Properties

Label 9450.2.a.ef.1.1
Level $9450$
Weight $2$
Character 9450.1
Self dual yes
Analytic conductor $75.459$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.a (trivial)

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: yes
Analytic conductor: \(75.4586299101\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 9450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} -1.16228 q^{11} -2.16228 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -0.837722 q^{19} +1.16228 q^{22} -0.162278 q^{23} +2.16228 q^{26} -1.00000 q^{28} -2.16228 q^{29} -8.48683 q^{31} -1.00000 q^{32} -5.00000 q^{34} +9.16228 q^{37} +0.837722 q^{38} +2.32456 q^{41} +7.00000 q^{43} -1.16228 q^{44} +0.162278 q^{46} -2.00000 q^{47} +1.00000 q^{49} -2.16228 q^{52} -12.4868 q^{53} +1.00000 q^{56} +2.16228 q^{58} -7.00000 q^{59} -8.32456 q^{61} +8.48683 q^{62} +1.00000 q^{64} +15.3246 q^{67} +5.00000 q^{68} +6.48683 q^{71} +10.0000 q^{73} -9.16228 q^{74} -0.837722 q^{76} +1.16228 q^{77} +6.83772 q^{79} -2.32456 q^{82} -8.32456 q^{83} -7.00000 q^{86} +1.16228 q^{88} -15.3246 q^{89} +2.16228 q^{91} -0.162278 q^{92} +2.00000 q^{94} +1.16228 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 4 q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 10 q^{17} - 8 q^{19} - 4 q^{22} + 6 q^{23} - 2 q^{26} - 2 q^{28} + 2 q^{29} + 2 q^{31} - 2 q^{32} - 10 q^{34} + 12 q^{37} + 8 q^{38} - 8 q^{41} + 14 q^{43} + 4 q^{44} - 6 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{52} - 6 q^{53} + 2 q^{56} - 2 q^{58} - 14 q^{59} - 4 q^{61} - 2 q^{62} + 2 q^{64} + 18 q^{67} + 10 q^{68} - 6 q^{71} + 20 q^{73} - 12 q^{74} - 8 q^{76} - 4 q^{77} + 20 q^{79} + 8 q^{82} - 4 q^{83} - 14 q^{86} - 4 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} + 4 q^{94} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −1.16228 −0.350440 −0.175220 0.984529i \(-0.556064\pi\)
−0.175220 + 0.984529i \(0.556064\pi\)
\(12\) 0 0
\(13\) −2.16228 −0.599708 −0.299854 0.953985i \(-0.596938\pi\)
−0.299854 + 0.953985i \(0.596938\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −0.837722 −0.192187 −0.0960933 0.995372i \(-0.530635\pi\)
−0.0960933 + 0.995372i \(0.530635\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.16228 0.247798
\(23\) −0.162278 −0.0338372 −0.0169186 0.999857i \(-0.505386\pi\)
−0.0169186 + 0.999857i \(0.505386\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.16228 0.424058
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.16228 −0.401525 −0.200762 0.979640i \(-0.564342\pi\)
−0.200762 + 0.979640i \(0.564342\pi\)
\(30\) 0 0
\(31\) −8.48683 −1.52428 −0.762140 0.647412i \(-0.775851\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 0 0
\(37\) 9.16228 1.50627 0.753135 0.657866i \(-0.228541\pi\)
0.753135 + 0.657866i \(0.228541\pi\)
\(38\) 0.837722 0.135897
\(39\) 0 0
\(40\) 0 0
\(41\) 2.32456 0.363035 0.181517 0.983388i \(-0.441899\pi\)
0.181517 + 0.983388i \(0.441899\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) −1.16228 −0.175220
\(45\) 0 0
\(46\) 0.162278 0.0239265
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −2.16228 −0.299854
\(53\) −12.4868 −1.71520 −0.857599 0.514319i \(-0.828045\pi\)
−0.857599 + 0.514319i \(0.828045\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.16228 0.283921
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −8.32456 −1.06585 −0.532925 0.846162i \(-0.678907\pi\)
−0.532925 + 0.846162i \(0.678907\pi\)
\(62\) 8.48683 1.07783
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 15.3246 1.87219 0.936096 0.351744i \(-0.114411\pi\)
0.936096 + 0.351744i \(0.114411\pi\)
\(68\) 5.00000 0.606339
\(69\) 0 0
\(70\) 0 0
\(71\) 6.48683 0.769845 0.384923 0.922949i \(-0.374228\pi\)
0.384923 + 0.922949i \(0.374228\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −9.16228 −1.06509
\(75\) 0 0
\(76\) −0.837722 −0.0960933
\(77\) 1.16228 0.132454
\(78\) 0 0
\(79\) 6.83772 0.769304 0.384652 0.923062i \(-0.374321\pi\)
0.384652 + 0.923062i \(0.374321\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.32456 −0.256704
\(83\) −8.32456 −0.913739 −0.456869 0.889534i \(-0.651029\pi\)
