Properties

Label 7200.2.f.bg.6049.1
Level $7200$
Weight $2$
Character 7200.6049
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(6049,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.6049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.f (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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 800)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6049.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 7200.6049
Dual form 7200.2.f.bg.6049.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.47214i q^{7} +O(q^{10})\) \(q-4.47214i q^{7} -2.23607 q^{11} +4.00000i q^{13} -7.00000i q^{17} +6.70820 q^{19} +4.47214i q^{23} +4.47214 q^{31} -2.00000i q^{37} -5.00000 q^{41} -8.94427i q^{47} -13.0000 q^{49} -6.00000i q^{53} +8.94427 q^{59} +10.0000 q^{61} +2.23607i q^{67} +8.94427 q^{71} -9.00000i q^{73} +10.0000i q^{77} +4.47214 q^{79} +11.1803i q^{83} -5.00000 q^{89} +17.8885 q^{91} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{41} - 52 q^{49} + 40 q^{61} - 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(7\) − 4.47214i − 1.69031i −0.534522 0.845154i \(-0.679509\pi\)
0.534522 0.845154i \(-0.320491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) 6.70820 1.53897 0.769484 0.638666i \(-0.220514\pi\)
0.769484 + 0.638666i \(0.220514\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.47214i 0.932505i 0.884652 + 0.466252i \(0.154396\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.47214 0.803219 0.401610 0.915811i \(-0.368451\pi\)
0.401610 + 0.915811i \(0.368451\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.94427i − 1.30466i −0.757937 0.652328i \(-0.773792\pi\)
0.757937 0.652328i \(-0.226208\pi\)
\(48\) 0 0
\(49\) −13.0000 −1.85714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.23607i 0.273179i 0.990628 + 0.136590i \(0.0436142\pi\)
−0.990628 + 0.136590i \(0.956386\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) − 9.00000i − 1.05337i −0.850060 0.526685i \(-0.823435\pi\)
0.850060 0.526685i \(-0.176565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.0000i 1.13961i
\(78\) 0 0
\(79\) 4.47214 0.503155 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.1803i 1.22720i 0.789616 + 0.613601i \(0.210280\pi\)
−0.789616 + 0.613601i \(0.789720\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 17.8885 1.87523
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) − 8.94427i − 0.881305i −0.897678 0.440653i \(-0.854747\pi\)
0.897678 0.440653i \(-0.145253\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.23607i 0.216169i 0.994142 + 0.108084i \(0.0344717\pi\)
−0.994142 + 0.108084i \(0.965528\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.00000i 0.0940721i 0.998893 + 0.0470360i \(0.0149776\pi\)
−0.998893 + 0.0470360i \(0.985022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −31.3050 −2.86972
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.47214i − 0.396838i −0.980117 0.198419i \(-0.936419\pi\)
0.980117 0.198419i \(-0.0635807\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −17.8885 −1.56293 −0.781465 0.623949i \(-0.785527\pi\)
−0.781465 + 0.623949i \(0.785527\pi\)
\(132\) 0 0
\(133\) − 30.0000i − 2.60133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) −2.23607 −0.189661 −0.0948304 0.995493i \(-0.530231\pi\)
−0.0948304 + 0.995493i \(0.530231\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 8.94427i − 0.747958i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −13.4164 −1.09181 −0.545906 0.837846i \(-0.683814\pi\)
−0.545906 + 0.837846i \(0.683814\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.0000 1.57622
\(162\) 0 0
\(163\) − 2.23607i − 0.175142i −0.996158 0.0875712i \(-0.972089\pi\)
0.996158 0.0875712i \(-0.0279105\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.94427i 0.692129i 0.938211 + 0.346064i \(0.112482\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 16.0000i − 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.23607 0.167132 0.0835658 0.996502i \(-0.473369\pi\)
0.0835658 + 0.996502i \(0.473369\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.6525i 1.14462i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.4164 −0.970777 −0.485389 0.874299i \(-0.661322\pi\)
