Properties

Label 1950.2.f.p.649.4
Level $1950$
Weight $2$
Character 1950.649
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
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] = [1950,2,Mod(649,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1950.649
Dual form 1950.2.f.p.649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +4.56155 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +4.56155 q^{7} +1.00000 q^{8} -1.00000 q^{9} +2.56155i q^{11} +1.00000i q^{12} +(-3.56155 + 0.561553i) q^{13} +4.56155 q^{14} +1.00000 q^{16} +5.68466i q^{17} -1.00000 q^{18} +4.68466i q^{19} +4.56155i q^{21} +2.56155i q^{22} -5.12311i q^{23} +1.00000i q^{24} +(-3.56155 + 0.561553i) q^{26} -1.00000i q^{27} +4.56155 q^{28} +1.00000 q^{29} -1.43845i q^{31} +1.00000 q^{32} -2.56155 q^{33} +5.68466i q^{34} -1.00000 q^{36} +0.438447 q^{37} +4.68466i q^{38} +(-0.561553 - 3.56155i) q^{39} +0.438447i q^{41} +4.56155i q^{42} +2.87689i q^{43} +2.56155i q^{44} -5.12311i q^{46} -0.123106 q^{47} +1.00000i q^{48} +13.8078 q^{49} -5.68466 q^{51} +(-3.56155 + 0.561553i) q^{52} +5.00000i q^{53} -1.00000i q^{54} +4.56155 q^{56} -4.68466 q^{57} +1.00000 q^{58} -5.43845i q^{59} +11.6847 q^{61} -1.43845i q^{62} -4.56155 q^{63} +1.00000 q^{64} -2.56155 q^{66} +4.12311 q^{67} +5.68466i q^{68} +5.12311 q^{69} -14.9309i q^{71} -1.00000 q^{72} -1.12311 q^{73} +0.438447 q^{74} +4.68466i q^{76} +11.6847i q^{77} +(-0.561553 - 3.56155i) q^{78} +14.6847 q^{79} +1.00000 q^{81} +0.438447i q^{82} -13.9309 q^{83} +4.56155i q^{84} +2.87689i q^{86} +1.00000i q^{87} +2.56155i q^{88} +3.12311i q^{89} +(-16.2462 + 2.56155i) q^{91} -5.12311i q^{92} +1.43845 q^{93} -0.123106 q^{94} +1.00000i q^{96} -4.87689 q^{97} +13.8078 q^{98} -2.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 10 q^{7} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 10 q^{7} + 4 q^{8} - 4 q^{9} - 6 q^{13} + 10 q^{14} + 4 q^{16} - 4 q^{18} - 6 q^{26} + 10 q^{28} + 4 q^{29} + 4 q^{32} - 2 q^{33} - 4 q^{36} + 10 q^{37} + 6 q^{39} + 16 q^{47} + 14 q^{49} + 2 q^{51} - 6 q^{52} + 10 q^{56} + 6 q^{57} + 4 q^{58} + 22 q^{61} - 10 q^{63} + 4 q^{64} - 2 q^{66} + 4 q^{69} - 4 q^{72} + 12 q^{73} + 10 q^{74} + 6 q^{78} + 34 q^{79} + 4 q^{81} + 2 q^{83} - 32 q^{91} + 14 q^{93} + 16 q^{94} - 36 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000i 0.408248i
\(7\) 4.56155 1.72410 0.862052 0.506819i \(-0.169179\pi\)
0.862052 + 0.506819i \(0.169179\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.56155i 0.772337i 0.922428 + 0.386169i \(0.126202\pi\)
−0.922428 + 0.386169i \(0.873798\pi\)
\(12\) 1.00000i 0.288675i
\(13\) −3.56155 + 0.561553i −0.987797 + 0.155747i
\(14\) 4.56155 1.21913
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.68466i 1.37873i 0.724413 + 0.689366i \(0.242111\pi\)
−0.724413 + 0.689366i \(0.757889\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.68466i 1.07473i 0.843348 + 0.537367i \(0.180581\pi\)
−0.843348 + 0.537367i \(0.819419\pi\)
\(20\) 0 0
\(21\) 4.56155i 0.995412i
\(22\) 2.56155i 0.546125i
\(23\) 5.12311i 1.06824i −0.845408 0.534121i \(-0.820643\pi\)
0.845408 0.534121i \(-0.179357\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 0 0