−0.456869 + 0.889534i \(0.651029\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) 0 0
\(88\) 1.16228 0.123899
\(89\) −15.3246 −1.62440 −0.812200 0.583379i \(-0.801730\pi\)
−0.812200 + 0.583379i \(0.801730\pi\)
\(90\) 0 0
\(91\) 2.16228 0.226668
\(92\) −0.162278 −0.0169186
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 0 0
\(97\) 1.16228 0.118011 0.0590057 0.998258i \(-0.481207\pi\)
0.0590057 + 0.998258i \(0.481207\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) 1.16228 0.115651 0.0578255 0.998327i \(-0.481583\pi\)
0.0578255 + 0.998327i \(0.481583\pi\)
\(102\) 0 0
\(103\) −4.48683 −0.442101 −0.221050 0.975262i \(-0.570949\pi\)
−0.221050 + 0.975262i \(0.570949\pi\)
\(104\) 2.16228 0.212029
\(105\) 0 0
\(106\) 12.4868 1.21283
\(107\) 5.16228 0.499056 0.249528 0.968368i \(-0.419725\pi\)
0.249528 + 0.968368i \(0.419725\pi\)
\(108\) 0 0
\(109\) 3.16228 0.302891 0.151446 0.988466i \(-0.451607\pi\)
0.151446 + 0.988466i \(0.451607\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.83772 −0.643239 −0.321619 0.946869i \(-0.604227\pi\)
−0.321619 + 0.946869i \(0.604227\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.16228 −0.200762
\(117\) 0 0
\(118\) 7.00000 0.644402
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −9.64911 −0.877192
\(122\) 8.32456 0.753670
\(123\) 0 0
\(124\) −8.48683 −0.762140
\(125\) 0 0
\(126\) 0 0
\(127\) 5.67544 0.503614 0.251807 0.967777i \(-0.418975\pi\)
0.251807 + 0.967777i \(0.418975\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 7.32456 0.639949 0.319975 0.947426i \(-0.396326\pi\)
0.319975 + 0.947426i \(0.396326\pi\)
\(132\) 0 0
\(133\) 0.837722 0.0726397
\(134\) −15.3246 −1.32384
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 4.32456 0.369472 0.184736 0.982788i \(-0.440857\pi\)
0.184736 + 0.982788i \(0.440857\pi\)
\(138\) 0 0
\(139\) 1.48683 0.126112 0.0630558 0.998010i \(-0.479915\pi\)
0.0630558 + 0.998010i \(0.479915\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.48683 −0.544363
\(143\) 2.51317 0.210162
\(144\) 0 0
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 9.16228 0.753135
\(149\) −20.8114 −1.70494 −0.852468 0.522780i \(-0.824895\pi\)
−0.852468 + 0.522780i \(0.824895\pi\)
\(150\) 0 0
\(151\) 17.8114 1.44947 0.724735 0.689028i \(-0.241962\pi\)
0.724735 + 0.689028i \(0.241962\pi\)
\(152\) 0.837722 0.0679483
\(153\) 0 0
\(154\) −1.16228 −0.0936590
\(155\) 0 0
\(156\) 0 0
\(157\) −1.83772 −0.146666 −0.0733331 0.997308i \(-0.523364\pi\)
−0.0733331 + 0.997308i \(0.523364\pi\)
\(158\) −6.83772 −0.543980
\(159\) 0 0
\(160\) 0 0
\(161\) 0.162278 0.0127893
\(162\) 0 0
\(163\) 13.6491 1.06908 0.534540 0.845143i \(-0.320485\pi\)
0.534540 + 0.845143i \(0.320485\pi\)
\(164\) 2.32456 0.181517
\(165\) 0 0
\(166\) 8.32456 0.646111
\(167\) −2.51317 −0.194475 −0.0972374 0.995261i \(-0.531001\pi\)
−0.0972374 + 0.995261i \(0.531001\pi\)
\(168\) 0 0
\(169\) −8.32456 −0.640350
\(170\) 0 0
\(171\) 0 0
\(172\) 7.00000 0.533745
\(173\) 11.8114 0.898003 0.449002 0.893531i \(-0.351780\pi\)
0.449002 + 0.893531i \(0.351780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.16228 −0.0876100
\(177\) 0 0
\(178\) 15.3246 1.14862
\(179\) 18.3246 1.36964 0.684821 0.728712i \(-0.259880\pi\)
0.684821 + 0.728712i \(0.259880\pi\)
\(180\) 0 0
\(181\) 14.4868 1.07680 0.538399 0.842690i \(-0.319029\pi\)
0.538399 + 0.842690i \(0.319029\pi\)
\(182\) −2.16228 −0.160279
\(183\) 0 0
\(184\) 0.162278 0.0119633
\(185\) 0 0
\(186\) 0 0
\(187\) −5.81139 −0.424971
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 0 0
\(191\) 13.6754 0.989520 0.494760 0.869030i \(-0.335256\pi\)
0.494760 + 0.869030i \(0.335256\pi\)
\(192\) 0 0
\(193\) −17.9737 −1.29377 −0.646886 0.762586i \(-0.723929\pi\)
−0.646886 + 0.762586i \(0.723929\pi\)
\(194\) −1.16228 −0.0834467
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.3246 −1.30557 −0.652785 0.757543i \(-0.726400\pi\)
−0.652785 + 0.757543i \(0.726400\pi\)
\(198\) 0 0
\(199\) −5.51317 −0.390818 −0.195409 0.980722i \(-0.562603\pi\)
−0.195409 + 0.980722i \(0.562603\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1.16228 −0.0817776
