Properties

Label 1350.2.c.d.649.1
Level $1350$
Weight $2$
Character 1350.649
Analytic conductor $10.780$
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] = [1350,2,Mod(649,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1350.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.c (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: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.2.c.d.649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -5.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +3.00000i q^{22} +3.00000i q^{23} -5.00000 q^{26} -2.00000i q^{28} +2.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} +11.0000i q^{37} -4.00000i q^{38} +6.00000 q^{41} +4.00000i q^{43} +3.00000 q^{44} +3.00000 q^{46} +3.00000i q^{47} +3.00000 q^{49} +5.00000i q^{52} +12.0000i q^{53} -2.00000 q^{56} -9.00000 q^{59} +11.0000 q^{61} -2.00000i q^{62} -1.00000 q^{64} +14.0000i q^{67} -6.00000i q^{68} -15.0000 q^{71} -2.00000i q^{73} +11.0000 q^{74} -4.00000 q^{76} -6.00000i q^{77} +10.0000 q^{79} -6.00000i q^{82} +4.00000 q^{86} -3.00000i q^{88} -6.00000 q^{89} +10.0000 q^{91} -3.00000i q^{92} +3.00000 q^{94} -7.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} - 10 q^{26} + 4 q^{31} + 12 q^{34} + 12 q^{41} + 6 q^{44} + 6 q^{46} + 6 q^{49} - 4 q^{56} - 18 q^{59} + 22 q^{61} - 2 q^{64} - 30 q^{71} + 22 q^{74} - 8 q^{76} + 20 q^{79} + 8 q^{86} - 12 q^{89} + 20 q^{91} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000i 0.639602i
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 11.0000i 1.80839i 0.427121 + 0.904194i \(0.359528\pi\)
−0.427121 + 0.904194i \(0.640472\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 5.00000i 0.693375i
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 11.0000 1.27872
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 6.00000i − 0.683763i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 6.00000i − 0.662589i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) − 3.00000i − 0.319801i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) − 3.00000i − 0.312772i
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 15.0000i 1.45010i 0.688694 + 0.725052i \(0.258184\pi\)
−0.688694 + 0.725052i \(0.741816\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 9.00000i 0.828517i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 11.0000i − 0.995893i
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.0000i 1.25877i
\(143\) 15.0000i 1.25436i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) − 11.0000i − 0.904194i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 4.00000i 0.324443i
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) − 2.00000i − 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) − 10.0000i − 0.741249i
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) − 18.0000i − 1.31629i
\(188\) − 3.00000i − 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) −7.00000 −0.502571
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 18.0000i − 1.26648i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) − 5.00000i − 0.346688i
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 12.0000i − 0.824163i
\(213\) 0 0
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) − 10.0000i − 0.677285i
\(219\) 0 0
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 21.0000i 1.39382i 0.717159 + 0.696909i \(0.245442\pi\)
−0.717159 + 0.696909i \(0.754558\pi\)
\(228\) 0 0
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) 0 0
\(238\) 12.0000i 0.777844i
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 0 0
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) 0 0
\(247\) − 20.0000i − 1.27257i
\(248\) 2.00000i 0.127000i
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) − 9.00000i − 0.565825i
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −22.0000 −1.36701
\(260\) 0 0
\(261\) 0 0
\(262\) − 21.0000i − 1.29738i
\(263\) 3.00000i 0.184988i 0.995713 + 0.0924940i \(0.0294839\pi\)
−0.995713 + 0.0924940i \(0.970516\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) − 14.0000i − 0.855186i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 14.0000i 0.839664i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) 15.0000 0.890086
\(285\) 0 0
\(286\) 15.0000 0.886969
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000i 0.117041i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −11.0000 −0.639362
\(297\) 0 0
\(298\) 18.0000i 1.04271i
\(299\) 15.0000 0.867472
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 16.0000i 0.920697i
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) − 10.0000i − 0.570730i −0.958419 0.285365i \(-0.907885\pi\)
0.958419 0.285365i \(-0.0921148\pi\)
\(308\) 6.00000i 0.341882i
\(309\) 0 0
\(310\) 0 0
\(311\) −27.0000 −1.53103 −0.765515 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 6.00000i 0.334367i
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000i 0.159901i
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) − 15.0000i − 0.792775i
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 7.00000i 0.367912i
\(363\) 0 0
\(364\) −10.0000 −0.524142
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.0000i 1.07305i 0.843884 + 0.536525i \(0.180263\pi\)
−0.843884 + 0.536525i \(0.819737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 7.00000i 0.355371i
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 3.00000i 0.151523i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.00000i − 0.0501886i −0.999685 0.0250943i \(-0.992011\pi\)
0.999685 0.0250943i \(-0.00798860\pi\)
\(398\) 2.00000i 0.100251i