−0.485389 + 0.874299i \(0.661322\pi\)
\(192\) 0 0
\(193\) 9.00000i 0.647834i 0.946085 + 0.323917i \(0.105000\pi\)
−0.946085 + 0.323917i \(0.895000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −8.94427 −0.634043 −0.317021 0.948418i \(-0.602683\pi\)
−0.317021 + 0.948418i \(0.602683\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −20.1246 −1.38544 −0.692718 0.721209i \(-0.743587\pi\)
−0.692718 + 0.721209i \(0.743587\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 20.0000i − 1.35769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 28.0000 1.88348
\(222\) 0 0
\(223\) − 26.8328i − 1.79686i −0.439119 0.898429i \(-0.644709\pi\)
0.439119 0.898429i \(-0.355291\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 17.8885i − 1.18730i −0.804722 0.593652i \(-0.797686\pi\)
0.804722 0.593652i \(-0.202314\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.8885 1.15711 0.578557 0.815642i \(-0.303616\pi\)
0.578557 + 0.815642i \(0.303616\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 26.8328i 1.70733i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.1803 0.705697 0.352848 0.935681i \(-0.385213\pi\)
0.352848 + 0.935681i \(0.385213\pi\)
\(252\) 0 0
\(253\) − 10.0000i − 0.628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) −8.94427 −0.555770
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.3607i 1.37882i 0.724372 + 0.689409i \(0.242130\pi\)
−0.724372 + 0.689409i \(0.757870\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −13.4164 −0.814989 −0.407494 0.913208i \(-0.633597\pi\)
−0.407494 + 0.913208i \(0.633597\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 6.70820i 0.398761i 0.979922 + 0.199381i \(0.0638930\pi\)
−0.979922 + 0.199381i \(0.936107\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.3607i 1.31991i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −17.8885 −1.03452
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 15.6525i 0.893334i 0.894700 + 0.446667i \(0.147389\pi\)
−0.894700 + 0.446667i \(0.852611\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.3607 1.26796 0.633979 0.773350i \(-0.281421\pi\)
0.633979 + 0.773350i \(0.281421\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 46.9574i − 2.61278i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −40.0000 −2.20527
\(330\) 0 0
\(331\) 11.1803 0.614527 0.307264 0.951624i \(-0.400587\pi\)
0.307264 + 0.951624i \(0.400587\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 27.0000i − 1.47078i −0.677642 0.735392i \(-0.736998\pi\)
0.677642 0.735392i \(-0.263002\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 26.8328i 1.44884i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.1246i 1.08035i 0.841554 + 0.540173i \(0.181641\pi\)
−0.841554 + 0.540173i \(0.818359\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.3050 1.65221 0.826106 0.563515i \(-0.190551\pi\)
0.826106 + 0.563515i \(0.190551\pi\)
\(360\) 0 0
\(361\) 26.0000 1.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 26.8328i − 1.40066i −0.713818 0.700331i \(-0.753036\pi\)
0.713818 0.700331i \(-0.246964\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −26.8328 −1.39309
\(372\) 0 0
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −15.6525 −0.804014 −0.402007 0.915637i \(-0.631687\pi\)
−0.402007 + 0.915637i \(0.631687\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.47214i 0.228515i 0.993451 + 0.114258i \(0.0364490\pi\)
−0.993451 + 0.114258i \(0.963551\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 31.3050 1.58316
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000i 1.40528i 0.711546 + 0.702640i \(0.247995\pi\)
−0.711546 + 0.702640i \(0.752005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) 0 0
\(403\) 17.8885i 0.891092i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.47214i 0.221676i
\(408\) 0 0
\(409\) −29.0000 −1.43396 −0.716979 0.697095i \(-0.754476\pi\)
−0.716979 + 0.697095i \(0.754476\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 40.0000i − 1.96827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.6525 −0.764673 −0.382337 0.924023i \(-0.624881\pi\)
−0.382337 + 0.924023i \(0.624881\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 44.7214i − 2.16422i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.47214 0.215415 0.107708 0.994183i \(-0.465649\pi\)