\(203\) 2.16228 0.151762
\(204\) 0 0
\(205\) 0 0
\(206\) 4.48683 0.312612
\(207\) 0 0
\(208\) −2.16228 −0.149927
\(209\) 0.973666 0.0673499
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −12.4868 −0.857599
\(213\) 0 0
\(214\) −5.16228 −0.352886
\(215\) 0 0
\(216\) 0 0
\(217\) 8.48683 0.576124
\(218\) −3.16228 −0.214176
\(219\) 0 0
\(220\) 0 0
\(221\) −10.8114 −0.727253
\(222\) 0 0
\(223\) 24.6491 1.65063 0.825313 0.564675i \(-0.190999\pi\)
0.825313 + 0.564675i \(0.190999\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.83772 0.454839
\(227\) 0.675445 0.0448308 0.0224154 0.999749i \(-0.492864\pi\)
0.0224154 + 0.999749i \(0.492864\pi\)
\(228\) 0 0
\(229\) 18.3246 1.21092 0.605460 0.795875i \(-0.292989\pi\)
0.605460 + 0.795875i \(0.292989\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.16228 0.141960
\(233\) 28.1359 1.84325 0.921623 0.388085i \(-0.126863\pi\)
0.921623 + 0.388085i \(0.126863\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 0 0
\(238\) 5.00000 0.324102
\(239\) −6.32456 −0.409101 −0.204551 0.978856i \(-0.565573\pi\)
−0.204551 + 0.978856i \(0.565573\pi\)
\(240\) 0 0
\(241\) −7.16228 −0.461363 −0.230681 0.973029i \(-0.574096\pi\)
−0.230681 + 0.973029i \(0.574096\pi\)
\(242\) 9.64911 0.620268
\(243\) 0 0
\(244\) −8.32456 −0.532925
\(245\) 0 0
\(246\) 0 0
\(247\) 1.81139 0.115256
\(248\) 8.48683 0.538914
\(249\) 0 0
\(250\) 0 0
\(251\) 3.35089 0.211506 0.105753 0.994392i \(-0.466275\pi\)
0.105753 + 0.994392i \(0.466275\pi\)
\(252\) 0 0
\(253\) 0.188612 0.0118579
\(254\) −5.67544 −0.356109
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −9.16228 −0.569316
\(260\) 0 0
\(261\) 0 0
\(262\) −7.32456 −0.452513
\(263\) 4.16228 0.256657 0.128329 0.991732i \(-0.459039\pi\)
0.128329 + 0.991732i \(0.459039\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.837722 −0.0513641
\(267\) 0 0
\(268\) 15.3246 0.936096
\(269\) 22.3246 1.36115 0.680576 0.732677i \(-0.261730\pi\)
0.680576 + 0.732677i \(0.261730\pi\)
\(270\) 0 0
\(271\) −16.1623 −0.981788 −0.490894 0.871219i \(-0.663330\pi\)
−0.490894 + 0.871219i \(0.663330\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) −4.32456 −0.261256
\(275\) 0 0
\(276\) 0 0
\(277\) 12.6491 0.760011 0.380006 0.924984i \(-0.375922\pi\)
0.380006 + 0.924984i \(0.375922\pi\)
\(278\) −1.48683 −0.0891743
\(279\) 0 0
\(280\) 0 0
\(281\) −2.51317 −0.149923 −0.0749615 0.997186i \(-0.523883\pi\)
−0.0749615 + 0.997186i \(0.523883\pi\)
\(282\) 0 0
\(283\) 21.8114 1.29655 0.648276 0.761405i \(-0.275490\pi\)
0.648276 + 0.761405i \(0.275490\pi\)
\(284\) 6.48683 0.384923
\(285\) 0 0
\(286\) −2.51317 −0.148607
\(287\) −2.32456 −0.137214
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −24.9737 −1.45898 −0.729489 0.683993i \(-0.760242\pi\)
−0.729489 + 0.683993i \(0.760242\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.16228 −0.532547
\(297\) 0 0
\(298\) 20.8114 1.20557
\(299\) 0.350889 0.0202925
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) −17.8114 −1.02493
\(303\) 0 0
\(304\) −0.837722 −0.0480467
\(305\) 0 0
\(306\) 0 0
\(307\) 11.1623 0.637065 0.318532 0.947912i \(-0.396810\pi\)
0.318532 + 0.947912i \(0.396810\pi\)
\(308\) 1.16228 0.0662269
\(309\) 0 0
\(310\) 0 0
\(311\) −21.4868 −1.21841 −0.609203 0.793014i \(-0.708511\pi\)
−0.609203 + 0.793014i \(0.708511\pi\)
\(312\) 0 0
\(313\) 32.4605 1.83478 0.917388 0.397994i \(-0.130294\pi\)
0.917388 + 0.397994i \(0.130294\pi\)
\(314\) 1.83772 0.103709
\(315\) 0 0
\(316\) 6.83772 0.384652
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 2.51317 0.140710
\(320\) 0 0
\(321\) 0 0
\(322\) −0.162278 −0.00904338
\(323\) −4.18861 −0.233061
\(324\) 0 0
\(325\) 0 0
\(326\) −13.6491 −0.755954
\(327\) 0 0
\(328\) −2.32456 −0.128352
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 27.6491 1.51973 0.759866 0.650079i \(-0.225264\pi\)
0.759866 + 0.650079i \(0.225264\pi\)
\(332\) −8.32456 −0.456869
\(333\) 0 0
\(334\) 2.51317 0.137514
\(335\) 0 0
\(336\) 0 0
\(337\) −9.32456 −0.507941 −0.253970 0.967212i \(-0.581737\pi\)
−0.253970 + 0.967212i \(0.581737\pi\)
\(338\) 8.32456 0.452796
\(339\) 0 0
\(340\) 0 0
