Properties

Label 3600.3.l.d.1601.2
Level $3600$
Weight $3$
Character 3600.1601
Analytic conductor $98.093$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,3,Mod(1601,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.l (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: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1601.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 3600.1601
Dual form 3600.3.l.d.1601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{7} +O(q^{10})\) \(q-4.00000 q^{7} +16.9706i q^{11} -8.00000 q^{13} -12.7279i q^{17} +16.0000 q^{19} +16.9706i q^{23} -4.24264i q^{29} -44.0000 q^{31} +34.0000 q^{37} -46.6690i q^{41} -40.0000 q^{43} +84.8528i q^{47} -33.0000 q^{49} +38.1838i q^{53} +33.9411i q^{59} +50.0000 q^{61} +8.00000 q^{67} -50.9117i q^{71} +16.0000 q^{73} -67.8823i q^{77} +76.0000 q^{79} -118.794i q^{83} -12.7279i q^{89} +32.0000 q^{91} -176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 16 q^{13} + 32 q^{19} - 88 q^{31} + 68 q^{37} - 80 q^{43} - 66 q^{49} + 100 q^{61} + 16 q^{67} + 32 q^{73} + 152 q^{79} + 64 q^{91} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −0.571429 −0.285714 0.958315i \(-0.592231\pi\)
−0.285714 + 0.958315i \(0.592231\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 16.9706i 1.54278i 0.636364 + 0.771389i \(0.280438\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) −8.00000 −0.615385 −0.307692 0.951486i \(-0.599557\pi\)
−0.307692 + 0.951486i \(0.599557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 12.7279i − 0.748701i −0.927287 0.374351i \(-0.877866\pi\)
0.927287 0.374351i \(-0.122134\pi\)
\(18\) 0 0
\(19\) 16.0000 0.842105 0.421053 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 16.9706i 0.737851i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.146298i −0.997321 0.0731490i \(-0.976695\pi\)
0.997321 0.0731490i \(-0.0233049\pi\)
\(30\) 0 0
\(31\) −44.0000 −1.41935 −0.709677 0.704527i \(-0.751159\pi\)
−0.709677 + 0.704527i \(0.751159\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000 0.918919 0.459459 0.888199i \(-0.348043\pi\)
0.459459 + 0.888199i \(0.348043\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 46.6690i − 1.13827i −0.822244 0.569135i \(-0.807278\pi\)
0.822244 0.569135i \(-0.192722\pi\)
\(42\) 0 0
\(43\) −40.0000 −0.930233 −0.465116 0.885250i \(-0.653987\pi\)
−0.465116 + 0.885250i \(0.653987\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 84.8528i 1.80538i 0.430293 + 0.902690i \(0.358410\pi\)
−0.430293 + 0.902690i \(0.641590\pi\)
\(48\) 0 0
\(49\) −33.0000 −0.673469
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 38.1838i 0.720448i 0.932866 + 0.360224i \(0.117300\pi\)
−0.932866 + 0.360224i \(0.882700\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) 50.0000 0.819672 0.409836 0.912159i \(-0.365586\pi\)
0.409836 + 0.912159i \(0.365586\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.119403 0.0597015 0.998216i \(-0.480985\pi\)
0.0597015 + 0.998216i \(0.480985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 50.9117i − 0.717066i −0.933517 0.358533i \(-0.883277\pi\)
0.933517 0.358533i \(-0.116723\pi\)
\(72\) 0 0
\(73\) 16.0000 0.219178 0.109589 0.993977i \(-0.465047\pi\)
0.109589 + 0.993977i \(0.465047\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 67.8823i − 0.881588i
\(78\) 0 0
\(79\) 76.0000 0.962025 0.481013 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 118.794i − 1.43125i −0.698484 0.715626i \(-0.746141\pi\)
0.698484 0.715626i \(-0.253859\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12.7279i − 0.143010i −0.997440 0.0715052i \(-0.977220\pi\)
0.997440 0.0715052i \(-0.0227802\pi\)
\(90\) 0 0
\(91\) 32.0000 0.351648
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −176.000 −1.81443 −0.907216 0.420664i \(-0.861797\pi\)
−0.907216 + 0.420664i \(0.861797\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 29.6985i − 0.294044i −0.989133 0.147022i \(-0.953031\pi\)
0.989133 0.147022i \(-0.0469689\pi\)