0.107708 + 0.994183i \(0.465649\pi\)
\(432\) 0 0
\(433\) 1.00000i 0.0480569i 0.999711 + 0.0240285i \(0.00764923\pi\)
−0.999711 + 0.0240285i \(0.992351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.0000i 1.43509i
\(438\) 0 0
\(439\) 35.7771 1.70755 0.853774 0.520644i \(-0.174308\pi\)
0.853774 + 0.520644i \(0.174308\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 38.0132i 1.80606i 0.429578 + 0.903030i \(0.358662\pi\)
−0.429578 + 0.903030i \(0.641338\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.0000 −0.896665 −0.448333 0.893867i \(-0.647982\pi\)
−0.448333 + 0.893867i \(0.647982\pi\)
\(450\) 0 0
\(451\) 11.1803 0.526462
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.00000i 0.327446i 0.986506 + 0.163723i \(0.0523504\pi\)
−0.986506 + 0.163723i \(0.947650\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 26.8328i 1.24703i 0.781813 + 0.623513i \(0.214295\pi\)
−0.781813 + 0.623513i \(0.785705\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.8885i − 0.827783i −0.910326 0.413892i \(-0.864169\pi\)
0.910326 0.413892i \(-0.135831\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.4164 0.613011 0.306506 0.951869i \(-0.400840\pi\)
0.306506 + 0.951869i \(0.400840\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.47214i − 0.202652i −0.994853 0.101326i \(-0.967692\pi\)
0.994853 0.101326i \(-0.0323085\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 35.7771 1.61460 0.807299 0.590143i \(-0.200929\pi\)
0.807299 + 0.590143i \(0.200929\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 40.0000i − 1.79425i
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 26.8328i − 1.19642i −0.801341 0.598208i \(-0.795880\pi\)
0.801341 0.598208i \(-0.204120\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −40.2492 −1.78052
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 20.0000i 0.879599i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 0 0
\(523\) − 6.70820i − 0.293329i −0.989186 0.146665i \(-0.953146\pi\)
0.989186 0.146665i \(-0.0468538\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 31.3050i − 1.36367i
\(528\) 0 0
\(529\) 3.00000 0.130435
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 29.0689 1.25209
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.1246i − 0.860466i −0.902718 0.430233i \(-0.858431\pi\)
0.902718 0.430233i \(-0.141569\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 20.0000i − 0.850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.0000i − 1.18640i −0.805056 0.593199i \(-0.797865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 44.7214i − 1.88478i −0.334515 0.942390i \(-0.608573\pi\)
0.334515 0.942390i \(-0.391427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.00000 0.0419222 0.0209611 0.999780i \(-0.493327\pi\)
0.0209611 + 0.999780i \(0.493327\pi\)
\(570\) 0 0
\(571\) −17.8885 −0.748612 −0.374306 0.927305i \(-0.622119\pi\)
−0.374306 + 0.927305i \(0.622119\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 43.0000i − 1.79011i −0.445952 0.895057i \(-0.647135\pi\)
0.445952 0.895057i \(-0.352865\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 50.0000 2.07435
\(582\) 0 0
\(583\) 13.4164i 0.555651i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 20.1246i − 0.830632i −0.909677 0.415316i \(-0.863671\pi\)
0.909677 0.415316i \(-0.136329\pi\)
\(588\) 0 0
\(589\) 30.0000 1.23613
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.00000i 0.369586i 0.982777 + 0.184793i \(0.0591614\pi\)
−0.982777 + 0.184793i \(0.940839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.47214 −0.182727 −0.0913633 0.995818i \(-0.529122\pi\)
−0.0913633 + 0.995818i \(0.529122\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 17.8885i − 0.726074i −0.931775 0.363037i \(-0.881740\pi\)
0.931775 0.363037i \(-0.118260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.7771 1.44739
\(612\) 0 0
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) −17.8885 −0.719001 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.3607i 0.895862i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.0000 −0.558217
\(630\) 0 0
\(631\) −13.4164 −0.534099 −0.267049 0.963683i \(-0.586049\pi\)