\(341\) 9.86406 0.534169
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) −11.8114 −0.634984
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) 1.83772 0.0983710 0.0491855 0.998790i \(-0.484337\pi\)
0.0491855 + 0.998790i \(0.484337\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.16228 0.0619496
\(353\) 5.32456 0.283397 0.141699 0.989910i \(-0.454744\pi\)
0.141699 + 0.989910i \(0.454744\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −15.3246 −0.812200
\(357\) 0 0
\(358\) −18.3246 −0.968483
\(359\) 14.1623 0.747456 0.373728 0.927538i \(-0.378079\pi\)
0.373728 + 0.927538i \(0.378079\pi\)
\(360\) 0 0
\(361\) −18.2982 −0.963064
\(362\) −14.4868 −0.761411
\(363\) 0 0
\(364\) 2.16228 0.113334
\(365\) 0 0
\(366\) 0 0
\(367\) −0.811388 −0.0423541 −0.0211771 0.999776i \(-0.506741\pi\)
−0.0211771 + 0.999776i \(0.506741\pi\)
\(368\) −0.162278 −0.00845931
\(369\) 0 0
\(370\) 0 0
\(371\) 12.4868 0.648284
\(372\) 0 0
\(373\) −1.81139 −0.0937901 −0.0468951 0.998900i \(-0.514933\pi\)
−0.0468951 + 0.998900i \(0.514933\pi\)
\(374\) 5.81139 0.300500
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 4.67544 0.240798
\(378\) 0 0
\(379\) 5.35089 0.274857 0.137428 0.990512i \(-0.456116\pi\)
0.137428 + 0.990512i \(0.456116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −13.6754 −0.699697
\(383\) 26.3246 1.34512 0.672561 0.740042i \(-0.265194\pi\)
0.672561 + 0.740042i \(0.265194\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.9737 0.914836
\(387\) 0 0
\(388\) 1.16228 0.0590057
\(389\) −21.6754 −1.09899 −0.549494 0.835497i \(-0.685180\pi\)
−0.549494 + 0.835497i \(0.685180\pi\)
\(390\) 0 0
\(391\) −0.811388 −0.0410337
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.3246 0.923178
\(395\) 0 0
\(396\) 0 0
\(397\) 5.67544 0.284842 0.142421 0.989806i \(-0.454511\pi\)
0.142421 + 0.989806i \(0.454511\pi\)
\(398\) 5.51317 0.276350
\(399\) 0 0
\(400\) 0 0
\(401\) 29.6228 1.47929 0.739645 0.672997i \(-0.234993\pi\)
0.739645 + 0.672997i \(0.234993\pi\)
\(402\) 0 0
\(403\) 18.3509 0.914123
\(404\) 1.16228 0.0578255
\(405\) 0 0
\(406\) −2.16228 −0.107312
\(407\) −10.6491 −0.527857
\(408\) 0 0
\(409\) 31.2982 1.54760 0.773799 0.633432i \(-0.218354\pi\)
0.773799 + 0.633432i \(0.218354\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.48683 −0.221050
\(413\) 7.00000 0.344447
\(414\) 0 0
\(415\) 0 0
\(416\) 2.16228 0.106014
\(417\) 0 0
\(418\) −0.973666 −0.0476236
\(419\) 28.6228 1.39831 0.699157 0.714968i \(-0.253559\pi\)
0.699157 + 0.714968i \(0.253559\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 3.00000 0.146038
\(423\) 0 0
\(424\) 12.4868 0.606414
\(425\) 0 0
\(426\) 0 0
\(427\) 8.32456 0.402853
\(428\) 5.16228 0.249528
\(429\) 0 0
\(430\) 0 0
\(431\) −19.2982 −0.929563 −0.464781 0.885426i \(-0.653867\pi\)
−0.464781 + 0.885426i \(0.653867\pi\)
\(432\) 0 0
\(433\) −13.2982 −0.639072 −0.319536 0.947574i \(-0.603527\pi\)
−0.319536 + 0.947574i \(0.603527\pi\)
\(434\) −8.48683 −0.407381
\(435\) 0 0
\(436\) 3.16228 0.151446
\(437\) 0.135944 0.00650307
\(438\) 0 0
\(439\) 10.8114 0.515999 0.258000 0.966145i \(-0.416937\pi\)
0.258000 + 0.966145i \(0.416937\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.8114 0.514245
\(443\) −9.48683 −0.450733 −0.225367 0.974274i \(-0.572358\pi\)
−0.225367 + 0.974274i \(0.572358\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.6491 −1.16717
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 29.2982 1.38267 0.691334 0.722535i \(-0.257023\pi\)
0.691334 + 0.722535i \(0.257023\pi\)
\(450\) 0 0
\(451\) −2.70178 −0.127222
\(452\) −6.83772 −0.321619
\(453\) 0 0
\(454\) −0.675445 −0.0317002
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) −18.3246 −0.856250
\(459\) 0 0
\(460\) 0 0
\(461\) −35.2982 −1.64400 −0.822001 0.569486i \(-0.807142\pi\)
−0.822001 + 0.569486i \(0.807142\pi\)
\(462\) 0 0
\(463\) −17.1623 −0.797599 −0.398799 0.917038i \(-0.630573\pi\)
−0.398799 + 0.917038i \(0.630573\pi\)
\(464\) −2.16228 −0.100381
\(465\) 0 0
\(466\) −28.1359 −1.30337
\(467\) −14.6491 −0.677880 −0.338940 0.940808i \(-0.610068\pi\)
−0.338940 + 0.940808i \(0.610068\pi\)
