i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −29893.6 + 29893.6i −0.00894524 + 0.00894524i
\(408\) 0 0
\(409\) 5.01805e6i 1.48329i −0.670792 0.741645i \(-0.734046\pi\)
0.670792 0.741645i \(-0.265954\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −254492. 254492.i −0.0734174 0.0734174i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.71304e6 −0.476685 −0.238342 0.971181i \(-0.576604\pi\)
−0.238342 + 0.971181i \(0.576604\pi\)
\(420\) 0 0
\(421\) −1.12412e6 −0.309105 −0.154552 0.987985i \(-0.549394\pi\)
−0.154552 + 0.987985i \(0.549394\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.34646e6 1.34646e6i −0.357375 0.357375i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.16531e6i 0.820773i −0.911912 0.410386i \(-0.865394\pi\)
0.911912 0.410386i \(-0.134606\pi\)
\(432\) 0 0
\(433\) 520810. 520810.i 0.133493 0.133493i −0.637203 0.770696i \(-0.719909\pi\)
0.770696 + 0.637203i \(0.219909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −639959. + 639959.i −0.160305 + 0.160305i
\(438\) 0 0
\(439\) 2.63513e6i 0.652589i 0.945268 + 0.326295i \(0.105800\pi\)
−0.945268 + 0.326295i \(0.894200\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.89945e6 4.89945e6i −1.18615 1.18615i −0.978124 0.208023i \(-0.933297\pi\)
−0.208023 0.978124i \(-0.566703\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.20484e6 0.516133 0.258066 0.966127i \(-0.416915\pi\)
0.258066 + 0.966127i \(0.416915\pi\)
\(450\) 0 0
\(451\) −4.32791e6 −1.00193
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.27070e6 3.27070e6i −0.732571 0.732571i 0.238558 0.971128i \(-0.423325\pi\)
−0.971128 + 0.238558i \(0.923325\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.76391e6i 1.70148i −0.525583 0.850742i \(-0.676153\pi\)
0.525583 0.850742i \(-0.323847\pi\)
\(462\) 0 0
\(463\) −171191. + 171191.i −0.0371133 + 0.0371133i −0.725420 0.688307i \(-0.758354\pi\)
0.688307 + 0.725420i \(0.258354\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −921034. + 921034.i −0.195426 + 0.195426i −0.798036 0.602610i \(-0.794128\pi\)
0.602610 + 0.798036i \(0.294128\pi\)
\(468\) 0 0
\(469\) 2.28611e6i 0.479917i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.74635e6 + 2.74635e6i 0.564421 + 0.564421i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 157207. 0.0313064 0.0156532 0.999877i \(-0.495017\pi\)
0.0156532 + 0.999877i \(0.495017\pi\)
\(480\) 0 0
\(481\) −3275.61 −0.000645550
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −690885. 690885.i −0.132003 0.132003i 0.638018 0.770021i \(-0.279754\pi\)
−0.770021 + 0.638018i \(0.779754\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.17012e6i 0.593434i −0.954965 0.296717i \(-0.904108\pi\)
0.954965 0.296717i \(-0.0958918\pi\)
\(492\) 0 0
\(493\) −2.36838e6 + 2.36838e6i −0.438868 + 0.438868i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.31095e6 + 1.31095e6i −0.238065 + 0.238065i
\(498\) 0 0
\(499\) 3.10049e6i 0.557415i −0.960376 0.278708i \(-0.910094\pi\)
0.960376 0.278708i \(-0.0899061\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.20877e6 3.20877e6i −0.565481 0.565481i 0.365378 0.930859i \(-0.380940\pi\)
−0.930859 + 0.365378i \(0.880940\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.43063e6 1.10017 0.550084 0.835109i \(-0.314596\pi\)
0.550084 + 0.835109i \(0.314596\pi\)
\(510\) 0 0
\(511\) −5.08032e6 −0.860674
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 440202. + 440202.i 0.0724312 + 0.0724312i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.25599e6i 0.525519i 0.964861 + 0.262759i \(0.0846325\pi\)
−0.964861 + 0.262759i \(0.915367\pi\)
\(522\) 0 0
\(523\) 259544. 259544.i 0.0414912 0.0414912i −0.686057 0.727548i \(-0.740660\pi\)