−0.267049 + 0.963683i \(0.586049\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 52.0000i − 2.06032i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) − 17.8885i − 0.705455i −0.935726 0.352728i \(-0.885254\pi\)
0.935726 0.352728i \(-0.114746\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.94427i 0.351636i 0.984423 + 0.175818i \(0.0562570\pi\)
−0.984423 + 0.175818i \(0.943743\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.0000i 1.33052i 0.746611 + 0.665261i \(0.231680\pi\)
−0.746611 + 0.665261i \(0.768320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.70820 0.261315 0.130657 0.991428i \(-0.458291\pi\)
0.130657 + 0.991428i \(0.458291\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −22.3607 −0.863224
\(672\) 0 0
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 32.0000i 1.22986i 0.788582 + 0.614930i \(0.210816\pi\)
−0.788582 + 0.614930i \(0.789184\pi\)
\(678\) 0 0
\(679\) −8.94427 −0.343250
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 11.1803i − 0.427804i −0.976855 0.213902i \(-0.931383\pi\)
0.976855 0.213902i \(-0.0686173\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 38.0132 1.44609 0.723044 0.690802i \(-0.242742\pi\)
0.723044 + 0.690802i \(0.242742\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 35.0000i 1.32572i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) − 13.4164i − 0.506009i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 8.94427i − 0.336384i
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.0000i 0.749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.47214 0.166783 0.0833913 0.996517i \(-0.473425\pi\)
0.0833913 + 0.996517i \(0.473425\pi\)
\(720\) 0 0
\(721\) −40.0000 −1.48968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 44.0000i − 1.62518i −0.582838 0.812589i \(-0.698058\pi\)
0.582838 0.812589i \(-0.301942\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.00000i − 0.184177i
\(738\) 0 0
\(739\) 35.7771 1.31608 0.658041 0.752982i \(-0.271385\pi\)
0.658041 + 0.752982i \(0.271385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 31.3050i − 1.14847i −0.818691 0.574234i \(-0.805300\pi\)
0.818691 0.574234i \(-0.194700\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 53.6656 1.95829 0.979143 0.203171i \(-0.0651246\pi\)
0.979143 + 0.203171i \(0.0651246\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 8.00000i − 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 43.0000 1.55875 0.779374 0.626559i \(-0.215537\pi\)
0.779374 + 0.626559i \(0.215537\pi\)
\(762\) 0 0
\(763\) 26.8328i 0.971413i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 35.7771i 1.29184i
\(768\) 0 0
\(769\) 15.0000 0.540914 0.270457 0.962732i \(-0.412825\pi\)
0.270457 + 0.962732i \(0.412825\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33.5410 −1.20173
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.47214 0.159011
\(792\) 0 0
\(793\) 40.0000i 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) −62.6099 −2.21498
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.1246i 0.710182i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −17.8885 −0.628152 −0.314076 0.949398i \(-0.601695\pi\)
−0.314076 + 0.949398i \(0.601695\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) 26.8328i 0.935333i 0.883905 + 0.467667i \(0.154905\pi\)
−0.883905 + 0.467667i \(0.845095\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.4296i 1.78838i 0.447687 + 0.894191i \(0.352248\pi\)
−0.447687 + 0.894191i \(0.647752\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 91.0000i 3.15296i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −17.8885 −0.617581 −0.308791 0.951130i \(-0.599924\pi\)
−0.308791 + 0.951130i \(0.599924\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 26.8328i 0.921986i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.94427 0.306606
\(852\) 0 0
\(853\) 34.0000i 1.16414i 0.813139 + 0.582069i \(0.197757\pi\)
−0.813139 + 0.582069i \(0.802243\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) 0 0
\(859\) 33.5410 1.14440 0.572202 0.820112i \(-0.306089\pi\)
0.572202 + 0.820112i \(0.306089\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −8.94427 −0.303065
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 28.0000i − 0.945493i −0.881199 0.472746i \(-0.843263\pi\)