\(26\) −3.56155 + 0.561553i −0.698478 + 0.110130i
\(27\) 1.00000i 0.192450i
\(28\) 4.56155 0.862052
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 1.43845i 0.258353i −0.991622 0.129176i \(-0.958767\pi\)
0.991622 0.129176i \(-0.0412333\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.56155 −0.445909
\(34\) 5.68466i 0.974911i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 0.438447 0.0720803 0.0360401 0.999350i \(-0.488526\pi\)
0.0360401 + 0.999350i \(0.488526\pi\)
\(38\) 4.68466i 0.759952i
\(39\) −0.561553 3.56155i −0.0899204 0.570305i
\(40\) 0 0
\(41\) 0.438447i 0.0684739i 0.999414 + 0.0342370i \(0.0109001\pi\)
−0.999414 + 0.0342370i \(0.989100\pi\)
\(42\) 4.56155i 0.703863i
\(43\) 2.87689i 0.438722i 0.975644 + 0.219361i \(0.0703973\pi\)
−0.975644 + 0.219361i \(0.929603\pi\)
\(44\) 2.56155i 0.386169i
\(45\) 0 0
\(46\) 5.12311i 0.755361i
\(47\) −0.123106 −0.0179568 −0.00897840 0.999960i \(-0.502858\pi\)
−0.00897840 + 0.999960i \(0.502858\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 13.8078 1.97254
\(50\) 0 0
\(51\) −5.68466 −0.796011
\(52\) −3.56155 + 0.561553i −0.493899 + 0.0778734i
\(53\) 5.00000i 0.686803i 0.939189 + 0.343401i \(0.111579\pi\)
−0.939189 + 0.343401i \(0.888421\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) 4.56155 0.609563
\(57\) −4.68466 −0.620498
\(58\) 1.00000 0.131306
\(59\) 5.43845i 0.708026i −0.935241 0.354013i \(-0.884817\pi\)
0.935241 0.354013i \(-0.115183\pi\)
\(60\) 0 0
\(61\) 11.6847 1.49607 0.748034 0.663661i \(-0.230998\pi\)
0.748034 + 0.663661i \(0.230998\pi\)
\(62\) 1.43845i 0.182683i
\(63\) −4.56155 −0.574702
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.56155 −0.315305
\(67\) 4.12311 0.503718 0.251859 0.967764i \(-0.418958\pi\)
0.251859 + 0.967764i \(0.418958\pi\)
\(68\) 5.68466i 0.689366i
\(69\) 5.12311 0.616749
\(70\) 0 0
\(71\) 14.9309i 1.77197i −0.463716 0.885984i \(-0.653484\pi\)
0.463716 0.885984i \(-0.346516\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.12311 −0.131450 −0.0657248 0.997838i \(-0.520936\pi\)
−0.0657248 + 0.997838i \(0.520936\pi\)
\(74\) 0.438447 0.0509685
\(75\) 0 0
\(76\) 4.68466i 0.537367i
\(77\) 11.6847i 1.33159i
\(78\) −0.561553 3.56155i −0.0635833 0.403266i
\(79\) 14.6847 1.65215 0.826077 0.563558i \(-0.190568\pi\)
0.826077 + 0.563558i \(0.190568\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.438447i 0.0484184i
\(83\) −13.9309 −1.52911 −0.764556 0.644558i \(-0.777042\pi\)
−0.764556 + 0.644558i \(0.777042\pi\)
\(84\) 4.56155i 0.497706i
\(85\) 0 0
\(86\) 2.87689i 0.310223i
\(87\) 1.00000i 0.107211i
\(88\) 2.56155i 0.273062i
\(89\) 3.12311i 0.331049i 0.986206 + 0.165524i \(0.0529316\pi\)
−0.986206 + 0.165524i \(0.947068\pi\)
\(90\) 0 0
\(91\) −16.2462 + 2.56155i −1.70307 + 0.268524i
\(92\) 5.12311i 0.534121i
\(93\) 1.43845 0.149160
\(94\) −0.123106 −0.0126974
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) −4.87689 −0.495174 −0.247587 0.968866i \(-0.579638\pi\)
−0.247587 + 0.968866i \(0.579638\pi\)
\(98\) 13.8078 1.39479
\(99\) 2.56155i 0.257446i
\(100\) 0 0
\(101\) −16.5616 −1.64794 −0.823968 0.566636i \(-0.808244\pi\)