\(468\) 0 0
\(469\) −15.3246 −0.707622
\(470\) 0 0
\(471\) 0 0
\(472\) 7.00000 0.322201
\(473\) −8.13594 −0.374091
\(474\) 0 0
\(475\) 0 0
\(476\) −5.00000 −0.229175
\(477\) 0 0
\(478\) 6.32456 0.289278
\(479\) −28.6491 −1.30901 −0.654506 0.756057i \(-0.727123\pi\)
−0.654506 + 0.756057i \(0.727123\pi\)
\(480\) 0 0
\(481\) −19.8114 −0.903322
\(482\) 7.16228 0.326233
\(483\) 0 0
\(484\) −9.64911 −0.438596
\(485\) 0 0
\(486\) 0 0
\(487\) −14.6491 −0.663815 −0.331907 0.943312i \(-0.607692\pi\)
−0.331907 + 0.943312i \(0.607692\pi\)
\(488\) 8.32456 0.376835
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −10.8114 −0.486920
\(494\) −1.81139 −0.0814982
\(495\) 0 0
\(496\) −8.48683 −0.381070
\(497\) −6.48683 −0.290974
\(498\) 0 0
\(499\) −28.3246 −1.26798 −0.633991 0.773341i \(-0.718584\pi\)
−0.633991 + 0.773341i \(0.718584\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.35089 −0.149558
\(503\) 12.8377 0.572406 0.286203 0.958169i \(-0.407607\pi\)
0.286203 + 0.958169i \(0.407607\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.188612 −0.00838481
\(507\) 0 0
\(508\) 5.67544 0.251807
\(509\) −17.1623 −0.760705 −0.380352 0.924842i \(-0.624197\pi\)
−0.380352 + 0.924842i \(0.624197\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 2.32456 0.102234
\(518\) 9.16228 0.402567
\(519\) 0 0
\(520\) 0 0
\(521\) 0.675445 0.0295918 0.0147959 0.999891i \(-0.495290\pi\)
0.0147959 + 0.999891i \(0.495290\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 7.32456 0.319975
\(525\) 0 0
\(526\) −4.16228 −0.181484
\(527\) −42.4342 −1.84846
\(528\) 0 0
\(529\) −22.9737 −0.998855
\(530\) 0 0
\(531\) 0 0
\(532\) 0.837722 0.0363199
\(533\) −5.02633 −0.217715
\(534\) 0 0
\(535\) 0 0
\(536\) −15.3246 −0.661920
\(537\) 0 0
\(538\) −22.3246 −0.962480
\(539\) −1.16228 −0.0500628
\(540\) 0 0
\(541\) 15.1623 0.651877 0.325939 0.945391i \(-0.394320\pi\)
0.325939 + 0.945391i \(0.394320\pi\)
\(542\) 16.1623 0.694229
\(543\) 0 0
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) −0.324555 −0.0138770 −0.00693849 0.999976i \(-0.502209\pi\)
−0.00693849 + 0.999976i \(0.502209\pi\)
\(548\) 4.32456 0.184736
\(549\) 0 0
\(550\) 0 0
\(551\) 1.81139 0.0771677
\(552\) 0 0
\(553\) −6.83772 −0.290770
\(554\) −12.6491 −0.537409
\(555\) 0 0
\(556\) 1.48683 0.0630558
\(557\) −4.81139 −0.203865 −0.101933 0.994791i \(-0.532503\pi\)
−0.101933 + 0.994791i \(0.532503\pi\)
\(558\) 0 0
\(559\) −15.1359 −0.640182
\(560\) 0 0
\(561\) 0 0
\(562\) 2.51317 0.106012
\(563\) −41.9737 −1.76898 −0.884490 0.466560i \(-0.845493\pi\)
−0.884490 + 0.466560i \(0.845493\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −21.8114 −0.916801
\(567\) 0 0
\(568\) −6.48683 −0.272181
\(569\) −25.1623 −1.05486 −0.527429 0.849599i \(-0.676844\pi\)
−0.527429 + 0.849599i \(0.676844\pi\)
\(570\) 0 0
\(571\) 28.6228 1.19783 0.598913 0.800814i \(-0.295600\pi\)
0.598913 + 0.800814i \(0.295600\pi\)
\(572\) 2.51317 0.105081
\(573\) 0 0
\(574\) 2.32456 0.0970251
\(575\) 0 0
\(576\) 0 0
\(577\) 37.4868 1.56060 0.780299 0.625407i \(-0.215067\pi\)
0.780299 + 0.625407i \(0.215067\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 0 0
\(581\) 8.32456 0.345361
\(582\) 0 0
\(583\) 14.5132 0.601074
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 24.9737 1.03165
\(587\) 45.2719 1.86857 0.934285 0.356526i \(-0.116039\pi\)
0.934285 + 0.356526i \(0.116039\pi\)
\(588\) 0 0
\(589\) 7.10961 0.292946
\(590\) 0 0
\(591\) 0 0
\(592\) 9.16228 0.376567
\(593\) 18.9737 0.779155 0.389578 0.920994i \(-0.372621\pi\)
0.389578 + 0.920994i \(0.372621\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.8114 −0.852468
\(597\) 0 0
\(598\) −0.350889 −0.0143489
\(599\) 30.4868 1.24566 0.622829 0.782358i \(-0.285983\pi\)
0.622829 + 0.782358i \(0.285983\pi\)
\(600\) 0 0
\(601\) 1.02633 0.0418650 0.0209325 0.999781i \(-0.493336\pi\)
0.0209325 + 0.999781i \(0.493336\pi\)
\(602\) 7.00000 0.285299
\(603\) 0 0
\(604\) 17.8114 0.724735
\(605\) 0 0
\(606\) 0 0
\(607\) −36.1623 −1.46778 −0.733891 0.679267i \(-0.762298\pi\)
−0.733891 + 0.679267i \(0.762298\pi\)
\(608\) 0.837722 0.0339741
\(609\) 0 0
\(610\) 0 0