\(102\) 0 0
\(103\) −28.0000 −0.271845 −0.135922 0.990719i \(-0.543400\pi\)
−0.135922 + 0.990719i \(0.543400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 56.0000 0.513761 0.256881 0.966443i \(-0.417305\pi\)
0.256881 + 0.966443i \(0.417305\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 156.978i − 1.38918i −0.719404 0.694592i \(-0.755585\pi\)
0.719404 0.694592i \(-0.244415\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 50.9117i 0.427829i
\(120\) 0 0
\(121\) −167.000 −1.38017
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 92.0000 0.724409 0.362205 0.932099i \(-0.382024\pi\)
0.362205 + 0.932099i \(0.382024\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 169.706i − 1.29546i −0.761869 0.647731i \(-0.775718\pi\)
0.761869 0.647731i \(-0.224282\pi\)
\(132\) 0 0
\(133\) −64.0000 −0.481203
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 156.978i 1.14582i 0.819617 + 0.572911i \(0.194186\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(138\) 0 0
\(139\) −152.000 −1.09353 −0.546763 0.837288i \(-0.684140\pi\)
−0.546763 + 0.837288i \(0.684140\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 135.765i − 0.949402i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 275.772i − 1.85082i −0.378972 0.925408i \(-0.623722\pi\)
0.378972 0.925408i \(-0.376278\pi\)
\(150\) 0 0
\(151\) 148.000 0.980132 0.490066 0.871685i \(-0.336973\pi\)
0.490066 + 0.871685i \(0.336973\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 82.0000 0.522293 0.261146 0.965299i \(-0.415899\pi\)
0.261146 + 0.965299i \(0.415899\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 67.8823i − 0.421629i
\(162\) 0 0
\(163\) 56.0000 0.343558 0.171779 0.985135i \(-0.445048\pi\)
0.171779 + 0.985135i \(0.445048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 33.9411i − 0.203240i −0.994823 0.101620i \(-0.967597\pi\)
0.994823 0.101620i \(-0.0324026\pi\)
\(168\) 0 0
\(169\) −105.000 −0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 173.948i − 1.00548i −0.864437 0.502741i \(-0.832325\pi\)
0.864437 0.502741i \(-0.167675\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 203.647i − 1.13769i −0.822444 0.568846i \(-0.807390\pi\)
0.822444 0.568846i \(-0.192610\pi\)
\(180\) 0 0
\(181\) −232.000 −1.28177 −0.640884 0.767638i \(-0.721432\pi\)
−0.640884 + 0.767638i \(0.721432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 216.000 1.15508
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 33.9411i − 0.177702i −0.996045 0.0888511i \(-0.971680\pi\)
0.996045 0.0888511i \(-0.0283195\pi\)
\(192\) 0 0
\(193\) −206.000 −1.06736 −0.533679 0.845687i \(-0.679191\pi\)
−0.533679 + 0.845687i \(0.679191\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 165.463i 0.839914i 0.907544 + 0.419957i \(0.137955\pi\)
−0.907544 + 0.419957i \(0.862045\pi\)
\(198\) 0 0
\(199\) −20.0000 −0.100503 −0.0502513 0.998737i \(-0.516002\pi\)
−0.0502513 + 0.998737i \(0.516002\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.9706i 0.0835988i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 271.529i 1.29918i
\(210\) 0 0
\(211\) −296.000 −1.40284 −0.701422 0.712746i \(-0.747451\pi\)
−0.701422 + 0.712746i \(0.747451\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 176.000 0.811060
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 101.823i 0.460739i
\(222\) 0 0
\(223\) −436.000 −1.95516 −0.977578 0.210571i \(-0.932468\pi\)
−0.977578 + 0.210571i \(0.932468\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 16.9706i − 0.0747602i −0.999301 0.0373801i \(-0.988099\pi\)
0.999301 0.0373801i \(-0.0119012\pi\)
\(228\) 0 0
\(229\) 8.00000 0.0349345 0.0174672 0.999847i \(-0.494440\pi\)
0.0174672 + 0.999847i \(0.494440\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 12.7279i − 0.0546263i −0.999627 0.0273131i \(-0.991305\pi\)
0.999627 0.0273131i \(-0.00869512\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 135.765i 0.568052i 0.958817 + 0.284026i \(0.0916703\pi\)