−0.823968 + 0.566636i \(0.808244\pi\)
\(102\) −5.68466 −0.562865
\(103\) 1.12311i 0.110663i −0.998468 0.0553314i \(-0.982378\pi\)
0.998468 0.0553314i \(-0.0176215\pi\)
\(104\) −3.56155 + 0.561553i −0.349239 + 0.0550648i
\(105\) 0 0
\(106\) 5.00000i 0.485643i
\(107\) 12.6847i 1.22627i 0.789977 + 0.613136i \(0.210092\pi\)
−0.789977 + 0.613136i \(0.789908\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 16.4384i 1.57452i −0.616623 0.787259i \(-0.711500\pi\)
0.616623 0.787259i \(-0.288500\pi\)
\(110\) 0 0
\(111\) 0.438447i 0.0416156i
\(112\) 4.56155 0.431026
\(113\) 12.2462i 1.15203i 0.817440 + 0.576013i \(0.195392\pi\)
−0.817440 + 0.576013i \(0.804608\pi\)
\(114\) −4.68466 −0.438758
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 3.56155 0.561553i 0.329266 0.0519156i
\(118\) 5.43845i 0.500650i
\(119\) 25.9309i 2.37708i
\(120\) 0 0
\(121\) 4.43845 0.403495
\(122\) 11.6847 1.05788
\(123\) −0.438447 −0.0395335
\(124\) 1.43845i 0.129176i
\(125\) 0 0
\(126\) −4.56155 −0.406375
\(127\) 15.5616i 1.38086i −0.723397 0.690432i \(-0.757420\pi\)
0.723397 0.690432i \(-0.242580\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.87689 −0.253296
\(130\) 0 0
\(131\) −18.6847 −1.63249 −0.816243 0.577709i \(-0.803947\pi\)
−0.816243 + 0.577709i \(0.803947\pi\)
\(132\) −2.56155 −0.222955
\(133\) 21.3693i 1.85295i
\(134\) 4.12311 0.356182
\(135\) 0 0
\(136\) 5.68466i 0.487455i
\(137\) 2.43845 0.208331 0.104165 0.994560i \(-0.466783\pi\)
0.104165 + 0.994560i \(0.466783\pi\)
\(138\) 5.12311 0.436108
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 0 0
\(141\) 0.123106i 0.0103674i
\(142\) 14.9309i 1.25297i
\(143\) −1.43845 9.12311i −0.120289 0.762912i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −1.12311 −0.0929489
\(147\) 13.8078i 1.13885i
\(148\) 0.438447 0.0360401
\(149\) 5.36932i 0.439872i −0.975514 0.219936i \(-0.929415\pi\)
0.975514 0.219936i \(-0.0705848\pi\)
\(150\) 0 0
\(151\) 15.6847i 1.27640i 0.769871 + 0.638200i \(0.220321\pi\)
−0.769871 + 0.638200i \(0.779679\pi\)
\(152\) 4.68466i 0.379976i
\(153\) 5.68466i 0.459577i
\(154\) 11.6847i 0.941577i
\(155\) 0 0
\(156\) −0.561553 3.56155i −0.0449602 0.285152i
\(157\) 1.68466i 0.134450i −0.997738 0.0672252i \(-0.978585\pi\)
0.997738 0.0672252i \(-0.0214146\pi\)
\(158\) 14.6847 1.16825
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 23.3693i 1.84176i
\(162\) 1.00000 0.0785674
\(163\) −16.4924 −1.29179 −0.645893 0.763428i \(-0.723515\pi\)
−0.645893 + 0.763428i \(0.723515\pi\)
\(164\) 0.438447i 0.0342370i
\(165\) 0 0
\(166\) −13.9309 −1.08125
\(167\) 17.5616 1.35895 0.679477 0.733697i \(-0.262207\pi\)
0.679477 + 0.733697i \(0.262207\pi\)
\(168\) 4.56155i 0.351931i
\(169\) 12.3693 4.00000i 0.951486 0.307692i
\(170\) 0 0
\(171\) 4.68466i 0.358245i
\(172\) 2.87689i 0.219361i
\(173\) 8.12311i 0.617588i 0.951129 + 0.308794i \(0.0999254\pi\)
−0.951129 + 0.308794i \(0.900075\pi\)
\(174\) 1.00000i 0.0758098i
\(175\) 0 0
\(176\) 2.56155i 0.193084i
\(177\) 5.43845 0.408779
\(178\) 3.12311i 0.234087i
\(179\) 10.8769 0.812977 0.406489 0.913656i \(-0.366753\pi\)
0.406489 + 0.913656i \(0.366753\pi\)
\(180\) 0 0