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) − 10.0000i − 0.498135i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) − 33.0000i − 1.63575i
\(408\) 0 0
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14.0000i 0.689730i
\(413\) − 18.0000i − 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 12.0000i 0.586939i
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 22.0000i 1.06465i
\(428\) − 15.0000i − 0.725052i
\(429\) 0 0
\(430\) 0 0
\(431\) −9.00000 −0.433515 −0.216757 0.976226i \(-0.569548\pi\)
−0.216757 + 0.976226i \(0.569548\pi\)
\(432\) 0 0
\(433\) − 35.0000i − 1.68199i −0.541041 0.840996i \(-0.681970\pi\)
0.541041 0.840996i \(-0.318030\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 30.0000i − 1.42695i
\(443\) 27.0000i 1.28281i 0.767203 + 0.641404i \(0.221648\pi\)
−0.767203 + 0.641404i \(0.778352\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) 42.0000 1.98210 0.991051 0.133482i \(-0.0426157\pi\)
0.991051 + 0.133482i \(0.0426157\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 18.0000i 0.846649i
\(453\) 0 0
\(454\) 21.0000 0.985579
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) − 7.00000i − 0.327089i
\(459\) 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
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 21.0000i 0.971764i 0.874024 + 0.485882i \(0.161502\pi\)
−0.874024 + 0.485882i \(0.838498\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) − 9.00000i − 0.414259i
\(473\) − 12.0000i − 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) − 9.00000i − 0.411650i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 55.0000 2.50778
\(482\) 19.0000i 0.865426i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) 11.0000i 0.497947i
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −20.0000 −0.899843
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) − 30.0000i − 1.34568i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 27.0000i 1.20507i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) − 2.00000i − 0.0887357i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) − 9.00000i − 0.395820i
\(518\) 22.0000i 0.966625i
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −21.0000 −0.917389
\(525\) 0 0
\(526\) 3.00000 0.130806
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) − 8.00000i − 0.346844i
\(533\) − 30.0000i − 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 12.0000i 0.512615i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 39.0000i − 1.64365i −0.569737 0.821827i \(-0.692955\pi\)
0.569737 0.821827i \(-0.307045\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) − 15.0000i − 0.629386i
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) − 15.0000i − 0.627182i
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 23.0000i 0.957503i 0.877951 + 0.478751i \(0.158910\pi\)
−0.877951 + 0.478751i \(0.841090\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 11.0000i 0.452097i
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) − 15.0000i − 0.613396i
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) − 11.0000i − 0.444286i −0.975014 0.222143i \(-0.928695\pi\)
0.975014 0.222143i \(-0.0713052\pi\)
\(614\) −10.0000 −0.403567
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 27.0000i 1.08260i
\(623\) − 12.0000i − 0.480770i
\(624\) 0 0
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −66.0000 −2.63159
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) − 15.0000i − 0.594322i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) − 50.0000i − 1.97181i −0.167313 0.985904i \(-0.553509\pi\)
0.167313 0.985904i \(-0.446491\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) − 33.0000i − 1.29736i −0.761060 0.648682i \(-0.775321\pi\)
0.761060 0.648682i \(-0.224679\pi\)
\(648\) 0 0
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) 0 0
\(652\) 2.00000i 0.0783260i
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 6.00000i 0.233904i
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 10.0000i 0.388661i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 3.00000i − 0.116073i
\(669\) 0 0
\(670\) 0 0
\(671\) −33.0000 −1.27395
\(672\) 0 0
\(673\) 1.00000i 0.0385472i 0.999814 + 0.0192736i \(0.00613535\pi\)
−0.999814 + 0.0192736i \(0.993865\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 6.00000i 0.229752i
\(683\) − 15.0000i − 0.573959i −0.957937 0.286980i \(-0.907349\pi\)
0.957937 0.286980i \(-0.0926512\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 4.00000i 0.152499i
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 26.0000i 0.984115i
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 44.0000i 1.65949i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 36.0000i 1.35392i
\(708\) 0 0
\(709\) 43.0000 1.61490 0.807449 0.589937i \(-0.200847\pi\)
0.807449 + 0.589937i \(0.200847\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) 6.00000i 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) −15.0000 −0.560576
\(717\) 0 0
\(718\) 3.00000i 0.111959i
\(719\) 21.0000 0.783168 0.391584 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 3.00000i 0.111648i
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) 10.0000i 0.370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 31.0000i 1.14501i 0.819901 + 0.572506i \(0.194029\pi\)
−0.819901 + 0.572506i \(0.805971\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) − 42.0000i − 1.54709i
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 24.0000i 0.881068i
\(743\) 3.00000i 0.110059i 0.998485 + 0.0550297i \(0.0175253\pi\)
−0.998485 + 0.0550297i \(0.982475\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 18.0000i 0.658145i