0.727548 + 0.686057i \(0.240660\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.63698e6 4.63698e6i 0.727292 0.727292i
\(528\) 0 0
\(529\) 5.42202e6i 0.842406i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −237117. 237117.i −0.0361531 0.0361531i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.12260e6 −0.611223
\(540\) 0 0
\(541\) −8.89249e6 −1.30626 −0.653131 0.757245i \(-0.726545\pi\)
−0.653131 + 0.757245i \(0.726545\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.90236e6 1.90236e6i −0.271846 0.271846i 0.557997 0.829843i \(-0.311570\pi\)
−0.829843 + 0.557997i \(0.811570\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.13739e6i 0.159599i
\(552\) 0 0
\(553\) 7.29309e6 7.29309e6i 1.01414 1.01414i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.92347e6 6.92347e6i 0.945553 0.945553i −0.0530399 0.998592i \(-0.516891\pi\)
0.998592 + 0.0530399i \(0.0168910\pi\)
\(558\) 0 0
\(559\) 300933.i 0.0407325i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.52384e6 + 9.52384e6i 1.26631 + 1.26631i 0.947979 + 0.318334i \(0.103123\pi\)
0.318334 + 0.947979i \(0.396877\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.04760e7 −1.35648 −0.678241 0.734839i \(-0.737257\pi\)
−0.678241 + 0.734839i \(0.737257\pi\)
\(570\) 0 0
\(571\) 2.27504e6 0.292011 0.146006 0.989284i \(-0.453358\pi\)
0.146006 + 0.989284i \(0.453358\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −541993. 541993.i −0.0677726 0.0677726i 0.672408 0.740181i \(-0.265260\pi\)
−0.740181 + 0.672408i \(0.765260\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.53345e7i 1.88464i
\(582\) 0 0
\(583\) −2.74510e6 + 2.74510e6i −0.334493 + 0.334493i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.04755e7 1.04755e7i 1.25482 1.25482i 0.301286 0.953534i \(-0.402584\pi\)
0.953534 0.301286i \(-0.0974158\pi\)
\(588\) 0 0
\(589\) 2.22686e6i 0.264487i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.39119e6 + 9.39119e6i 1.09669 + 1.09669i 0.994795 + 0.101894i \(0.0324904\pi\)
0.101894 + 0.994795i \(0.467510\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.38639e6 −0.385629 −0.192814 0.981235i \(-0.561762\pi\)
−0.192814 + 0.981235i \(0.561762\pi\)
\(600\) 0 0
\(601\) 7.38299e6 0.833770 0.416885 0.908959i \(-0.363122\pi\)
0.416885 + 0.908959i \(0.363122\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.09377e7 1.09377e7i −1.20491 1.20491i −0.972654 0.232259i \(-0.925388\pi\)
−0.232259 0.972654i \(-0.574612\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48235.5i 0.00522713i
\(612\) 0 0
\(613\) 9.57139e6 9.57139e6i 1.02878 1.02878i 0.0292102 0.999573i \(-0.490701\pi\)
0.999573 0.0292102i \(-0.00929921\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.65393e6 + 2.65393e6i −0.280657 + 0.280657i −0.833371 0.552714i \(-0.813592\pi\)
0.552714 + 0.833371i \(0.313592\pi\)
\(618\) 0 0
\(619\) 2.23480e6i 0.234429i 0.993107 + 0.117214i \(0.0373965\pi\)
−0.993107 + 0.117214i \(0.962604\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.40274e6 + 5.40274e6i 0.557691 + 0.557691i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 110427. 0.0111288
\(630\) 0 0
\(631\) 9.17502e6 0.917347 0.458674 0.888605i \(-0.348325\pi\)
0.458674 + 0.888605i \(0.348325\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −225869. 225869.i −0.0220550 0.0220550i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 949377.i 0.0912628i 0.998958 + 0.0456314i \(0.0145300\pi\)
−0.998958 + 0.0456314i \(0.985470\pi\)
\(642\) 0 0
\(643\) −2.12833e6 + 2.12833e6i −0.203007 + 0.203007i −0.801287 0.598280i \(-0.795851\pi\)
0.598280 + 0.801287i \(0.295851\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.02885e7 1.02885e7i 0.966253 0.966253i −0.0331962 0.999449i \(-0.510569\pi\)