\(181\) 20.8078 1.54663 0.773314 0.634023i \(-0.218597\pi\)
0.773314 + 0.634023i \(0.218597\pi\)
\(182\) −16.2462 + 2.56155i −1.20425 + 0.189875i
\(183\) 11.6847i 0.863755i
\(184\) 5.12311i 0.377680i
\(185\) 0 0
\(186\) 1.43845 0.105472
\(187\) −14.5616 −1.06485
\(188\) −0.123106 −0.00897840
\(189\) 4.56155i 0.331804i
\(190\) 0 0
\(191\) 16.4924 1.19335 0.596675 0.802483i \(-0.296488\pi\)
0.596675 + 0.802483i \(0.296488\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) −13.1231 −0.944622 −0.472311 0.881432i \(-0.656580\pi\)
−0.472311 + 0.881432i \(0.656580\pi\)
\(194\) −4.87689 −0.350141
\(195\) 0 0
\(196\) 13.8078 0.986269
\(197\) −4.24621 −0.302530 −0.151265 0.988493i \(-0.548335\pi\)
−0.151265 + 0.988493i \(0.548335\pi\)
\(198\) 2.56155i 0.182042i
\(199\) −22.9309 −1.62553 −0.812763 0.582594i \(-0.802038\pi\)
−0.812763 + 0.582594i \(0.802038\pi\)
\(200\) 0 0
\(201\) 4.12311i 0.290821i
\(202\) −16.5616 −1.16527
\(203\) 4.56155 0.320158
\(204\) −5.68466 −0.398006
\(205\) 0 0
\(206\) 1.12311i 0.0782505i
\(207\) 5.12311i 0.356080i
\(208\) −3.56155 + 0.561553i −0.246949 + 0.0389367i
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 7.36932 0.507325 0.253662 0.967293i \(-0.418365\pi\)
0.253662 + 0.967293i \(0.418365\pi\)
\(212\) 5.00000i 0.343401i
\(213\) 14.9309 1.02305
\(214\) 12.6847i 0.867105i
\(215\) 0 0
\(216\) 1.00000i 0.0680414i
\(217\) 6.56155i 0.445427i
\(218\) 16.4384i 1.11335i
\(219\) 1.12311i 0.0758924i
\(220\) 0 0
\(221\) −3.19224 20.2462i −0.214733 1.36191i
\(222\) 0.438447i 0.0294266i
\(223\) −9.36932 −0.627416 −0.313708 0.949520i \(-0.601571\pi\)
−0.313708 + 0.949520i \(0.601571\pi\)
\(224\) 4.56155 0.304782
\(225\) 0 0
\(226\) 12.2462i 0.814606i
\(227\) −15.6847 −1.04103 −0.520514 0.853853i \(-0.674260\pi\)
−0.520514 + 0.853853i \(0.674260\pi\)
\(228\) −4.68466 −0.310249
\(229\) 27.8078i 1.83759i −0.394736 0.918794i \(-0.629164\pi\)
0.394736 0.918794i \(-0.370836\pi\)
\(230\) 0 0
\(231\) −11.6847 −0.768794
\(232\) 1.00000 0.0656532
\(233\) 5.12311i 0.335626i −0.985819 0.167813i \(-0.946330\pi\)
0.985819 0.167813i \(-0.0536704\pi\)
\(234\) 3.56155 0.561553i 0.232826 0.0367099i
\(235\) 0 0
\(236\) 5.43845i 0.354013i
\(237\) 14.6847i 0.953871i
\(238\) 25.9309i 1.68085i
\(239\) 21.9309i 1.41859i −0.704912 0.709295i \(-0.749013\pi\)
0.704912 0.709295i \(-0.250987\pi\)
\(240\) 0 0
\(241\) 3.75379i 0.241803i −0.992665 0.120901i \(-0.961422\pi\)
0.992665 0.120901i \(-0.0385785\pi\)
\(242\) 4.43845 0.285314
\(243\) 1.00000i 0.0641500i
\(244\) 11.6847 0.748034
\(245\) 0 0
\(246\) −0.438447 −0.0279544
\(247\) −2.63068 16.6847i −0.167386 1.06162i
\(248\) 1.43845i 0.0913415i
\(249\) 13.9309i 0.882833i
\(250\) 0 0
\(251\) 15.5616 0.982237 0.491118 0.871093i \(-0.336588\pi\)
0.491118 + 0.871093i \(0.336588\pi\)
\(252\) −4.56155 −0.287351
\(253\) 13.1231 0.825043
\(254\) 15.5616i 0.976419i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.80776i 0.299900i −0.988694 0.149950i \(-0.952089\pi\)
0.988694 0.149950i \(-0.0479113\pi\)
\(258\) −2.87689 −0.179108
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −18.6847 −1.15434
\(263\) 24.7386i 1.52545i −0.646723 0.762725i \(-0.723861\pi\)