−0.958817 + 0.284026i \(0.908330\pi\)
\(240\) 0 0
\(241\) 32.0000 0.132780 0.0663900 0.997794i \(-0.478852\pi\)
0.0663900 + 0.997794i \(0.478852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −128.000 −0.518219
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 50.9117i 0.202835i 0.994844 + 0.101418i \(0.0323379\pi\)
−0.994844 + 0.101418i \(0.967662\pi\)
\(252\) 0 0
\(253\) −288.000 −1.13834
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 182.434i − 0.709858i −0.934893 0.354929i \(-0.884505\pi\)
0.934893 0.354929i \(-0.115495\pi\)
\(258\) 0 0
\(259\) −136.000 −0.525097
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 373.352i − 1.41959i −0.704408 0.709795i \(-0.748787\pi\)
0.704408 0.709795i \(-0.251213\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 343.654i 1.27752i 0.769404 + 0.638762i \(0.220553\pi\)
−0.769404 + 0.638762i \(0.779447\pi\)
\(270\) 0 0
\(271\) −380.000 −1.40221 −0.701107 0.713056i \(-0.747310\pi\)
−0.701107 + 0.713056i \(0.747310\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 328.000 1.18412 0.592058 0.805896i \(-0.298316\pi\)
0.592058 + 0.805896i \(0.298316\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 284.257i − 1.01159i −0.862654 0.505795i \(-0.831199\pi\)
0.862654 0.505795i \(-0.168801\pi\)
\(282\) 0 0
\(283\) −208.000 −0.734982 −0.367491 0.930027i \(-0.619783\pi\)
−0.367491 + 0.930027i \(0.619783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 186.676i 0.650440i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 436.992i − 1.49144i −0.666259 0.745720i \(-0.732106\pi\)
0.666259 0.745720i \(-0.267894\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 135.765i − 0.454062i
\(300\) 0 0
\(301\) 160.000 0.531561
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −520.000 −1.69381 −0.846906 0.531743i \(-0.821537\pi\)
−0.846906 + 0.531743i \(0.821537\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 373.352i − 1.20049i −0.799816 0.600245i \(-0.795070\pi\)
0.799816 0.600245i \(-0.204930\pi\)
\(312\) 0 0
\(313\) 94.0000 0.300319 0.150160 0.988662i \(-0.452021\pi\)
0.150160 + 0.988662i \(0.452021\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 335.169i 1.05731i 0.848835 + 0.528657i \(0.177304\pi\)
−0.848835 + 0.528657i \(0.822696\pi\)
\(318\) 0 0
\(319\) 72.0000 0.225705
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 203.647i − 0.630485i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 339.411i − 1.03165i
\(330\) 0 0
\(331\) −536.000 −1.61934 −0.809668 0.586889i \(-0.800353\pi\)
−0.809668 + 0.586889i \(0.800353\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 208.000 0.617211 0.308605 0.951190i \(-0.400138\pi\)
0.308605 + 0.951190i \(0.400138\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 746.705i − 2.18975i
\(342\) 0 0
\(343\) 328.000 0.956268
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 288.500i − 0.831411i −0.909499 0.415705i \(-0.863535\pi\)
0.909499 0.415705i \(-0.136465\pi\)
\(348\) 0 0
\(349\) −238.000 −0.681948 −0.340974 0.940073i \(-0.610757\pi\)
−0.340974 + 0.940073i \(0.610757\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 224.860i − 0.636997i −0.947923 0.318499i \(-0.896821\pi\)
0.947923 0.318499i \(-0.103179\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 560.029i − 1.55997i −0.625799 0.779984i \(-0.715227\pi\)
0.625799 0.779984i \(-0.284773\pi\)
\(360\) 0 0
\(361\) −105.000 −0.290859
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 284.000 0.773842 0.386921 0.922113i \(-0.373539\pi\)
0.386921 + 0.922113i \(0.373539\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 152.735i − 0.411685i
\(372\) 0 0
\(373\) 190.000 0.509383 0.254692 0.967022i \(-0.418026\pi\)
0.254692 + 0.967022i \(0.418026\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.9411i 0.0900295i
\(378\) 0 0
\(379\) 160.000 0.422164 0.211082 0.977468i \(-0.432301\pi\)
0.211082 + 0.977468i \(0.432301\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 271.529i 0.708953i 0.935065 + 0.354477i \(0.115341\pi\)
