Properties

Label 3600.2.h.d.1151.4
Level $3600$
Weight $2$
Character 3600.1151
Analytic conductor $28.746$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,2,Mod(1151,3600)]
 
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([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.h (of order \(2\), degree \(1\), 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: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 13x^{2} - 12x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.4
Root \(0.500000 - 3.35071i\) of defining polynomial
Character \(\chi\) \(=\) 3600.1151
Dual form 3600.2.h.d.1151.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.87298i q^{7} +O(q^{10})\) \(q+3.87298i q^{7} +5.47723 q^{11} -1.00000 q^{13} +4.24264i q^{17} +3.87298i q^{19} -5.47723 q^{23} +4.24264i q^{29} +3.87298i q^{31} -8.00000 q^{37} -8.48528i q^{41} -3.87298i q^{43} +5.47723 q^{47} -8.00000 q^{49} +8.48528i q^{53} +5.47723 q^{59} +5.00000 q^{61} -11.6190i q^{67} -10.9545 q^{71} -4.00000 q^{73} +21.2132i q^{77} -15.4919i q^{79} +16.4317 q^{83} +16.9706i q^{89} -3.87298i q^{91} -13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{13} - 32 q^{37} - 32 q^{49} + 20 q^{61} - 16 q^{73} - 52 q^{97}+O(q^{100}) \) 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
\(7\) 3.87298i 1.46385i 0.681385 + 0.731925i \(0.261378\pi\)
−0.681385 + 0.731925i \(0.738622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.47723 1.65145 0.825723 0.564076i \(-0.190768\pi\)
0.825723 + 0.564076i \(0.190768\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.24264i 1.02899i 0.857493 + 0.514496i \(0.172021\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) 3.87298i 0.888523i 0.895897 + 0.444262i \(0.146534\pi\)
−0.895897 + 0.444262i \(0.853466\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.47723 −1.14208 −0.571040 0.820922i \(-0.693460\pi\)
−0.571040 + 0.820922i \(0.693460\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 3.87298i 0.695608i 0.937567 + 0.347804i \(0.113073\pi\)
−0.937567 + 0.347804i \(0.886927\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.48528i − 1.32518i −0.748983 0.662589i \(-0.769458\pi\)
0.748983 0.662589i \(-0.230542\pi\)
\(42\) 0 0
\(43\) − 3.87298i − 0.590624i −0.955401 0.295312i \(-0.904576\pi\)
0.955401 0.295312i \(-0.0954237\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.47723 0.798935 0.399468 0.916747i \(-0.369195\pi\)
0.399468 + 0.916747i \(0.369195\pi\)
\(48\) 0 0
\(49\) −8.00000 −1.14286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528i 1.16554i 0.812636 + 0.582772i \(0.198032\pi\)
−0.812636 + 0.582772i \(0.801968\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.47723 0.713074 0.356537 0.934281i \(-0.383957\pi\)
0.356537 + 0.934281i \(0.383957\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.6190i − 1.41948i −0.704463 0.709740i \(-0.748812\pi\)
0.704463 0.709740i \(-0.251188\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.9545 −1.30005 −0.650027 0.759911i \(-0.725242\pi\)
−0.650027 + 0.759911i \(0.725242\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 21.2132i 2.41747i
\(78\) 0 0
\(79\) − 15.4919i − 1.74298i −0.490414 0.871489i \(-0.663155\pi\)
0.490414 0.871489i \(-0.336845\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.4317 1.80361 0.901805 0.432142i \(-0.142242\pi\)
0.901805 + 0.432142i \(0.142242\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.9706i 1.79888i 0.437048 + 0.899438i \(0.356024\pi\)
−0.437048 + 0.899438i \(0.643976\pi\)
\(90\) 0 0
\(91\) − 3.87298i − 0.405999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 8.48528i − 0.844317i −0.906522 0.422159i \(-0.861273\pi\)
0.906522 0.422159i \(-0.138727\pi\)
\(102\) 0 0
\(103\) 7.74597i 0.763233i 0.924321 + 0.381616i \(0.124632\pi\)
−0.924321 + 0.381616i \(0.875368\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.9545 −1.05901 −0.529503 0.848308i \(-0.677622\pi\)
−0.529503 + 0.848308i \(0.677622\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.4317 −1.50629
\(120\) 0 0