0.646723 0.762725i \(-0.276139\pi\)
\(264\) −2.56155 −0.157653
\(265\) 0 0
\(266\) 21.3693i 1.31024i
\(267\) −3.12311 −0.191131
\(268\) 4.12311 0.251859
\(269\) −23.4924 −1.43236 −0.716179 0.697916i \(-0.754111\pi\)
−0.716179 + 0.697916i \(0.754111\pi\)
\(270\) 0 0
\(271\) 24.8078i 1.50696i −0.657468 0.753482i \(-0.728373\pi\)
0.657468 0.753482i \(-0.271627\pi\)
\(272\) 5.68466i 0.344683i
\(273\) −2.56155 16.2462i −0.155032 0.983265i
\(274\) 2.43845 0.147312
\(275\) 0 0
\(276\) 5.12311 0.308375
\(277\) 25.6155i 1.53909i −0.638594 0.769544i \(-0.720484\pi\)
0.638594 0.769544i \(-0.279516\pi\)
\(278\) 16.4924 0.989150
\(279\) 1.43845i 0.0861176i
\(280\) 0 0
\(281\) 22.9309i 1.36794i −0.729510 0.683970i \(-0.760252\pi\)
0.729510 0.683970i \(-0.239748\pi\)
\(282\) 0.123106i 0.00733083i
\(283\) 17.3693i 1.03250i 0.856438 + 0.516249i \(0.172672\pi\)
−0.856438 + 0.516249i \(0.827328\pi\)
\(284\) 14.9309i 0.885984i
\(285\) 0 0
\(286\) −1.43845 9.12311i −0.0850572 0.539461i
\(287\) 2.00000i 0.118056i
\(288\) −1.00000 −0.0589256
\(289\) −15.3153 −0.900902
\(290\) 0 0
\(291\) 4.87689i 0.285889i
\(292\) −1.12311 −0.0657248
\(293\) 25.3693 1.48209 0.741046 0.671455i \(-0.234330\pi\)
0.741046 + 0.671455i \(0.234330\pi\)
\(294\) 13.8078i 0.805285i
\(295\) 0 0
\(296\) 0.438447 0.0254842
\(297\) 2.56155 0.148636
\(298\) 5.36932i 0.311036i
\(299\) 2.87689 + 18.2462i 0.166375 + 1.05521i
\(300\) 0 0
\(301\) 13.1231i 0.756403i
\(302\) 15.6847i 0.902551i
\(303\) 16.5616i 0.951436i
\(304\) 4.68466i 0.268684i
\(305\) 0 0
\(306\) 5.68466i 0.324970i
\(307\) 30.9309 1.76532 0.882659 0.470014i \(-0.155751\pi\)
0.882659 + 0.470014i \(0.155751\pi\)
\(308\) 11.6847i 0.665795i
\(309\) 1.12311 0.0638912
\(310\) 0 0
\(311\) 23.1231 1.31119 0.655596 0.755112i \(-0.272418\pi\)
0.655596 + 0.755112i \(0.272418\pi\)
\(312\) −0.561553 3.56155i −0.0317917 0.201633i
\(313\) 32.8617i 1.85746i −0.370763 0.928728i \(-0.620904\pi\)
0.370763 0.928728i \(-0.379096\pi\)
\(314\) 1.68466i 0.0950708i
\(315\) 0 0
\(316\) 14.6847 0.826077
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −5.00000 −0.280386
\(319\) 2.56155i 0.143419i
\(320\) 0 0
\(321\) −12.6847 −0.707989
\(322\) 23.3693i 1.30232i
\(323\) −26.6307 −1.48177
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.4924 −0.913431
\(327\) 16.4384 0.909048
\(328\) 0.438447i 0.0242092i
\(329\) −0.561553 −0.0309594
\(330\) 0 0
\(331\) 8.49242i 0.466786i 0.972383 + 0.233393i \(0.0749828\pi\)
−0.972383 + 0.233393i \(0.925017\pi\)
\(332\) −13.9309 −0.764556
\(333\) −0.438447 −0.0240268
\(334\) 17.5616 0.960925
\(335\) 0 0
\(336\) 4.56155i 0.248853i
\(337\) 1.68466i 0.0917692i −0.998947 0.0458846i \(-0.985389\pi\)
0.998947 0.0458846i \(-0.0146107\pi\)
\(338\) 12.3693 4.00000i 0.672802 0.217571i
\(339\) −12.2462 −0.665123
\(340\) 0 0
\(341\) 3.68466 0.199535
\(342\) 4.68466i 0.253317i
\(343\) 31.0540 1.67676
\(344\) 2.87689i 0.155112i
\(345\) 0 0
\(346\) 8.12311i 0.436701i
\(347\) 14.4384i 0.775096i −0.921849 0.387548i \(-0.873322\pi\)
0.921849 0.387548i \(-0.126678\pi\)
\(348\) 1.00000i 0.0536056i
\(349\) 22.0000i 1.17763i 0.808267 + 0.588817i \(0.200406\pi\)