\(611\) 4.32456 0.174953
\(612\) 0 0
\(613\) 48.7851 1.97041 0.985205 0.171381i \(-0.0548228\pi\)
0.985205 + 0.171381i \(0.0548228\pi\)
\(614\) −11.1623 −0.450473
\(615\) 0 0
\(616\) −1.16228 −0.0468295
\(617\) −11.6754 −0.470036 −0.235018 0.971991i \(-0.575515\pi\)
−0.235018 + 0.971991i \(0.575515\pi\)
\(618\) 0 0
\(619\) −30.6491 −1.23189 −0.615946 0.787788i \(-0.711226\pi\)
−0.615946 + 0.787788i \(0.711226\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 21.4868 0.861544
\(623\) 15.3246 0.613965
\(624\) 0 0
\(625\) 0 0
\(626\) −32.4605 −1.29738
\(627\) 0 0
\(628\) −1.83772 −0.0733331
\(629\) 45.8114 1.82662
\(630\) 0 0
\(631\) 9.67544 0.385173 0.192587 0.981280i \(-0.438312\pi\)
0.192587 + 0.981280i \(0.438312\pi\)
\(632\) −6.83772 −0.271990
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 0 0
\(637\) −2.16228 −0.0856726
\(638\) −2.51317 −0.0994972
\(639\) 0 0
\(640\) 0 0
\(641\) 31.4868 1.24366 0.621828 0.783154i \(-0.286390\pi\)
0.621828 + 0.783154i \(0.286390\pi\)
\(642\) 0 0
\(643\) 32.7851 1.29292 0.646458 0.762949i \(-0.276249\pi\)
0.646458 + 0.762949i \(0.276249\pi\)
\(644\) 0.162278 0.00639464
\(645\) 0 0
\(646\) 4.18861 0.164799
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 8.13594 0.319364
\(650\) 0 0
\(651\) 0 0
\(652\) 13.6491 0.534540
\(653\) 41.1359 1.60977 0.804887 0.593428i \(-0.202226\pi\)
0.804887 + 0.593428i \(0.202226\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.32456 0.0907586
\(657\) 0 0
\(658\) −2.00000 −0.0779681
\(659\) −29.8114 −1.16129 −0.580643 0.814158i \(-0.697199\pi\)
−0.580643 + 0.814158i \(0.697199\pi\)
\(660\) 0 0
\(661\) 0.324555 0.0126237 0.00631187 0.999980i \(-0.497991\pi\)
0.00631187 + 0.999980i \(0.497991\pi\)
\(662\) −27.6491 −1.07461
\(663\) 0 0
\(664\) 8.32456 0.323055
\(665\) 0 0
\(666\) 0 0
\(667\) 0.350889 0.0135865
\(668\) −2.51317 −0.0972374
\(669\) 0 0
\(670\) 0 0
\(671\) 9.67544 0.373516
\(672\) 0 0
\(673\) 23.6491 0.911606 0.455803 0.890081i \(-0.349352\pi\)
0.455803 + 0.890081i \(0.349352\pi\)
\(674\) 9.32456 0.359168
\(675\) 0 0
\(676\) −8.32456 −0.320175
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) 0 0
\(679\) −1.16228 −0.0446041
\(680\) 0 0
\(681\) 0 0
\(682\) −9.86406 −0.377714
\(683\) −0.513167 −0.0196358 −0.00981790 0.999952i \(-0.503125\pi\)
−0.00981790 + 0.999952i \(0.503125\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 7.00000 0.266872
\(689\) 27.0000 1.02862
\(690\) 0 0
\(691\) 10.9737 0.417458 0.208729 0.977974i \(-0.433067\pi\)
0.208729 + 0.977974i \(0.433067\pi\)
\(692\) 11.8114 0.449002
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) 0 0
\(697\) 11.6228 0.440244
\(698\) −1.83772 −0.0695588
\(699\) 0 0
\(700\) 0 0
\(701\) 30.6491 1.15760 0.578800 0.815469i \(-0.303521\pi\)
0.578800 + 0.815469i \(0.303521\pi\)
\(702\) 0 0
\(703\) −7.67544 −0.289485
\(704\) −1.16228 −0.0438050
\(705\) 0 0
\(706\) −5.32456 −0.200392
\(707\) −1.16228 −0.0437120
\(708\) 0 0
\(709\) 45.4868 1.70829 0.854147 0.520032i \(-0.174080\pi\)
0.854147 + 0.520032i \(0.174080\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.3246 0.574312
\(713\) 1.37722 0.0515774
\(714\) 0 0
\(715\) 0 0
\(716\) 18.3246 0.684821
\(717\) 0 0
\(718\) −14.1623 −0.528532
\(719\) 39.1623 1.46051 0.730253 0.683177i \(-0.239402\pi\)
0.730253 + 0.683177i \(0.239402\pi\)
\(720\) 0 0
\(721\) 4.48683 0.167098
\(722\) 18.2982 0.680989
\(723\) 0 0
\(724\) 14.4868 0.538399
\(725\) 0 0
\(726\) 0 0
\(727\) −2.16228 −0.0801944 −0.0400972 0.999196i \(-0.512767\pi\)
−0.0400972 + 0.999196i \(0.512767\pi\)
\(728\) −2.16228 −0.0801393
\(729\) 0 0
\(730\) 0 0
\(731\) 35.0000 1.29452
\(732\) 0 0
\(733\) −5.18861 −0.191646 −0.0958229 0.995398i \(-0.530548\pi\)
−0.0958229 + 0.995398i \(0.530548\pi\)
\(734\) 0.811388 0.0299489
\(735\) 0 0
\(736\) 0.162278 0.00598163
\(737\) −17.8114 −0.656091
\(738\) 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
\(742\) −12.4868 −0.458406
\(743\) 12.4868 0.458097 0.229049 0.973415i \(-0.426438\pi\)
0.229049 + 0.973415i \(0.426438\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.81139 0.0663196
\(747\) 0 0