−0.935065 + 0.354477i \(0.884659\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 403.051i 1.03612i 0.855344 + 0.518060i \(0.173346\pi\)
−0.855344 + 0.518060i \(0.826654\pi\)
\(390\) 0 0
\(391\) 216.000 0.552430
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −146.000 −0.367758 −0.183879 0.982949i \(-0.558865\pi\)
−0.183879 + 0.982949i \(0.558865\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 326.683i − 0.814672i −0.913278 0.407336i \(-0.866458\pi\)
0.913278 0.407336i \(-0.133542\pi\)
\(402\) 0 0
\(403\) 352.000 0.873449
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 576.999i 1.41769i
\(408\) 0 0
\(409\) 368.000 0.899756 0.449878 0.893090i \(-0.351468\pi\)
0.449878 + 0.893090i \(0.351468\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 135.765i − 0.328728i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 390.323i 0.931558i 0.884901 + 0.465779i \(0.154226\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(420\) 0 0
\(421\) −40.0000 −0.0950119 −0.0475059 0.998871i \(-0.515127\pi\)
−0.0475059 + 0.998871i \(0.515127\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −200.000 −0.468384
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 152.735i − 0.354374i −0.984177 0.177187i \(-0.943300\pi\)
0.984177 0.177187i \(-0.0566997\pi\)
\(432\) 0 0
\(433\) −542.000 −1.25173 −0.625866 0.779931i \(-0.715254\pi\)
−0.625866 + 0.779931i \(0.715254\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 271.529i 0.621348i
\(438\) 0 0
\(439\) 4.00000 0.00911162 0.00455581 0.999990i \(-0.498550\pi\)
0.00455581 + 0.999990i \(0.498550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 322.441i − 0.727857i −0.931427 0.363929i \(-0.881435\pi\)
0.931427 0.363929i \(-0.118565\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 216.375i − 0.481904i −0.970537 0.240952i \(-0.922540\pi\)
0.970537 0.240952i \(-0.0774596\pi\)
\(450\) 0 0
\(451\) 792.000 1.75610
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 400.000 0.875274 0.437637 0.899152i \(-0.355816\pi\)
0.437637 + 0.899152i \(0.355816\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 301.227i 0.653422i 0.945124 + 0.326711i \(0.105940\pi\)
−0.945124 + 0.326711i \(0.894060\pi\)
\(462\) 0 0
\(463\) −604.000 −1.30454 −0.652268 0.757989i \(-0.726182\pi\)
−0.652268 + 0.757989i \(0.726182\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 356.382i − 0.763130i −0.924342 0.381565i \(-0.875385\pi\)
0.924342 0.381565i \(-0.124615\pi\)
\(468\) 0 0
\(469\) −32.0000 −0.0682303
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 678.823i − 1.43514i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 526.087i 1.09830i 0.835723 + 0.549152i \(0.185049\pi\)
−0.835723 + 0.549152i \(0.814951\pi\)
\(480\) 0 0
\(481\) −272.000 −0.565489
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 596.000 1.22382 0.611910 0.790928i \(-0.290402\pi\)
0.611910 + 0.790928i \(0.290402\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 271.529i − 0.553012i −0.961012 0.276506i \(-0.910823\pi\)
0.961012 0.276506i \(-0.0891766\pi\)
\(492\) 0 0
\(493\) −54.0000 −0.109533
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 203.647i 0.409752i
\(498\) 0 0
\(499\) −224.000 −0.448898 −0.224449 0.974486i \(-0.572058\pi\)
−0.224449 + 0.974486i \(0.572058\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 865.499i − 1.72067i −0.509726 0.860337i \(-0.670253\pi\)
0.509726 0.860337i \(-0.329747\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 479.418i − 0.941883i −0.882164 0.470941i \(-0.843914\pi\)
0.882164 0.470941i \(-0.156086\pi\)
\(510\) 0 0
\(511\) −64.0000 −0.125245
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1440.00 −2.78530
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 521.845i 1.00162i 0.865557 + 0.500811i \(0.166965\pi\)
−0.865557 + 0.500811i \(0.833035\pi\)
\(522\) 0 0
\(523\) −736.000 −1.40727 −0.703633 0.710564i \(-0.748440\pi\)