−0.808267 + 0.588817i \(0.799594\pi\)
\(350\) 0 0
\(351\) 0.561553 + 3.56155i 0.0299735 + 0.190102i
\(352\) 2.56155i 0.136531i
\(353\) −19.5616 −1.04116 −0.520578 0.853814i \(-0.674284\pi\)
−0.520578 + 0.853814i \(0.674284\pi\)
\(354\) 5.43845 0.289050
\(355\) 0 0
\(356\) 3.12311i 0.165524i
\(357\) −25.9309 −1.37241
\(358\) 10.8769 0.574862
\(359\) 12.1231i 0.639833i 0.947446 + 0.319917i \(0.103655\pi\)
−0.947446 + 0.319917i \(0.896345\pi\)
\(360\) 0 0
\(361\) −2.94602 −0.155054
\(362\) 20.8078 1.09363
\(363\) 4.43845i 0.232958i
\(364\) −16.2462 + 2.56155i −0.851533 + 0.134262i
\(365\) 0 0
\(366\) 11.6847i 0.610767i
\(367\) 7.31534i 0.381858i −0.981604 0.190929i \(-0.938850\pi\)
0.981604 0.190929i \(-0.0611500\pi\)
\(368\) 5.12311i 0.267060i
\(369\) 0.438447i 0.0228246i
\(370\) 0 0
\(371\) 22.8078i 1.18412i
\(372\) 1.43845 0.0745800
\(373\) 36.5616i 1.89309i −0.322578 0.946543i \(-0.604550\pi\)
0.322578 0.946543i \(-0.395450\pi\)
\(374\) −14.5616 −0.752960
\(375\) 0 0
\(376\) −0.123106 −0.00634869
\(377\) −3.56155 + 0.561553i −0.183429 + 0.0289214i
\(378\) 4.56155i 0.234621i
\(379\) 19.6847i 1.01113i 0.862788 + 0.505566i \(0.168716\pi\)
−0.862788 + 0.505566i \(0.831284\pi\)
\(380\) 0 0
\(381\) 15.5616 0.797243
\(382\) 16.4924 0.843826
\(383\) −14.9309 −0.762932 −0.381466 0.924383i \(-0.624581\pi\)
−0.381466 + 0.924383i \(0.624581\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) −13.1231 −0.667948
\(387\) 2.87689i 0.146241i
\(388\) −4.87689 −0.247587
\(389\) −13.8078 −0.700081 −0.350041 0.936734i \(-0.613832\pi\)
−0.350041 + 0.936734i \(0.613832\pi\)
\(390\) 0 0
\(391\) 29.1231 1.47282
\(392\) 13.8078 0.697397
\(393\) 18.6847i 0.942516i
\(394\) −4.24621 −0.213921
\(395\) 0 0
\(396\) 2.56155i 0.128723i
\(397\) 2.05398 0.103086 0.0515430 0.998671i \(-0.483586\pi\)
0.0515430 + 0.998671i \(0.483586\pi\)
\(398\) −22.9309 −1.14942
\(399\) −21.3693 −1.06980
\(400\) 0 0
\(401\) 22.0000i 1.09863i 0.835616 + 0.549314i \(0.185111\pi\)
−0.835616 + 0.549314i \(0.814889\pi\)
\(402\) 4.12311i 0.205642i
\(403\) 0.807764 + 5.12311i 0.0402376 + 0.255200i
\(404\) −16.5616 −0.823968
\(405\) 0 0
\(406\) 4.56155 0.226386
\(407\) 1.12311i 0.0556703i
\(408\) −5.68466 −0.281433
\(409\) 13.6155i 0.673245i −0.941640 0.336622i \(-0.890715\pi\)
0.941640 0.336622i \(-0.109285\pi\)
\(410\) 0 0
\(411\) 2.43845i 0.120280i
\(412\) 1.12311i 0.0553314i
\(413\) 24.8078i 1.22071i
\(414\) 5.12311i 0.251787i
\(415\) 0 0
\(416\) −3.56155 + 0.561553i −0.174619 + 0.0275324i
\(417\) 16.4924i 0.807637i
\(418\) −12.0000 −0.586939
\(419\) −6.68466 −0.326567 −0.163283 0.986579i \(-0.552209\pi\)
−0.163283 + 0.986579i \(0.552209\pi\)
\(420\) 0 0
\(421\) 29.3693i 1.43137i −0.698422 0.715686i \(-0.746114\pi\)
0.698422 0.715686i \(-0.253886\pi\)
\(422\) 7.36932 0.358733
\(423\) 0.123106 0.00598560
\(424\) 5.00000i 0.242821i
\(425\) 0 0
\(426\) 14.9309 0.723403
\(427\) 53.3002 2.57938
\(428\) 12.6847i 0.613136i
\(429\) 9.12311 1.43845i 0.440468 0.0694489i
\(430\) 0 0
\(431\) 37.1771i 1.79076i 0.445306 + 0.895378i \(0.353095\pi\)
−0.445306 + 0.895378i \(0.646905\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 32.0540i 1.54042i 0.637793 + 0.770208i \(0.279848\pi\)