\(121\) 19.0000 1.72727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.9545 −0.957095 −0.478547 0.878062i \(-0.658837\pi\)
−0.478547 + 0.878062i \(0.658837\pi\)
\(132\) 0 0
\(133\) −15.0000 −1.30066
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.2132i 1.81237i 0.422885 + 0.906183i \(0.361017\pi\)
−0.422885 + 0.906183i \(0.638983\pi\)
\(138\) 0 0
\(139\) 15.4919i 1.31401i 0.753887 + 0.657004i \(0.228177\pi\)
−0.753887 + 0.657004i \(0.771823\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.47723 −0.458029
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264i 0.347571i 0.984784 + 0.173785i \(0.0555999\pi\)
−0.984784 + 0.173785i \(0.944400\pi\)
\(150\) 0 0
\(151\) − 19.3649i − 1.57589i −0.615743 0.787947i \(-0.711144\pi\)
0.615743 0.787947i \(-0.288856\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.0000 −1.35675 −0.678374 0.734717i \(-0.737315\pi\)
−0.678374 + 0.734717i \(0.737315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 21.2132i − 1.67183i
\(162\) 0 0
\(163\) 11.6190i 0.910066i 0.890474 + 0.455033i \(0.150373\pi\)
−0.890474 + 0.455033i \(0.849627\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.9545 −0.847681 −0.423840 0.905737i \(-0.639318\pi\)
−0.423840 + 0.905737i \(0.639318\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.2132i 1.61281i 0.591364 + 0.806405i \(0.298590\pi\)
−0.591364 + 0.806405i \(0.701410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.47723 0.409387 0.204694 0.978826i \(-0.434380\pi\)
0.204694 + 0.978826i \(0.434380\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 23.2379i 1.69932i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.4317 1.18895 0.594477 0.804112i \(-0.297359\pi\)
0.594477 + 0.804112i \(0.297359\pi\)
\(192\) 0 0
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i 0.654931 + 0.755689i \(0.272698\pi\)
−0.654931 + 0.755689i \(0.727302\pi\)
\(198\) 0 0
\(199\) 11.6190i 0.823646i 0.911264 + 0.411823i \(0.135108\pi\)
−0.911264 + 0.411823i \(0.864892\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.4317 −1.15328
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21.2132i 1.46735i
\(210\) 0 0
\(211\) 11.6190i 0.799882i 0.916541 + 0.399941i \(0.130969\pi\)
−0.916541 + 0.399941i \(0.869031\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −15.0000 −1.01827
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.24264i − 0.285391i
\(222\) 0 0
\(223\) − 27.1109i − 1.81548i −0.419534 0.907740i \(-0.637807\pi\)
0.419534 0.907740i \(-0.362193\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.9545 0.727072 0.363536 0.931580i \(-0.381569\pi\)
0.363536 + 0.931580i \(0.381569\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 8.48528i − 0.555889i −0.960597 0.277945i \(-0.910347\pi\)
0.960597 0.277945i \(-0.0896532\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9545 0.708585 0.354292 0.935135i \(-0.384722\pi\)
0.354292 + 0.935135i \(0.384722\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.87298i − 0.246432i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.9089 1.38288 0.691439 0.722435i \(-0.256977\pi\)
0.691439 + 0.722435i \(0.256977\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) − 30.9839i − 1.92524i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.9545 0.675480 0.337740 0.941239i \(-0.390337\pi\)
0.337740 + 0.941239i \(0.390337\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.24264i − 0.258678i −0.991600 0.129339i \(-0.958714\pi\)
0.991600 0.129339i \(-0.0412856\pi\)
\(270\) 0 0
\(271\) 30.9839i 1.88214i 0.338218 + 0.941068i \(0.390176\pi\)
−0.338218 + 0.941068i \(0.609824\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 12.7279i − 0.759284i −0.925133 0.379642i \(-0.876047\pi\)
0.925133 0.379642i \(-0.123953\pi\)
\(282\) 0 0
\(283\) 3.87298i 0.230225i 0.993352 + 0.115112i \(0.0367228\pi\)
−0.993352 + 0.115112i \(0.963277\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 32.8634 1.93986
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.2132i 1.23929i 0.784883 + 0.619644i \(0.212723\pi\)
−0.784883 + 0.619644i \(0.787277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.47723 0.316756