−0.703633 + 0.710564i \(0.748440\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 560.029i 1.06267i
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 373.352i 0.700474i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 560.029i − 1.03901i
\(540\) 0 0
\(541\) −808.000 −1.49353 −0.746765 0.665088i \(-0.768394\pi\)
−0.746765 + 0.665088i \(0.768394\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 536.000 0.979890 0.489945 0.871753i \(-0.337017\pi\)
0.489945 + 0.871753i \(0.337017\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 67.8823i − 0.123198i
\(552\) 0 0
\(553\) −304.000 −0.549729
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 165.463i − 0.297061i −0.988908 0.148531i \(-0.952546\pi\)
0.988908 0.148531i \(-0.0474543\pi\)
\(558\) 0 0
\(559\) 320.000 0.572451
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 322.441i 0.572719i 0.958122 + 0.286359i \(0.0924451\pi\)
−0.958122 + 0.286359i \(0.907555\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 156.978i − 0.275883i −0.990440 0.137942i \(-0.955951\pi\)
0.990440 0.137942i \(-0.0440487\pi\)
\(570\) 0 0
\(571\) −368.000 −0.644483 −0.322242 0.946657i \(-0.604436\pi\)
−0.322242 + 0.946657i \(0.604436\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 142.000 0.246101 0.123050 0.992400i \(-0.460732\pi\)
0.123050 + 0.992400i \(0.460732\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 475.176i 0.817858i
\(582\) 0 0
\(583\) −648.000 −1.11149
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 373.352i − 0.636035i −0.948085 0.318017i \(-0.896983\pi\)
0.948085 0.318017i \(-0.103017\pi\)
\(588\) 0 0
\(589\) −704.000 −1.19525
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1107.33i 1.86733i 0.358142 + 0.933667i \(0.383410\pi\)
−0.358142 + 0.933667i \(0.616590\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 797.616i 1.33158i 0.746139 + 0.665790i \(0.231905\pi\)
−0.746139 + 0.665790i \(0.768095\pi\)
\(600\) 0 0
\(601\) 158.000 0.262895 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 332.000 0.546952 0.273476 0.961879i \(-0.411827\pi\)
0.273476 + 0.961879i \(0.411827\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 678.823i − 1.11100i
\(612\) 0 0
\(613\) −578.000 −0.942904 −0.471452 0.881892i \(-0.656270\pi\)
−0.471452 + 0.881892i \(0.656270\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 55.1543i − 0.0893911i −0.999001 0.0446956i \(-0.985768\pi\)
0.999001 0.0446956i \(-0.0142318\pi\)
\(618\) 0 0
\(619\) −896.000 −1.44750 −0.723748 0.690064i \(-0.757582\pi\)
−0.723748 + 0.690064i \(0.757582\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 50.9117i 0.0817202i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 432.749i − 0.687996i
\(630\) 0 0
\(631\) −20.0000 −0.0316957 −0.0158479 0.999874i \(-0.505045\pi\)
−0.0158479 + 0.999874i \(0.505045\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 264.000 0.414443
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 258.801i − 0.403746i −0.979412 0.201873i \(-0.935297\pi\)
0.979412 0.201873i \(-0.0647028\pi\)
\(642\) 0 0
\(643\) 728.000 1.13219 0.566096 0.824339i \(-0.308453\pi\)
0.566096 + 0.824339i \(0.308453\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 458.205i 0.708200i 0.935208 + 0.354100i \(0.115213\pi\)
−0.935208 + 0.354100i \(0.884787\pi\)
\(648\) 0 0
\(649\) −576.000 −0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 301.227i 0.461298i 0.973037 + 0.230649i \(0.0740849\pi\)
−0.973037 + 0.230649i \(0.925915\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1052.17i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(660\) 0 0
\(661\) 62.0000 0.0937973 0.0468986 0.998900i \(-0.485066\pi\)
0.0468986 + 0.998900i \(0.485066\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 72.0000 0.107946
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 848.528i 1.26457i
\(672\) 0 0
\(673\) 670.000 0.995542 0.497771 0.867308i \(-0.334152\pi\)
0.497771 + 0.867308i \(0.334152\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1294.01i − 1.91138i −0.294372 0.955691i \(-0.595111\pi\)