\(748\) −5.81139 −0.212485
\(749\) −5.16228 −0.188626
\(750\) 0 0
\(751\) 40.7851 1.48827 0.744134 0.668031i \(-0.232862\pi\)
0.744134 + 0.668031i \(0.232862\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 0 0
\(754\) −4.67544 −0.170270
\(755\) 0 0
\(756\) 0 0
\(757\) 7.35089 0.267173 0.133586 0.991037i \(-0.457351\pi\)
0.133586 + 0.991037i \(0.457351\pi\)
\(758\) −5.35089 −0.194353
\(759\) 0 0
\(760\) 0 0
\(761\) 23.6491 0.857280 0.428640 0.903475i \(-0.358993\pi\)
0.428640 + 0.903475i \(0.358993\pi\)
\(762\) 0 0
\(763\) −3.16228 −0.114482
\(764\) 13.6754 0.494760
\(765\) 0 0
\(766\) −26.3246 −0.951145
\(767\) 15.1359 0.546527
\(768\) 0 0
\(769\) −1.81139 −0.0653203 −0.0326602 0.999467i \(-0.510398\pi\)
−0.0326602 + 0.999467i \(0.510398\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −17.9737 −0.646886
\(773\) −5.02633 −0.180785 −0.0903923 0.995906i \(-0.528812\pi\)
−0.0903923 + 0.995906i \(0.528812\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.16228 −0.0417233
\(777\) 0 0
\(778\) 21.6754 0.777102
\(779\) −1.94733 −0.0697704
\(780\) 0 0
\(781\) −7.53950 −0.269785
\(782\) 0.811388 0.0290152
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) −53.6228 −1.91145 −0.955723 0.294269i \(-0.904924\pi\)
−0.955723 + 0.294269i \(0.904924\pi\)
\(788\) −18.3246 −0.652785
\(789\) 0 0
\(790\) 0 0
\(791\) 6.83772 0.243121
\(792\) 0 0
\(793\) 18.0000 0.639199
\(794\) −5.67544 −0.201414
\(795\) 0 0
\(796\) −5.51317 −0.195409
\(797\) 20.9737 0.742925 0.371463 0.928448i \(-0.378856\pi\)
0.371463 + 0.928448i \(0.378856\pi\)
\(798\) 0 0
\(799\) −10.0000 −0.353775
\(800\) 0 0
\(801\) 0 0
\(802\) −29.6228 −1.04602
\(803\) −11.6228 −0.410159
\(804\) 0 0
\(805\) 0 0
\(806\) −18.3509 −0.646383
\(807\) 0 0
\(808\) −1.16228 −0.0408888
\(809\) 20.6491 0.725984 0.362992 0.931792i \(-0.381755\pi\)
0.362992 + 0.931792i \(0.381755\pi\)
\(810\) 0 0
\(811\) 39.2982 1.37995 0.689974 0.723835i \(-0.257622\pi\)
0.689974 + 0.723835i \(0.257622\pi\)
\(812\) 2.16228 0.0758811
\(813\) 0 0
\(814\) 10.6491 0.373251
\(815\) 0 0
\(816\) 0 0
\(817\) −5.86406 −0.205157
\(818\) −31.2982 −1.09432
\(819\) 0 0
\(820\) 0 0
\(821\) −38.8114 −1.35453 −0.677263 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(822\) 0 0
\(823\) −15.4868 −0.539837 −0.269919 0.962883i \(-0.586997\pi\)
−0.269919 + 0.962883i \(0.586997\pi\)
\(824\) 4.48683 0.156306
\(825\) 0 0
\(826\) −7.00000 −0.243561
\(827\) −11.6754 −0.405995 −0.202997 0.979179i \(-0.565068\pi\)
−0.202997 + 0.979179i \(0.565068\pi\)
\(828\) 0 0
\(829\) −54.2719 −1.88494 −0.942470 0.334290i \(-0.891504\pi\)
−0.942470 + 0.334290i \(0.891504\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.16228 −0.0749635
\(833\) 5.00000 0.173240
\(834\) 0 0
\(835\) 0 0
\(836\) 0.973666 0.0336749
\(837\) 0 0
\(838\) −28.6228 −0.988757
\(839\) 23.4868 0.810856 0.405428 0.914127i \(-0.367123\pi\)
0.405428 + 0.914127i \(0.367123\pi\)
\(840\) 0 0
\(841\) −24.3246 −0.838778
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) −3.00000 −0.103264
\(845\) 0 0
\(846\) 0 0
\(847\) 9.64911 0.331547
\(848\) −12.4868 −0.428800
\(849\) 0 0
\(850\) 0 0
\(851\) −1.48683 −0.0509680
\(852\) 0 0
\(853\) −0.162278 −0.00555628 −0.00277814 0.999996i \(-0.500884\pi\)
−0.00277814 + 0.999996i \(0.500884\pi\)
\(854\) −8.32456 −0.284860
\(855\) 0 0
\(856\) −5.16228 −0.176443
\(857\) −25.3246 −0.865070 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(858\) 0 0
\(859\) −30.7851 −1.05037 −0.525186 0.850987i \(-0.676004\pi\)
−0.525186 + 0.850987i \(0.676004\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 19.2982 0.657300
\(863\) −8.16228 −0.277847 −0.138924 0.990303i \(-0.544364\pi\)
−0.138924 + 0.990303i \(0.544364\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13.2982 0.451892
\(867\) 0 0
\(868\) 8.48683 0.288062
\(869\) −7.94733 −0.269595
\(870\) 0 0
\(871\) −33.1359 −1.12277
\(872\) −3.16228 −0.107088
\(873\) 0 0
\(874\) −0.135944 −0.00459836
\(875\) 0 0
\(876\) 0 0
\(877\) −10.9737 −0.370554 −0.185277 0.982686i \(-0.559318\pi\)
−0.185277 + 0.982686i \(0.559318\pi\)
\(878\) −10.8114 −0.364867
\(879\) 0 0