\(300\) 0 0
\(301\) 15.0000 0.864586
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 19.3649i − 1.10521i −0.833442 0.552607i \(-0.813633\pi\)
0.833442 0.552607i \(-0.186367\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.47723 0.310585 0.155292 0.987869i \(-0.450368\pi\)
0.155292 + 0.987869i \(0.450368\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.9706i 0.953162i 0.879131 + 0.476581i \(0.158124\pi\)
−0.879131 + 0.476581i \(0.841876\pi\)
\(318\) 0 0
\(319\) 23.2379i 1.30107i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.4317 −0.914283
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.2132i 1.16952i
\(330\) 0 0
\(331\) 15.4919i 0.851514i 0.904838 + 0.425757i \(0.139992\pi\)
−0.904838 + 0.425757i \(0.860008\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 21.2132i 1.14876i
\(342\) 0 0
\(343\) − 3.87298i − 0.209121i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.4317 0.882099 0.441049 0.897483i \(-0.354606\pi\)
0.441049 + 0.897483i \(0.354606\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 29.6985i − 1.58069i −0.612661 0.790345i \(-0.709901\pi\)
0.612661 0.790345i \(-0.290099\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.9545 0.578154 0.289077 0.957306i \(-0.406652\pi\)
0.289077 + 0.957306i \(0.406652\pi\)
\(360\) 0 0
\(361\) 4.00000 0.210526
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 11.6190i − 0.606504i −0.952910 0.303252i \(-0.901928\pi\)
0.952910 0.303252i \(-0.0980724\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −32.8634 −1.70618
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.24264i − 0.218507i
\(378\) 0 0
\(379\) − 11.6190i − 0.596825i −0.954437 0.298413i \(-0.903543\pi\)
0.954437 0.298413i \(-0.0964572\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.9089 1.11949 0.559746 0.828664i \(-0.310899\pi\)
0.559746 + 0.828664i \(0.310899\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 25.4558i − 1.29066i −0.763903 0.645331i \(-0.776719\pi\)
0.763903 0.645331i \(-0.223281\pi\)
\(390\) 0 0
\(391\) − 23.2379i − 1.17519i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 8.48528i − 0.423735i −0.977298 0.211867i \(-0.932046\pi\)
0.977298 0.211867i \(-0.0679545\pi\)
\(402\) 0 0
\(403\) − 3.87298i − 0.192927i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −43.8178 −2.17197
\(408\) 0 0
\(409\) −35.0000 −1.73064 −0.865319 0.501221i \(-0.832884\pi\)
−0.865319 + 0.501221i \(0.832884\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 21.2132i 1.04383i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.8634 −1.60548 −0.802740 0.596329i \(-0.796625\pi\)
−0.802740 + 0.596329i \(0.796625\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 19.3649i 0.937134i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.47723 −0.263829 −0.131914 0.991261i \(-0.542112\pi\)
−0.131914 + 0.991261i \(0.542112\pi\)
\(432\) 0 0
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 21.2132i − 1.01477i
\(438\) 0 0
\(439\) 3.87298i 0.184847i 0.995720 + 0.0924237i \(0.0294614\pi\)
−0.995720 + 0.0924237i \(0.970539\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.9089 −1.04092 −0.520462 0.853885i \(-0.674240\pi\)
−0.520462 + 0.853885i \(0.674240\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 16.9706i − 0.800890i −0.916321 0.400445i \(-0.868855\pi\)
0.916321 0.400445i \(-0.131145\pi\)
\(450\) 0 0
\(451\) − 46.4758i − 2.18846i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.48528i 0.395199i 0.980283 + 0.197599i \(0.0633145\pi\)
−0.980283 + 0.197599i \(0.936685\pi\)
\(462\) 0 0
\(463\) 30.9839i 1.43994i 0.694004 + 0.719971i \(0.255845\pi\)
−0.694004 + 0.719971i \(0.744155\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −38.3406 −1.77419 −0.887095 0.461587i \(-0.847280\pi\)
−0.887095 + 0.461587i \(0.847280\pi\)
\(468\) 0 0
\(469\) 45.0000 2.07791
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 21.2132i − 0.975384i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.4317 0.750782 0.375391 0.926866i \(-0.377508\pi\)
0.375391 + 0.926866i \(0.377508\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 11.6190i 0.526505i 0.964727 + 0.263252i \(0.0847952\pi\)