0.294372 0.955691i \(-0.404889\pi\)
\(678\) 0 0
\(679\) 704.000 1.03682
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 560.029i 0.819954i 0.912096 + 0.409977i \(0.134463\pi\)
−0.912096 + 0.409977i \(0.865537\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 305.470i − 0.443353i
\(690\) 0 0
\(691\) 40.0000 0.0578871 0.0289436 0.999581i \(-0.490786\pi\)
0.0289436 + 0.999581i \(0.490786\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −594.000 −0.852224
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 954.594i − 1.36176i −0.732395 0.680880i \(-0.761597\pi\)
0.732395 0.680880i \(-0.238403\pi\)
\(702\) 0 0
\(703\) 544.000 0.773826
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 118.794i 0.168025i
\(708\) 0 0
\(709\) 968.000 1.36530 0.682652 0.730744i \(-0.260827\pi\)
0.682652 + 0.730744i \(0.260827\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 746.705i − 1.04727i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1170.97i 1.62861i 0.580439 + 0.814304i \(0.302881\pi\)
−0.580439 + 0.814304i \(0.697119\pi\)
\(720\) 0 0
\(721\) 112.000 0.155340
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −508.000 −0.698762 −0.349381 0.936981i \(-0.613608\pi\)
−0.349381 + 0.936981i \(0.613608\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 509.117i 0.696466i
\(732\) 0 0
\(733\) 1144.00 1.56071 0.780355 0.625337i \(-0.215039\pi\)
0.780355 + 0.625337i \(0.215039\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 135.765i 0.184212i
\(738\) 0 0
\(739\) 304.000 0.411367 0.205683 0.978619i \(-0.434058\pi\)
0.205683 + 0.978619i \(0.434058\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 848.528i − 1.14203i −0.820940 0.571015i \(-0.806550\pi\)
0.820940 0.571015i \(-0.193450\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −188.000 −0.250333 −0.125166 0.992136i \(-0.539946\pi\)
−0.125166 + 0.992136i \(0.539946\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1240.00 1.63804 0.819022 0.573761i \(-0.194516\pi\)
0.819022 + 0.573761i \(0.194516\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 156.978i 0.206278i 0.994667 + 0.103139i \(0.0328887\pi\)
−0.994667 + 0.103139i \(0.967111\pi\)
\(762\) 0 0
\(763\) −224.000 −0.293578
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 271.529i − 0.354014i
\(768\) 0 0
\(769\) −910.000 −1.18336 −0.591678 0.806175i \(-0.701534\pi\)
−0.591678 + 0.806175i \(0.701534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1387.34i − 1.79475i −0.441266 0.897376i \(-0.645471\pi\)
0.441266 0.897376i \(-0.354529\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 746.705i − 0.958543i
\(780\) 0 0
\(781\) 864.000 1.10627
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1360.00 −1.72808 −0.864041 0.503422i \(-0.832074\pi\)
−0.864041 + 0.503422i \(0.832074\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 627.911i 0.793819i
\(792\) 0 0
\(793\) −400.000 −0.504414
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 106.066i − 0.133082i −0.997784 0.0665408i \(-0.978804\pi\)
0.997784 0.0665408i \(-0.0211963\pi\)
\(798\) 0 0
\(799\) 1080.00 1.35169
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 271.529i 0.338143i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1107.33i 1.36876i 0.729124 + 0.684381i \(0.239928\pi\)
−0.729124 + 0.684381i \(0.760072\pi\)
\(810\) 0 0
\(811\) 160.000 0.197287 0.0986436 0.995123i \(-0.468550\pi\)
0.0986436 + 0.995123i \(0.468550\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −640.000 −0.783354
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 436.992i − 0.532268i −0.963936 0.266134i \(-0.914254\pi\)
0.963936 0.266134i \(-0.0857464\pi\)
\(822\) 0 0
\(823\) 332.000 0.403402 0.201701 0.979447i \(-0.435353\pi\)
0.201701 + 0.979447i \(0.435353\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 101.823i − 0.123124i −0.998103 0.0615619i \(-0.980392\pi\)
0.998103 0.0615619i \(-0.0196082\pi\)
\(828\) 0 0