−0.637793 + 0.770208i \(0.720152\pi\)
\(434\) 6.56155i 0.314965i
\(435\) 0 0
\(436\) 16.4384i 0.787259i
\(437\) 24.0000 1.14808
\(438\) 1.12311i 0.0536641i
\(439\) −26.3002 −1.25524 −0.627620 0.778520i \(-0.715971\pi\)
−0.627620 + 0.778520i \(0.715971\pi\)
\(440\) 0 0
\(441\) −13.8078 −0.657513
\(442\) −3.19224 20.2462i −0.151839 0.963014i
\(443\) 24.4384i 1.16111i 0.814223 + 0.580553i \(0.197164\pi\)
−0.814223 + 0.580553i \(0.802836\pi\)
\(444\) 0.438447i 0.0208078i
\(445\) 0 0
\(446\) −9.36932 −0.443650
\(447\) 5.36932 0.253960
\(448\) 4.56155 0.215513
\(449\) 9.56155i 0.451238i −0.974216 0.225619i \(-0.927560\pi\)
0.974216 0.225619i \(-0.0724404\pi\)
\(450\) 0 0
\(451\) −1.12311 −0.0528850
\(452\) 12.2462i 0.576013i
\(453\) −15.6847 −0.736930
\(454\) −15.6847 −0.736117
\(455\) 0 0
\(456\) −4.68466 −0.219379
\(457\) 20.7386 0.970112 0.485056 0.874483i \(-0.338799\pi\)
0.485056 + 0.874483i \(0.338799\pi\)
\(458\) 27.8078i 1.29937i
\(459\) 5.68466 0.265337
\(460\) 0 0
\(461\) 5.75379i 0.267981i −0.990983 0.133990i \(-0.957221\pi\)
0.990983 0.133990i \(-0.0427791\pi\)
\(462\) −11.6847 −0.543620
\(463\) −7.43845 −0.345694 −0.172847 0.984949i \(-0.555297\pi\)
−0.172847 + 0.984949i \(0.555297\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 5.12311i 0.237323i
\(467\) 20.1922i 0.934385i 0.884156 + 0.467193i \(0.154735\pi\)
−0.884156 + 0.467193i \(0.845265\pi\)
\(468\) 3.56155 0.561553i 0.164633 0.0259578i
\(469\) 18.8078 0.868462
\(470\) 0 0
\(471\) 1.68466 0.0776250
\(472\) 5.43845i 0.250325i
\(473\) −7.36932 −0.338842
\(474\) 14.6847i 0.674489i
\(475\) 0 0
\(476\) 25.9309i 1.18854i
\(477\) 5.00000i 0.228934i
\(478\) 21.9309i 1.00309i
\(479\) 23.8769i 1.09096i 0.838123 + 0.545482i \(0.183653\pi\)
−0.838123 + 0.545482i \(0.816347\pi\)
\(480\) 0 0
\(481\) −1.56155 + 0.246211i −0.0712007 + 0.0112263i
\(482\) 3.75379i 0.170980i
\(483\) 23.3693 1.06334
\(484\) 4.43845 0.201748
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) 5.68466 0.257596 0.128798 0.991671i \(-0.458888\pi\)
0.128798 + 0.991671i \(0.458888\pi\)
\(488\) 11.6847 0.528940
\(489\) 16.4924i 0.745813i
\(490\) 0 0
\(491\) −10.7386 −0.484628 −0.242314 0.970198i \(-0.577906\pi\)
−0.242314 + 0.970198i \(0.577906\pi\)
\(492\) −0.438447 −0.0197667
\(493\) 5.68466i 0.256024i
\(494\) −2.63068 16.6847i −0.118360 0.750678i
\(495\) 0 0
\(496\) 1.43845i 0.0645882i
\(497\) 68.1080i 3.05506i
\(498\) 13.9309i 0.624257i
\(499\) 19.8769i 0.889812i −0.895577 0.444906i \(-0.853237\pi\)
0.895577 0.444906i \(-0.146763\pi\)
\(500\) 0 0
\(501\) 17.5616i 0.784592i
\(502\) 15.5616 0.694546
\(503\) 40.7386i 1.81645i −0.418487 0.908223i \(-0.637439\pi\)
0.418487 0.908223i \(-0.362561\pi\)
\(504\) −4.56155 −0.203188
\(505\) 0 0
\(506\) 13.1231 0.583393
\(507\) 4.00000 + 12.3693i 0.177646 + 0.549341i
\(508\) 15.5616i 0.690432i
\(509\) 10.4924i 0.465068i −0.972588 0.232534i \(-0.925298\pi\)
0.972588 0.232534i \(-0.0747018\pi\)
\(510\) 0 0
\(511\) −5.12311 −0.226633
\(512\) 1.00000 0.0441942
\(513\) 4.68466 0.206833
\(514\) 4.80776i 0.212061i
\(515\) 0 0
\(516\) −2.87689 −0.126648
\(517\) 0.315342i 0.0138687i