−0.964727 + 0.263252i \(0.915205\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 42.4264i − 1.90308i
\(498\) 0 0
\(499\) − 3.87298i − 0.173379i −0.996235 0.0866893i \(-0.972371\pi\)
0.996235 0.0866893i \(-0.0276287\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.8634 1.46530 0.732652 0.680603i \(-0.238282\pi\)
0.732652 + 0.680603i \(0.238282\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 4.24264i − 0.188052i −0.995570 0.0940259i \(-0.970026\pi\)
0.995570 0.0940259i \(-0.0299736\pi\)
\(510\) 0 0
\(511\) − 15.4919i − 0.685323i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 12.7279i − 0.557620i −0.960346 0.278810i \(-0.910060\pi\)
0.960346 0.278810i \(-0.0899400\pi\)
\(522\) 0 0
\(523\) − 3.87298i − 0.169354i −0.996408 0.0846769i \(-0.973014\pi\)
0.996408 0.0846769i \(-0.0269858\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −16.4317 −0.715775
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528i 0.367538i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −43.8178 −1.88737
\(540\) 0 0
\(541\) −35.0000 −1.50477 −0.752384 0.658725i \(-0.771096\pi\)
−0.752384 + 0.658725i \(0.771096\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.74597i 0.331194i 0.986194 + 0.165597i \(0.0529550\pi\)
−0.986194 + 0.165597i \(0.947045\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −16.4317 −0.700013
\(552\) 0 0
\(553\) 60.0000 2.55146
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.24264i − 0.179766i −0.995952 0.0898832i \(-0.971351\pi\)
0.995952 0.0898832i \(-0.0286494\pi\)
\(558\) 0 0
\(559\) 3.87298i 0.163810i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.3406 1.61586 0.807931 0.589277i \(-0.200587\pi\)
0.807931 + 0.589277i \(0.200587\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 4.24264i − 0.177861i −0.996038 0.0889304i \(-0.971655\pi\)
0.996038 0.0889304i \(-0.0283449\pi\)
\(570\) 0 0
\(571\) 3.87298i 0.162079i 0.996711 + 0.0810397i \(0.0258240\pi\)
−0.996711 + 0.0810397i \(0.974176\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 37.0000 1.54033 0.770165 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 63.6396i 2.64022i
\(582\) 0 0
\(583\) 46.4758i 1.92483i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.8634 1.35642 0.678208 0.734870i \(-0.262757\pi\)
0.678208 + 0.734870i \(0.262757\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 42.4264i − 1.74224i −0.491067 0.871122i \(-0.663393\pi\)
0.491067 0.871122i \(-0.336607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 15.4919i − 0.628798i −0.949291 0.314399i \(-0.898197\pi\)
0.949291 0.314399i \(-0.101803\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.47723 −0.221585
\(612\) 0 0
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 25.4558i − 1.02481i −0.858743 0.512407i \(-0.828754\pi\)
0.858743 0.512407i \(-0.171246\pi\)
\(618\) 0 0
\(619\) 27.1109i 1.08968i 0.838541 + 0.544839i \(0.183409\pi\)
−0.838541 + 0.544839i \(0.816591\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −65.7267 −2.63328
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 33.9411i − 1.35332i
\(630\) 0 0
\(631\) 34.8569i 1.38763i 0.720154 + 0.693815i \(0.244071\pi\)
−0.720154 + 0.693815i \(0.755929\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.00000 0.316972
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.9411i 1.34059i 0.742093 + 0.670297i \(0.233833\pi\)
−0.742093 + 0.670297i \(0.766167\pi\)
\(642\) 0 0
\(643\) − 46.4758i − 1.83283i −0.400233 0.916413i \(-0.631071\pi\)
0.400233 0.916413i \(-0.368929\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.9089 0.861328 0.430664 0.902512i \(-0.358279\pi\)
0.430664 + 0.902512i \(0.358279\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.7279i 0.498082i 0.968493 + 0.249041i \(0.0801154\pi\)
−0.968493 + 0.249041i \(0.919885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.8634 1.28017 0.640087 0.768302i \(-0.278898\pi\)
0.640087 + 0.768302i \(0.278898\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 23.2379i − 0.899775i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27.3861 1.05723