\(880\) 0 0
\(881\) −34.9473 −1.17741 −0.588703 0.808350i \(-0.700361\pi\)
−0.588703 + 0.808350i \(0.700361\pi\)
\(882\) 0 0
\(883\) 56.2982 1.89459 0.947293 0.320369i \(-0.103807\pi\)
0.947293 + 0.320369i \(0.103807\pi\)
\(884\) −10.8114 −0.363626
\(885\) 0 0
\(886\) 9.48683 0.318716
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −5.67544 −0.190348
\(890\) 0 0
\(891\) 0 0
\(892\) 24.6491 0.825313
\(893\) 1.67544 0.0560666
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −29.2982 −0.977694
\(899\) 18.3509 0.612036
\(900\) 0 0
\(901\) −62.4342 −2.07998
\(902\) 2.70178 0.0899594
\(903\) 0 0
\(904\) 6.83772 0.227419
\(905\) 0 0
\(906\) 0 0
\(907\) −29.2982 −0.972831 −0.486416 0.873727i \(-0.661696\pi\)
−0.486416 + 0.873727i \(0.661696\pi\)
\(908\) 0.675445 0.0224154
\(909\) 0 0
\(910\) 0 0
\(911\) 41.2982 1.36827 0.684136 0.729355i \(-0.260180\pi\)
0.684136 + 0.729355i \(0.260180\pi\)
\(912\) 0 0
\(913\) 9.67544 0.320210
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) 18.3246 0.605460
\(917\) −7.32456 −0.241878
\(918\) 0 0
\(919\) −14.6491 −0.483230 −0.241615 0.970372i \(-0.577677\pi\)
−0.241615 + 0.970372i \(0.577677\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 35.2982 1.16249
\(923\) −14.0263 −0.461682
\(924\) 0 0
\(925\) 0 0
\(926\) 17.1623 0.563987
\(927\) 0 0
\(928\) 2.16228 0.0709802
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) −0.837722 −0.0274552
\(932\) 28.1359 0.921623
\(933\) 0 0
\(934\) 14.6491 0.479334
\(935\) 0 0
\(936\) 0 0
\(937\) 35.4868 1.15930 0.579652 0.814864i \(-0.303188\pi\)
0.579652 + 0.814864i \(0.303188\pi\)
\(938\) 15.3246 0.500364
\(939\) 0 0
\(940\) 0 0
\(941\) 42.6491 1.39032 0.695161 0.718854i \(-0.255333\pi\)
0.695161 + 0.718854i \(0.255333\pi\)
\(942\) 0 0
\(943\) −0.377223 −0.0122841
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) 8.13594 0.264522
\(947\) 33.4868 1.08818 0.544088 0.839028i \(-0.316876\pi\)
0.544088 + 0.839028i \(0.316876\pi\)
\(948\) 0 0
\(949\) −21.6228 −0.701905
\(950\) 0 0
\(951\) 0 0
\(952\) 5.00000 0.162051
\(953\) −35.6228 −1.15393 −0.576967 0.816767i \(-0.695764\pi\)
−0.576967 + 0.816767i \(0.695764\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.32456 −0.204551
\(957\) 0 0
\(958\) 28.6491 0.925611
\(959\) −4.32456 −0.139647
\(960\) 0 0
\(961\) 41.0263 1.32343
\(962\) 19.8114 0.638745
\(963\) 0 0
\(964\) −7.16228 −0.230681
\(965\) 0 0
\(966\) 0 0
\(967\) −19.6228 −0.631026 −0.315513 0.948921i \(-0.602177\pi\)
−0.315513 + 0.948921i \(0.602177\pi\)
\(968\) 9.64911 0.310134
\(969\) 0 0
\(970\) 0 0
\(971\) 34.9473 1.12151 0.560757 0.827981i \(-0.310510\pi\)
0.560757 + 0.827981i \(0.310510\pi\)
\(972\) 0 0
\(973\) −1.48683 −0.0476657
\(974\) 14.6491 0.469388
\(975\) 0 0
\(976\) −8.32456 −0.266463
\(977\) 43.9473 1.40600 0.703000 0.711190i \(-0.251843\pi\)
0.703000 + 0.711190i \(0.251843\pi\)
\(978\) 0 0
\(979\) 17.8114 0.569254
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −42.3246 −1.34994 −0.674972 0.737843i \(-0.735844\pi\)
−0.674972 + 0.737843i \(0.735844\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 10.8114 0.344305
\(987\) 0 0
\(988\) 1.81139 0.0576279
\(989\) −1.13594 −0.0361209
\(990\) 0 0
\(991\) −15.9473 −0.506584 −0.253292 0.967390i \(-0.581513\pi\)
−0.253292 + 0.967390i \(0.581513\pi\)
\(992\) 8.48683 0.269457
\(993\) 0 0
\(994\) 6.48683 0.205750
\(995\) 0 0
\(996\) 0 0
\(997\) 9.83772 0.311564 0.155782 0.987791i \(-0.450210\pi\)
0.155782 + 0.987791i \(0.450210\pi\)
\(998\) 28.3246 0.896598
\(999\) 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 9450.2.a.ef.1.1 2
3.2 odd 2 9450.2.a.en.1.2 2
5.2 odd 4 1890.2.g.p.379.2 yes 4
5.3 odd 4 1890.2.g.p.379.4 yes 4
5.4 even 2 9450.2.a.ew.1.1 2
15.2 even 4 1890.2.g.o.379.3 yes 4
15.8 even 4 1890.2.g.o.379.1 4
15.14 odd 2 9450.2.a.eg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1890.2.g.o.379.1 4 15.8 even 4
1890.2.g.o.379.3 yes 4 15.2 even 4
1890.2.g.p.379.2 yes 4 5.2 odd 4
1890.2.g.p.379.4 yes 4 5.3 odd 4
9450.2.a.ef.1.1 2 1.1 even 1 trivial
9450.2.a.eg.1.2 2 15.14 odd 2
9450.2.a.en.1.2 2 3.2 odd 2
9450.2.a.ew.1.1 2 5.4 even 2