0.881199 0.472746i \(-0.156737\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 2.23607i 0.0752497i 0.999292 + 0.0376248i \(0.0119792\pi\)
−0.999292 + 0.0376248i \(0.988021\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.94427i − 0.300319i −0.988662 0.150160i \(-0.952021\pi\)
0.988662 0.150160i \(-0.0479788\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 60.0000i − 2.00782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −42.0000 −1.39922
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 53.6656i 1.78194i 0.454064 + 0.890969i \(0.349974\pi\)
−0.454064 + 0.890969i \(0.650026\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −44.7214 −1.48168 −0.740842 0.671679i \(-0.765573\pi\)
−0.740842 + 0.671679i \(0.765573\pi\)
\(912\) 0 0
\(913\) − 25.0000i − 0.827379i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 80.0000i 2.64183i
\(918\) 0 0
\(919\) −13.4164 −0.442566 −0.221283 0.975210i \(-0.571025\pi\)
−0.221283 + 0.975210i \(0.571025\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 35.7771i 1.17762i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −87.2067 −2.85808
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.0000i 0.555366i 0.960673 + 0.277683i \(0.0895665\pi\)
−0.960673 + 0.277683i \(0.910434\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) − 22.3607i − 0.728164i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8328i 0.871949i 0.899959 + 0.435975i \(0.143596\pi\)
−0.899959 + 0.435975i \(0.856404\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 9.00000i − 0.291539i −0.989319 0.145769i \(-0.953434\pi\)
0.989319 0.145769i \(-0.0465657\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.4164 −0.433238
\(960\) 0 0
\(961\) −11.0000 −0.354839
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 53.6656i 1.72577i 0.505400 + 0.862885i \(0.331345\pi\)
−0.505400 + 0.862885i \(0.668655\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −55.9017 −1.79397 −0.896985 0.442060i \(-0.854248\pi\)
−0.896985 + 0.442060i \(0.854248\pi\)
\(972\) 0 0
\(973\) 10.0000i 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 3.00000i − 0.0959785i −0.998848 0.0479893i \(-0.984719\pi\)
0.998848 0.0479893i \(-0.0152813\pi\)
\(978\) 0 0
\(979\) 11.1803 0.357325
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.47214i 0.142639i 0.997454 + 0.0713195i \(0.0227210\pi\)
−0.997454 + 0.0713195i \(0.977279\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 13.4164 0.426186 0.213093 0.977032i \(-0.431646\pi\)
0.213093 + 0.977032i \(0.431646\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.0000i 1.64686i 0.567420 + 0.823428i \(0.307941\pi\)
−0.567420 + 0.823428i \(0.692059\pi\)
\(998\) 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 7200.2.f.bg.6049.1 4
3.2 odd 2 800.2.c.g.449.4 4
4.3 odd 2 inner 7200.2.f.bg.6049.4 4
5.2 odd 4 7200.2.a.cn.1.2 2
5.3 odd 4 7200.2.a.cf.1.1 2
5.4 even 2 inner 7200.2.f.bg.6049.3 4
12.11 even 2 800.2.c.g.449.1 4
15.2 even 4 800.2.a.l.1.2 yes 2
15.8 even 4 800.2.a.k.1.1 2
15.14 odd 2 800.2.c.g.449.2 4
20.3 even 4 7200.2.a.cf.1.2 2
20.7 even 4 7200.2.a.cn.1.1 2
20.19 odd 2 inner 7200.2.f.bg.6049.2 4
24.5 odd 2 1600.2.c.o.449.1 4
24.11 even 2 1600.2.c.o.449.4 4
60.23 odd 4 800.2.a.k.1.2 yes 2
60.47 odd 4 800.2.a.l.1.1 yes 2
60.59 even 2 800.2.c.g.449.3 4
120.29 odd 2 1600.2.c.o.449.3 4
120.53 even 4 1600.2.a.bb.1.2 2
120.59 even 2 1600.2.c.o.449.2 4
120.77 even 4 1600.2.a.ba.1.1 2
120.83 odd 4 1600.2.a.bb.1.1 2
120.107 odd 4 1600.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.a.k.1.1 2 15.8 even 4
800.2.a.k.1.2 yes 2 60.23 odd 4
800.2.a.l.1.1 yes 2 60.47 odd 4
800.2.a.l.1.2 yes 2 15.2 even 4
800.2.c.g.449.1 4 12.11 even 2
800.2.c.g.449.2 4 15.14 odd 2
800.2.c.g.449.3 4 60.59 even 2
800.2.c.g.449.4 4 3.2 odd 2
1600.2.a.ba.1.1 2 120.77 even 4
1600.2.a.ba.1.2 2 120.107 odd 4
1600.2.a.bb.1.1 2 120.83 odd 4
1600.2.a.bb.1.2 2 120.53 even 4
1600.2.c.o.449.1 4 24.5 odd 2
1600.2.c.o.449.2 4 120.59 even 2
1600.2.c.o.449.3 4 120.29 odd 2
1600.2.c.o.449.4 4 24.11 even 2
7200.2.a.cf.1.1 2 5.3 odd 4
7200.2.a.cf.1.2 2 20.3 even 4
7200.2.a.cn.1.1 2 20.7 even 4
7200.2.a.cn.1.2 2 5.2 odd 4
7200.2.f.bg.6049.1 4 1.1 even 1 trivial
7200.2.f.bg.6049.2 4 20.19 odd 2 inner
7200.2.f.bg.6049.3 4 5.4 even 2 inner
7200.2.f.bg.6049.4 4 4.3 odd 2 inner