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 38.1838i − 1.46752i −0.679408 0.733761i \(-0.737763\pi\)
0.679408 0.733761i \(-0.262237\pi\)
\(678\) 0 0
\(679\) − 50.3488i − 1.93221i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.9089 0.838321 0.419160 0.907912i \(-0.362324\pi\)
0.419160 + 0.907912i \(0.362324\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 8.48528i − 0.323263i
\(690\) 0 0
\(691\) 23.2379i 0.884011i 0.897012 + 0.442006i \(0.145733\pi\)
−0.897012 + 0.442006i \(0.854267\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7279i 0.480727i 0.970683 + 0.240363i \(0.0772666\pi\)
−0.970683 + 0.240363i \(0.922733\pi\)
\(702\) 0 0
\(703\) − 30.9839i − 1.16858i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32.8634 1.23595
\(708\) 0 0
\(709\) 31.0000 1.16423 0.582115 0.813107i \(-0.302225\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 21.2132i − 0.794441i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27.3861 1.02133 0.510665 0.859780i \(-0.329399\pi\)
0.510665 + 0.859780i \(0.329399\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 27.1109i − 1.00549i −0.864436 0.502744i \(-0.832324\pi\)
0.864436 0.502744i \(-0.167676\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.4317 0.607748
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 63.6396i − 2.34420i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −21.9089 −0.803760 −0.401880 0.915692i \(-0.631643\pi\)
−0.401880 + 0.915692i \(0.631643\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 42.4264i − 1.55023i
\(750\) 0 0
\(751\) 30.9839i 1.13062i 0.824879 + 0.565309i \(0.191243\pi\)
−0.824879 + 0.565309i \(0.808757\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.9411i − 1.23036i −0.788385 0.615182i \(-0.789082\pi\)
0.788385 0.615182i \(-0.210918\pi\)
\(762\) 0 0
\(763\) − 19.3649i − 0.701057i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.47723 −0.197771
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8.48528i − 0.305194i −0.988288 0.152597i \(-0.951236\pi\)
0.988288 0.152597i \(-0.0487637\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.8634 1.17745
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19.3649i 0.690285i 0.938550 + 0.345142i \(0.112169\pi\)
−0.938550 + 0.345142i \(0.887831\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 23.2379i 0.822098i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −21.9089 −0.773148
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 25.4558i 0.894980i 0.894289 + 0.447490i \(0.147682\pi\)
−0.894289 + 0.447490i \(0.852318\pi\)
\(810\) 0 0
\(811\) − 3.87298i − 0.135999i −0.997685 0.0679994i \(-0.978338\pi\)
0.997685 0.0679994i \(-0.0216616\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 15.0000 0.524784
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 29.6985i 1.03648i 0.855234 + 0.518242i \(0.173413\pi\)
−0.855234 + 0.518242i \(0.826587\pi\)
\(822\) 0 0
\(823\) − 34.8569i − 1.21503i −0.794307 0.607517i \(-0.792166\pi\)
0.794307 0.607517i \(-0.207834\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.9545 0.380924 0.190462 0.981695i \(-0.439001\pi\)
0.190462 + 0.981695i \(0.439001\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 33.9411i − 1.17599i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.47723 −0.189095 −0.0945474 0.995520i \(-0.530140\pi\)
−0.0945474 + 0.995520i \(0.530140\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 73.5867i 2.52847i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 43.8178 1.50205
\(852\) 0 0
\(853\) −11.0000 −0.376633 −0.188316 0.982108i \(-0.560303\pi\)
−0.188316 + 0.982108i \(0.560303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 16.9706i − 0.579703i −0.957072 0.289852i \(-0.906394\pi\)
0.957072 0.289852i \(-0.0936060\pi\)
\(858\) 0 0
\(859\) − 7.74597i − 0.264289i −0.991230 0.132144i \(-0.957814\pi\)
0.991230 0.132144i \(-0.0421863\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 84.8528i − 2.87843i
\(870\) 0 0