\(829\) 632.000 0.762364 0.381182 0.924500i \(-0.375517\pi\)
0.381182 + 0.924500i \(0.375517\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 420.021i 0.504227i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 729.734i 0.869767i 0.900487 + 0.434883i \(0.143210\pi\)
−0.900487 + 0.434883i \(0.856790\pi\)
\(840\) 0 0
\(841\) 823.000 0.978597
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 668.000 0.788666
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 576.999i 0.678025i
\(852\) 0 0
\(853\) −446.000 −0.522860 −0.261430 0.965222i \(-0.584194\pi\)
−0.261430 + 0.965222i \(0.584194\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 428.507i − 0.500008i −0.968245 0.250004i \(-0.919568\pi\)
0.968245 0.250004i \(-0.0804319\pi\)
\(858\) 0 0
\(859\) −728.000 −0.847497 −0.423749 0.905780i \(-0.639286\pi\)
−0.423749 + 0.905780i \(0.639286\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 916.410i 1.06189i 0.847407 + 0.530945i \(0.178163\pi\)
−0.847407 + 0.530945i \(0.821837\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1289.76i 1.48419i
\(870\) 0 0
\(871\) −64.0000 −0.0734788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 910.000 1.03763 0.518814 0.854887i \(-0.326374\pi\)
0.518814 + 0.854887i \(0.326374\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 929.138i 1.05464i 0.849667 + 0.527320i \(0.176803\pi\)
−0.849667 + 0.527320i \(0.823197\pi\)
\(882\) 0 0
\(883\) 1064.00 1.20498 0.602492 0.798125i \(-0.294175\pi\)
0.602492 + 0.798125i \(0.294175\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1391.59i 1.56887i 0.620212 + 0.784434i \(0.287047\pi\)
−0.620212 + 0.784434i \(0.712953\pi\)
\(888\) 0 0
\(889\) −368.000 −0.413948
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1357.65i 1.52032i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 186.676i 0.207649i
\(900\) 0 0
\(901\) 486.000 0.539401
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1768.00 −1.94928 −0.974642 0.223771i \(-0.928163\pi\)
−0.974642 + 0.223771i \(0.928163\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 237.588i − 0.260799i −0.991462 0.130399i \(-0.958374\pi\)
0.991462 0.130399i \(-0.0416260\pi\)
\(912\) 0 0
\(913\) 2016.00 2.20811
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 678.823i 0.740264i
\(918\) 0 0
\(919\) −380.000 −0.413493 −0.206746 0.978395i \(-0.566288\pi\)
−0.206746 + 0.978395i \(0.566288\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 407.294i 0.441271i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 666.095i − 0.717002i −0.933529 0.358501i \(-0.883288\pi\)
0.933529 0.358501i \(-0.116712\pi\)
\(930\) 0 0
\(931\) −528.000 −0.567132
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 178.000 0.189968 0.0949840 0.995479i \(-0.469720\pi\)
0.0949840 + 0.995479i \(0.469720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 436.992i 0.464391i 0.972669 + 0.232196i \(0.0745909\pi\)
−0.972669 + 0.232196i \(0.925409\pi\)
\(942\) 0 0
\(943\) 792.000 0.839873
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1798.88i − 1.89956i −0.312924 0.949778i \(-0.601309\pi\)
0.312924 0.949778i \(-0.398691\pi\)
\(948\) 0 0
\(949\) −128.000 −0.134879
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1310.98i 1.37563i 0.725886 + 0.687815i \(0.241430\pi\)
−0.725886 + 0.687815i \(0.758570\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 627.911i − 0.654756i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1700.00 1.75801 0.879007 0.476808i \(-0.158206\pi\)
0.879007 + 0.476808i \(0.158206\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 458.205i 0.471890i 0.971766 + 0.235945i \(0.0758185\pi\)
−0.971766 + 0.235945i \(0.924181\pi\)
\(972\) 0 0
\(973\) 608.000 0.624872
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 759.433i 0.777311i 0.921383 + 0.388655i \(0.127060\pi\)
−0.921383 + 0.388655i \(0.872940\pi\)
\(978\) 0 0
\(979\) 216.000 0.220633
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1052.17i 1.07037i 0.844734 + 0.535186i \(0.179758\pi\)