0.999449 + 0.0331962i \(0.0105686\pi\)
\(648\) 0 0
\(649\) 608422.i 0.0567013i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.59428e6 + 4.59428e6i 0.421633 + 0.421633i 0.885766 0.464133i \(-0.153634\pi\)
−0.464133 + 0.885766i \(0.653634\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.22204e6 −0.468411 −0.234205 0.972187i \(-0.575249\pi\)
−0.234205 + 0.972187i \(0.575249\pi\)
\(660\) 0 0
\(661\) −2.16231e7 −1.92493 −0.962465 0.271405i \(-0.912512\pi\)
−0.962465 + 0.271405i \(0.912512\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.05378e7 1.05378e7i −0.917141 0.917141i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.21903e6i 0.276006i
\(672\) 0 0
\(673\) 1.51265e7 1.51265e7i 1.28736 1.28736i 0.350974 0.936385i \(-0.385850\pi\)
0.936385 0.350974i \(-0.114150\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.02380e6 2.02380e6i 0.169705 0.169705i −0.617144 0.786850i \(-0.711711\pi\)
0.786850 + 0.617144i \(0.211711\pi\)
\(678\) 0 0
\(679\) 1.86914e7i 1.55585i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.21866e6 8.21866e6i −0.674138 0.674138i 0.284529 0.958667i \(-0.408163\pi\)
−0.958667 + 0.284529i \(0.908163\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −300797. −0.0241393
\(690\) 0 0
\(691\) −1.62147e7 −1.29185 −0.645926 0.763400i \(-0.723529\pi\)
−0.645926 + 0.763400i \(0.723529\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.99366e6 + 7.99366e6i 0.623252 + 0.623252i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.49796e7i 1.91995i −0.280081 0.959976i \(-0.590361\pi\)
0.280081 0.959976i \(-0.409639\pi\)
\(702\) 0 0
\(703\) −26515.6 + 26515.6i −0.00202355 + 0.00202355i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.69580e7 1.69580e7i 1.27592 1.27592i
\(708\) 0 0
\(709\) 1.44518e7i 1.07971i −0.841760 0.539853i \(-0.818480\pi\)
0.841760 0.539853i \(-0.181520\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.06317e7 + 2.06317e7i 1.51989 + 1.51989i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.32849e6 0.312258 0.156129 0.987737i \(-0.450098\pi\)
0.156129 + 0.987737i \(0.450098\pi\)
\(720\) 0 0
\(721\) −1.68631e7 −1.20809
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.66934e7 1.66934e7i −1.17141 1.17141i −0.981873 0.189538i \(-0.939301\pi\)
−0.189538 0.981873i \(-0.560699\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.01450e7i 0.702198i
\(732\) 0 0
\(733\) −7.29281e6 + 7.29281e6i −0.501343 + 0.501343i −0.911855 0.410512i \(-0.865350\pi\)
0.410512 + 0.911855i \(0.365350\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.73274e6 2.73274e6i 0.185323 0.185323i
\(738\) 0 0
\(739\) 7.99531e6i 0.538548i −0.963064 0.269274i \(-0.913216\pi\)
0.963064 0.269274i \(-0.0867837\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.74977e7 1.74977e7i −1.16281 1.16281i −0.983858 0.178953i \(-0.942729\pi\)
−0.178953 0.983858i \(-0.557271\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.35509e6 0.348788
\(750\) 0 0
\(751\) −1.18625e7 −0.767494 −0.383747 0.923438i \(-0.625366\pi\)
−0.383747 + 0.923438i \(0.625366\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.18681e7 1.18681e7i −0.752733 0.752733i 0.222255 0.974989i \(-0.428658\pi\)
−0.974989 + 0.222255i \(0.928658\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.36406e7i 0.853832i −0.904291 0.426916i \(-0.859600\pi\)
0.904291 0.426916i \(-0.140400\pi\)
\(762\) 0 0
\(763\) −7.05102e6 + 7.05102e6i −0.438471 + 0.438471i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33334.2 33334.2i 0.00204598 0.00204598i
\(768\) 0 0
\(769\) 8.00583e6i 0.488192i 0.969751 + 0.244096i \(0.0784912\pi\)