\(871\) 11.6190i 0.393693i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.00000 0.236373 0.118187 0.992991i \(-0.462292\pi\)
0.118187 + 0.992991i \(0.462292\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 50.9117i 1.71526i 0.514269 + 0.857629i \(0.328063\pi\)
−0.514269 + 0.857629i \(0.671937\pi\)
\(882\) 0 0
\(883\) 50.3488i 1.69437i 0.531297 + 0.847186i \(0.321705\pi\)
−0.531297 + 0.847186i \(0.678295\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.3861 0.919536 0.459768 0.888039i \(-0.347933\pi\)
0.459768 + 0.888039i \(0.347933\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 21.2132i 0.709873i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.4317 −0.548027
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 30.9839i − 1.02880i −0.857550 0.514401i \(-0.828014\pi\)
0.857550 0.514401i \(-0.171986\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.47723 −0.181469 −0.0907343 0.995875i \(-0.528921\pi\)
−0.0907343 + 0.995875i \(0.528921\pi\)
\(912\) 0 0
\(913\) 90.0000 2.97857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 42.4264i − 1.40104i
\(918\) 0 0
\(919\) − 3.87298i − 0.127758i −0.997958 0.0638790i \(-0.979653\pi\)
0.997958 0.0638790i \(-0.0203472\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.9545 0.360570
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.24264i 0.139197i 0.997575 + 0.0695983i \(0.0221717\pi\)
−0.997575 + 0.0695983i \(0.977828\pi\)
\(930\) 0 0
\(931\) − 30.9839i − 1.01546i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −17.0000 −0.555366 −0.277683 0.960673i \(-0.589566\pi\)
−0.277683 + 0.960673i \(0.589566\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.7279i 0.414918i 0.978244 + 0.207459i \(0.0665194\pi\)
−0.978244 + 0.207459i \(0.933481\pi\)
\(942\) 0 0
\(943\) 46.4758i 1.51346i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49.2950 −1.60187 −0.800937 0.598749i \(-0.795665\pi\)
−0.800937 + 0.598749i \(0.795665\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −82.1584 −2.65303
\(960\) 0 0
\(961\) 16.0000 0.516129
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 15.4919i 0.498187i 0.968479 + 0.249093i \(0.0801326\pi\)
−0.968479 + 0.249093i \(0.919867\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.47723 0.175773 0.0878863 0.996131i \(-0.471989\pi\)
0.0878863 + 0.996131i \(0.471989\pi\)
\(972\) 0 0
\(973\) −60.0000 −1.92351
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 38.1838i − 1.22161i −0.791782 0.610803i \(-0.790847\pi\)
0.791782 0.610803i \(-0.209153\pi\)
\(978\) 0 0
\(979\) 92.9516i 2.97075i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 21.9089 0.698785 0.349393 0.936976i \(-0.386388\pi\)
0.349393 + 0.936976i \(0.386388\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.2132i 0.674541i
\(990\) 0 0
\(991\) − 50.3488i − 1.59938i −0.600412 0.799691i \(-0.704997\pi\)
0.600412 0.799691i \(-0.295003\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 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 3600.2.h.d.1151.4 yes 4
3.2 odd 2 inner 3600.2.h.d.1151.3 yes 4
4.3 odd 2 inner 3600.2.h.d.1151.1 4
5.2 odd 4 3600.2.o.e.3599.4 8
5.3 odd 4 3600.2.o.e.3599.7 8
5.4 even 2 3600.2.h.i.1151.2 yes 4
12.11 even 2 inner 3600.2.h.d.1151.2 yes 4
15.2 even 4 3600.2.o.e.3599.2 8
15.8 even 4 3600.2.o.e.3599.5 8
15.14 odd 2 3600.2.h.i.1151.1 yes 4
20.3 even 4 3600.2.o.e.3599.1 8
20.7 even 4 3600.2.o.e.3599.6 8
20.19 odd 2 3600.2.h.i.1151.3 yes 4
60.23 odd 4 3600.2.o.e.3599.3 8
60.47 odd 4 3600.2.o.e.3599.8 8
60.59 even 2 3600.2.h.i.1151.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3600.2.h.d.1151.1 4 4.3 odd 2 inner
3600.2.h.d.1151.2 yes 4 12.11 even 2 inner
3600.2.h.d.1151.3 yes 4 3.2 odd 2 inner
3600.2.h.d.1151.4 yes 4 1.1 even 1 trivial
3600.2.h.i.1151.1 yes 4 15.14 odd 2
3600.2.h.i.1151.2 yes 4 5.4 even 2
3600.2.h.i.1151.3 yes 4 20.19 odd 2
3600.2.h.i.1151.4 yes 4 60.59 even 2
3600.2.o.e.3599.1 8 20.3 even 4
3600.2.o.e.3599.2 8 15.2 even 4
3600.2.o.e.3599.3 8 60.23 odd 4
3600.2.o.e.3599.4 8 5.2 odd 4
3600.2.o.e.3599.5 8 15.8 even 4
3600.2.o.e.3599.6 8 20.7 even 4
3600.2.o.e.3599.7 8 5.3 odd 4
3600.2.o.e.3599.8 8 60.47 odd 4