\(749\) −30.0000 −1.09618
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41.0000i 1.49017i 0.666969 + 0.745085i \(0.267591\pi\)
−0.666969 + 0.745085i \(0.732409\pi\)
\(758\) − 34.0000i − 1.23494i
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 45.0000i 1.62486i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) − 12.0000i − 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7.00000 0.251285
\(777\) 0 0
\(778\) 12.0000i 0.430221i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 45.0000 1.61023
\(782\) 18.0000i 0.643679i
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000i 0.285169i 0.989783 + 0.142585i \(0.0455413\pi\)
−0.989783 + 0.142585i \(0.954459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) − 55.0000i − 1.95311i
\(794\) −1.00000 −0.0354887
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) − 12.0000i − 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) − 6.00000i − 0.211867i
\(803\) 6.00000i 0.211735i
\(804\) 0 0
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 0 0
\(808\) 18.0000i 0.633238i
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −33.0000 −1.15665
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) − 31.0000i − 1.08389i
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) − 32.0000i − 1.11545i −0.830026 0.557725i \(-0.811674\pi\)
0.830026 0.557725i \(-0.188326\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) − 51.0000i − 1.77344i −0.462303 0.886722i \(-0.652977\pi\)
0.462303 0.886722i \(-0.347023\pi\)
\(828\) 0 0
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.00000i 0.173344i
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) − 24.0000i − 0.829066i
\(839\) 9.00000 0.310715 0.155357 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 19.0000i 0.654783i
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.00000i − 0.137442i
\(848\) 12.0000i 0.412082i
\(849\) 0 0
\(850\) 0 0
\(851\) −33.0000 −1.13123
\(852\) 0 0
\(853\) 1.00000i 0.0342393i 0.999853 + 0.0171197i \(0.00544963\pi\)
−0.999853 + 0.0171197i \(0.994550\pi\)
\(854\) 22.0000 0.752825
\(855\) 0 0
\(856\) −15.0000 −0.512689
\(857\) − 48.0000i − 1.63965i −0.572615 0.819824i \(-0.694071\pi\)
0.572615 0.819824i \(-0.305929\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 9.00000i 0.306541i
\(863\) − 48.0000i − 1.63394i −0.576681 0.816970i \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −35.0000 −1.18935
\(867\) 0 0
\(868\) − 4.00000i − 0.135769i
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 70.0000 2.37186
\(872\) 10.0000i 0.338643i
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.00000i − 0.0337676i −0.999857 0.0168838i \(-0.994625\pi\)
0.999857 0.0168838i \(-0.00537454\pi\)
\(878\) 14.0000i 0.472477i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 22.0000i 0.740359i 0.928960 + 0.370179i \(0.120704\pi\)
−0.928960 + 0.370179i \(0.879296\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) 27.0000 0.907083
\(887\) 21.0000i 0.705111i 0.935791 + 0.352555i \(0.114687\pi\)
−0.935791 + 0.352555i \(0.885313\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) − 4.00000i − 0.133930i
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) − 42.0000i − 1.40156i
\(899\) 0 0
\(900\) 0 0
\(901\) −72.0000 −2.39867
\(902\) 18.0000i 0.599334i
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) − 21.0000i − 0.696909i
\(909\) 0 0
\(910\) 0 0
\(911\) −33.0000 −1.09334 −0.546669 0.837349i \(-0.684105\pi\)
−0.546669 + 0.837349i \(0.684105\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) −7.00000 −0.231287
\(917\) 42.0000i 1.38696i
\(918\) 0 0
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 12.0000i − 0.395199i
\(923\) 75.0000i 2.46866i
\(924\) 0 0
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) 0 0
\(937\) − 13.0000i − 0.424691i −0.977195 0.212346i \(-0.931890\pi\)
0.977195 0.212346i \(-0.0681103\pi\)
\(938\) 28.0000i 0.914232i
\(939\) 0 0
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) − 12.0000i − 0.388922i
\(953\) 12.0000i 0.388718i 0.980930 + 0.194359i \(0.0622627\pi\)
−0.980930 + 0.194359i \(0.937737\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 55.0000i − 1.77327i
\(963\) 0 0
\(964\) 19.0000 0.611949
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.00000i − 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 0 0
\(970\) 0 0
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 0 0
\(973\) − 28.0000i − 0.897639i
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 24.0000i 0.765871i
\(983\) 21.0000i 0.669796i 0.942254 + 0.334898i \(0.108702\pi\)
−0.942254 + 0.334898i \(0.891298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 20.0000i 0.636285i
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) 0 0
\(994\) −30.0000 −0.951542
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.00000i − 0.0316703i −0.999875 0.0158352i \(-0.994959\pi\)
0.999875 0.0158352i \(-0.00504070\pi\)
\(998\) 20.0000i 0.633089i
\(999\) 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 1350.2.c.d.649.1 2
3.2 odd 2 1350.2.c.i.649.2 2
5.2 odd 4 1350.2.a.n.1.1 yes 1
5.3 odd 4 1350.2.a.i.1.1 yes 1
5.4 even 2 inner 1350.2.c.d.649.2 2
15.2 even 4 1350.2.a.e.1.1 1
15.8 even 4 1350.2.a.t.1.1 yes 1
15.14 odd 2 1350.2.c.i.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1350.2.a.e.1.1 1 15.2 even 4
1350.2.a.i.1.1 yes 1 5.3 odd 4
1350.2.a.n.1.1 yes 1 5.2 odd 4
1350.2.a.t.1.1 yes 1 15.8 even 4
1350.2.c.d.649.1 2 1.1 even 1 trivial
1350.2.c.d.649.2 2 5.4 even 2 inner
1350.2.c.i.649.1 2 15.14 odd 2
1350.2.c.i.649.2 2 3.2 odd 2