−0.969751 + 0.244096i \(0.921509\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.94684e6 + 2.94684e6i 0.177381 + 0.177381i 0.790213 0.612832i \(-0.209970\pi\)
−0.612832 + 0.790213i \(0.709970\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.83886e6 −0.226652
\(780\) 0 0
\(781\) −3.13414e6 −0.183861
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.07512e6 + 5.07512e6i 0.292085 + 0.292085i 0.837904 0.545818i \(-0.183781\pi\)
−0.545818 + 0.837904i \(0.683781\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.53632e7i 0.873053i
\(792\) 0 0
\(793\) 176364. 176364.i 0.00995924 0.00995924i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.32317e7 + 1.32317e7i −0.737855 + 0.737855i −0.972162 0.234308i \(-0.924718\pi\)
0.234308 + 0.972162i \(0.424718\pi\)
\(798\) 0 0
\(799\) 1.62611e6i 0.0901120i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.07284e6 6.07284e6i −0.332355 0.332355i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.80318e6 0.204303 0.102152 0.994769i \(-0.467427\pi\)
0.102152 + 0.994769i \(0.467427\pi\)
\(810\) 0 0
\(811\) 1.12914e7 0.602829 0.301414 0.953493i \(-0.402541\pi\)
0.301414 + 0.953493i \(0.402541\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.43601e6 + 2.43601e6i 0.127681 + 0.127681i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.15427e7i 1.11543i −0.830032 0.557716i \(-0.811678\pi\)
0.830032 0.557716i \(-0.188322\pi\)
\(822\) 0 0
\(823\) 8.73412e6 8.73412e6i 0.449489 0.449489i −0.445695 0.895185i \(-0.647044\pi\)
0.895185 + 0.445695i \(0.147044\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.59254e7 1.59254e7i 0.809705 0.809705i −0.174884 0.984589i \(-0.555955\pi\)
0.984589 + 0.174884i \(0.0559552\pi\)
\(828\) 0 0
\(829\) 2.78306e7i 1.40649i −0.710948 0.703244i \(-0.751734\pi\)
0.710948 0.703244i \(-0.248266\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.61445e6 + 7.61445e6i 0.380212 + 0.380212i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.50255e6 −0.417008 −0.208504 0.978022i \(-0.566859\pi\)
−0.208504 + 0.978022i \(0.566859\pi\)
\(840\) 0 0
\(841\) −1.78247e6 −0.0869026
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.07933e6 + 9.07933e6i 0.434856 + 0.434856i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 491332.i 0.0232568i
\(852\) 0 0
\(853\) 4.51666e6 4.51666e6i 0.212542 0.212542i −0.592805 0.805346i \(-0.701979\pi\)
0.805346 + 0.592805i \(0.201979\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.32536e7 2.32536e7i 1.08153 1.08153i 0.0851602 0.996367i \(-0.472860\pi\)
0.996367 0.0851602i \(-0.0271402\pi\)
\(858\) 0 0
\(859\) 3.86671e7i 1.78796i 0.448105 + 0.893981i \(0.352099\pi\)
−0.448105 + 0.893981i \(0.647901\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.41644e7 + 1.41644e7i 0.647400 + 0.647400i 0.952364 0.304964i \(-0.0986444\pi\)
−0.304964 + 0.952364i \(0.598644\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.74358e7 0.783236
\(870\) 0 0
\(871\) 299442. 0.0133742
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.20885e6 + 6.20885e6i 0.272591 + 0.272591i 0.830143 0.557551i \(-0.188259\pi\)
−0.557551 + 0.830143i \(0.688259\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.69249e7i 0.734660i 0.930091 + 0.367330i \(0.119728\pi\)
−0.930091 + 0.367330i \(0.880272\pi\)
\(882\) 0 0
\(883\) −154530. + 154530.i −0.00666976 + 0.00666976i −0.710434 0.703764i \(-0.751501\pi\)
0.703764 + 0.710434i \(0.251501\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.66600e7 1.66600e7i 0.710992 0.710992i −0.255751 0.966743i \(-0.582323\pi\)
0.966743 + 0.255751i \(0.0823226\pi\)
\(888\) 0 0
\(889\) 1.48461e7i 0.630024i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 390460. + 390460.i 0.0163850 + 0.0163850i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.66683e7 −1.51318