−0.844734 + 0.535186i \(0.820242\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 678.823i − 0.686373i
\(990\) 0 0
\(991\) 772.000 0.779011 0.389506 0.921024i \(-0.372646\pi\)
0.389506 + 0.921024i \(0.372646\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −194.000 −0.194584 −0.0972919 0.995256i \(-0.531018\pi\)
−0.0972919 + 0.995256i \(0.531018\pi\)
\(998\) 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 3600.3.l.d.1601.2 2
3.2 odd 2 inner 3600.3.l.d.1601.1 2
4.3 odd 2 450.3.d.f.251.2 2
5.2 odd 4 3600.3.c.b.449.2 4
5.3 odd 4 3600.3.c.b.449.4 4
5.4 even 2 144.3.e.b.17.2 2
12.11 even 2 450.3.d.f.251.1 2
15.2 even 4 3600.3.c.b.449.1 4
15.8 even 4 3600.3.c.b.449.3 4
15.14 odd 2 144.3.e.b.17.1 2
20.3 even 4 450.3.b.b.449.3 4
20.7 even 4 450.3.b.b.449.2 4
20.19 odd 2 18.3.b.a.17.1 2
40.19 odd 2 576.3.e.c.449.1 2
40.29 even 2 576.3.e.f.449.1 2
45.4 even 6 1296.3.q.f.593.2 4
45.14 odd 6 1296.3.q.f.593.1 4
45.29 odd 6 1296.3.q.f.1025.2 4
45.34 even 6 1296.3.q.f.1025.1 4
60.23 odd 4 450.3.b.b.449.1 4
60.47 odd 4 450.3.b.b.449.4 4
60.59 even 2 18.3.b.a.17.2 yes 2
80.19 odd 4 2304.3.h.f.2177.4 4
80.29 even 4 2304.3.h.c.2177.4 4
80.59 odd 4 2304.3.h.f.2177.1 4
80.69 even 4 2304.3.h.c.2177.1 4
120.29 odd 2 576.3.e.f.449.2 2
120.59 even 2 576.3.e.c.449.2 2
140.19 even 6 882.3.s.d.557.1 4
140.39 odd 6 882.3.s.b.863.2 4
140.59 even 6 882.3.s.d.863.2 4
140.79 odd 6 882.3.s.b.557.1 4
140.139 even 2 882.3.b.a.197.1 2
180.59 even 6 162.3.d.b.107.1 4
180.79 odd 6 162.3.d.b.53.1 4
180.119 even 6 162.3.d.b.53.2 4
180.139 odd 6 162.3.d.b.107.2 4
220.219 even 2 2178.3.c.d.485.2 2
240.29 odd 4 2304.3.h.c.2177.2 4
240.59 even 4 2304.3.h.f.2177.3 4
240.149 odd 4 2304.3.h.c.2177.3 4
240.179 even 4 2304.3.h.f.2177.2 4
260.99 even 4 3042.3.d.a.3041.4 4
260.239 even 4 3042.3.d.a.3041.1 4
260.259 odd 2 3042.3.c.e.1691.2 2
420.59 odd 6 882.3.s.d.863.1 4
420.179 even 6 882.3.s.b.863.1 4
420.299 odd 6 882.3.s.d.557.2 4
420.359 even 6 882.3.s.b.557.2 4
420.419 odd 2 882.3.b.a.197.2 2
660.659 odd 2 2178.3.c.d.485.1 2
780.239 odd 4 3042.3.d.a.3041.3 4
780.359 odd 4 3042.3.d.a.3041.2 4
780.779 even 2 3042.3.c.e.1691.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 20.19 odd 2
18.3.b.a.17.2 yes 2 60.59 even 2
144.3.e.b.17.1 2 15.14 odd 2
144.3.e.b.17.2 2 5.4 even 2
162.3.d.b.53.1 4 180.79 odd 6
162.3.d.b.53.2 4 180.119 even 6
162.3.d.b.107.1 4 180.59 even 6
162.3.d.b.107.2 4 180.139 odd 6
450.3.b.b.449.1 4 60.23 odd 4
450.3.b.b.449.2 4 20.7 even 4
450.3.b.b.449.3 4 20.3 even 4
450.3.b.b.449.4 4 60.47 odd 4
450.3.d.f.251.1 2 12.11 even 2
450.3.d.f.251.2 2 4.3 odd 2
576.3.e.c.449.1 2 40.19 odd 2
576.3.e.c.449.2 2 120.59 even 2
576.3.e.f.449.1 2 40.29 even 2
576.3.e.f.449.2 2 120.29 odd 2
882.3.b.a.197.1 2 140.139 even 2
882.3.b.a.197.2 2 420.419 odd 2
882.3.s.b.557.1 4 140.79 odd 6
882.3.s.b.557.2 4 420.359 even 6
882.3.s.b.863.1 4 420.179 even 6
882.3.s.b.863.2 4 140.39 odd 6
882.3.s.d.557.1 4 140.19 even 6
882.3.s.d.557.2 4 420.299 odd 6
882.3.s.d.863.1 4 420.59 odd 6
882.3.s.d.863.2 4 140.59 even 6
1296.3.q.f.593.1 4 45.14 odd 6
1296.3.q.f.593.2 4 45.4 even 6
1296.3.q.f.1025.1 4 45.34 even 6
1296.3.q.f.1025.2 4 45.29 odd 6
2178.3.c.d.485.1 2 660.659 odd 2
2178.3.c.d.485.2 2 220.219 even 2
2304.3.h.c.2177.1 4 80.69 even 4
2304.3.h.c.2177.2 4 240.29 odd 4
2304.3.h.c.2177.3 4 240.149 odd 4
2304.3.h.c.2177.4 4 80.29 even 4
2304.3.h.f.2177.1 4 80.59 odd 4
2304.3.h.f.2177.2 4 240.179 even 4
2304.3.h.f.2177.3 4 240.59 even 4
2304.3.h.f.2177.4 4 80.19 odd 4
3042.3.c.e.1691.1 2 780.779 even 2
3042.3.c.e.1691.2 2 260.259 odd 2
3042.3.d.a.3041.1 4 260.239 even 4
3042.3.d.a.3041.2 4 780.359 odd 4
3042.3.d.a.3041.3 4 780.239 odd 4
3042.3.d.a.3041.4 4 260.99 even 4
3600.3.c.b.449.1 4 15.2 even 4
3600.3.c.b.449.2 4 5.2 odd 4
3600.3.c.b.449.3 4 15.8 even 4
3600.3.c.b.449.4 4 5.3 odd 4
3600.3.l.d.1601.1 2 3.2 odd 2 inner
3600.3.l.d.1601.2 2 1.1 even 1 trivial