\(900\) 0 0
\(901\) 1.01404e7 0.416145
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.88373e7 1.88373e7i −0.760328 0.760328i 0.216053 0.976382i \(-0.430681\pi\)
−0.976382 + 0.216053i \(0.930681\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.97112e7i 1.18611i 0.805163 + 0.593054i \(0.202078\pi\)
−0.805163 + 0.593054i \(0.797922\pi\)
\(912\) 0 0
\(913\) 1.83303e7 1.83303e7i 0.727767 0.727767i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.93800e7 + 3.93800e7i −1.54651 + 1.54651i
\(918\) 0 0
\(919\) 1.68807e7i 0.659329i −0.944098 0.329665i \(-0.893064\pi\)
0.944098 0.329665i \(-0.106936\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −171713. 171713.i −0.00663435 0.00663435i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.25540e7 1.23756 0.618779 0.785565i \(-0.287628\pi\)
0.618779 + 0.785565i \(0.287628\pi\)
\(930\) 0 0
\(931\) −3.65675e6 −0.138268
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.56214e6 6.56214e6i −0.244172 0.244172i 0.574401 0.818574i \(-0.305235\pi\)
−0.818574 + 0.574401i \(0.805235\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.46801e7i 1.64490i 0.568836 + 0.822451i \(0.307394\pi\)
−0.568836 + 0.822451i \(0.692606\pi\)
\(942\) 0 0
\(943\) −3.55668e7 + 3.55668e7i −1.30246 + 1.30246i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.74895e7 + 2.74895e7i −0.996075 + 0.996075i −0.999992 0.00391761i \(-0.998753\pi\)
0.00391761 + 0.999992i \(0.498753\pi\)
\(948\) 0 0
\(949\) 665436.i 0.0239851i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.00343e7 + 3.00343e7i 1.07123 + 1.07123i 0.997260 + 0.0739747i \(0.0235684\pi\)
0.0739747 + 0.997260i \(0.476432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.01725e7 −2.46389
\(960\) 0 0
\(961\) 4.31626e7 1.50765
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.78773e7 + 1.78773e7i 0.614803 + 0.614803i 0.944194 0.329391i \(-0.106843\pi\)
−0.329391 + 0.944194i \(0.606843\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.75062e7i 0.595861i 0.954588 + 0.297930i \(0.0962963\pi\)
−0.954588 + 0.297930i \(0.903704\pi\)
\(972\) 0 0
\(973\) 2.66992e7 2.66992e7i 0.904098 0.904098i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.05445e6 6.05445e6i 0.202926 0.202926i −0.598326 0.801253i \(-0.704167\pi\)
0.801253 + 0.598326i \(0.204167\pi\)
\(978\) 0 0
\(979\) 1.29165e7i 0.430713i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.92434e7 3.92434e7i −1.29534 1.29534i −0.931439 0.363898i \(-0.881446\pi\)
−0.363898 0.931439i \(-0.618554\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.51390e7 1.46744
\(990\) 0 0
\(991\) 9.21716e6 0.298135 0.149068 0.988827i \(-0.452373\pi\)
0.149068 + 0.988827i \(0.452373\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −3.03016e6 3.03016e6i −0.0965446 0.0965446i 0.657185 0.753729i \(-0.271747\pi\)
−0.753729 + 0.657185i \(0.771747\pi\)
\(998\) 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 900.6.j.c.557.11 yes 24
3.2 odd 2 inner 900.6.j.c.557.12 yes 24
5.2 odd 4 inner 900.6.j.c.593.2 yes 24
5.3 odd 4 inner 900.6.j.c.593.12 yes 24
5.4 even 2 inner 900.6.j.c.557.1 24
15.2 even 4 inner 900.6.j.c.593.1 yes 24
15.8 even 4 inner 900.6.j.c.593.11 yes 24
15.14 odd 2 inner 900.6.j.c.557.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.6.j.c.557.1 24 5.4 even 2 inner
900.6.j.c.557.2 yes 24 15.14 odd 2 inner
900.6.j.c.557.11 yes 24 1.1 even 1 trivial
900.6.j.c.557.12 yes 24 3.2 odd 2 inner
900.6.j.c.593.1 yes 24 15.2 even 4 inner
900.6.j.c.593.2 yes 24 5.2 odd 4 inner
900.6.j.c.593.11 yes 24 15.8 even 4 inner
900.6.j.c.593.12 yes 24 5.3 odd 4 inner