\(518\) 2.00000 0.0878750
\(519\) −8.12311 −0.356565
\(520\) 0 0
\(521\) 0.246211 0.0107867 0.00539336 0.999985i \(-0.498283\pi\)
0.00539336 + 0.999985i \(0.498283\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 25.3693i 1.10932i 0.832076 + 0.554661i \(0.187152\pi\)
−0.832076 + 0.554661i \(0.812848\pi\)
\(524\) −18.6847 −0.816243
\(525\) 0 0
\(526\) 24.7386i 1.07866i
\(527\) 8.17708 0.356199
\(528\) −2.56155 −0.111477
\(529\) −3.24621 −0.141140
\(530\) 0 0
\(531\) 5.43845i 0.236009i
\(532\) 21.3693i 0.926477i
\(533\) −0.246211 1.56155i −0.0106646 0.0676384i
\(534\) −3.12311 −0.135150
\(535\) 0 0
\(536\) 4.12311 0.178091
\(537\) 10.8769i 0.469373i
\(538\) −23.4924 −1.01283
\(539\) 35.3693i 1.52346i
\(540\) 0 0
\(541\) 15.7538i 0.677308i 0.940911 + 0.338654i \(0.109972\pi\)
−0.940911 + 0.338654i \(0.890028\pi\)
\(542\) 24.8078i 1.06558i
\(543\) 20.8078i 0.892947i
\(544\) 5.68466i 0.243728i
\(545\) 0 0
\(546\) −2.56155 16.2462i −0.109624 0.695274i
\(547\) 1.12311i 0.0480205i 0.999712 + 0.0240103i \(0.00764344\pi\)
−0.999712 + 0.0240103i \(0.992357\pi\)
\(548\) 2.43845 0.104165
\(549\) −11.6847 −0.498689
\(550\) 0 0
\(551\) 4.68466i 0.199573i
\(552\) 5.12311 0.218054
\(553\) 66.9848 2.84849
\(554\) 25.6155i 1.08830i
\(555\) 0 0
\(556\) 16.4924 0.699435
\(557\) −15.8617 −0.672083 −0.336042 0.941847i \(-0.609088\pi\)
−0.336042 + 0.941847i \(0.609088\pi\)
\(558\) 1.43845i 0.0608943i
\(559\) −1.61553 10.2462i −0.0683296 0.433369i
\(560\) 0 0
\(561\) 14.5616i 0.614789i
\(562\) 22.9309i 0.967280i
\(563\) 10.3002i 0.434101i −0.976160 0.217051i \(-0.930356\pi\)
0.976160 0.217051i \(-0.0696436\pi\)
\(564\) 0.123106i 0.00518368i
\(565\) 0 0
\(566\) 17.3693i 0.730087i
\(567\) 4.56155 0.191567
\(568\) 14.9309i 0.626485i
\(569\) 19.6847 0.825224 0.412612 0.910907i \(-0.364617\pi\)
0.412612 + 0.910907i \(0.364617\pi\)
\(570\) 0 0
\(571\) 18.8769 0.789973 0.394987 0.918687i \(-0.370749\pi\)
0.394987 + 0.918687i \(0.370749\pi\)
\(572\) −1.43845 9.12311i −0.0601445 0.381456i
\(573\) 16.4924i 0.688981i
\(574\) 2.00000i 0.0834784i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) −33.3693 −1.38918 −0.694591 0.719404i \(-0.744415\pi\)
−0.694591 + 0.719404i \(0.744415\pi\)
\(578\) −15.3153 −0.637034
\(579\) 13.1231i 0.545378i
\(580\) 0 0
\(581\) −63.5464 −2.63635
\(582\) 4.87689i 0.202154i
\(583\) −12.8078 −0.530443
\(584\) −1.12311 −0.0464744
\(585\) 0 0
\(586\) 25.3693 1.04800
\(587\) −9.93087 −0.409891 −0.204945 0.978773i \(-0.565702\pi\)
−0.204945 + 0.978773i \(0.565702\pi\)
\(588\) 13.8078i 0.569423i
\(589\) 6.73863 0.277661
\(590\) 0 0
\(591\) 4.24621i 0.174666i
\(592\) 0.438447 0.0180201
\(593\) −4.19224 −0.172154 −0.0860772 0.996288i \(-0.527433\pi\)
−0.0860772 + 0.996288i \(0.527433\pi\)
\(594\) 2.56155 0.105102
\(595\) 0 0
\(596\) 5.36932i 0.219936i
\(597\) 22.9309i 0.938498i
\(598\) 2.87689 + 18.2462i 0.117645 + 0.746143i
\(599\) −42.9848 −1.75631 −0.878157 0.478373i \(-0.841227\pi\)
−0.878157 + 0.478373i \(0.841227\pi\)
\(600\) 0 0
\(601\) −37.7386 −1.53939 −0.769695 0.638411i \(-0.779592\pi\)
−0.769695 + 0.638411i \(0.779592\pi\)
\(602\) 13.1231i 0.534858i
\(603\) −4.12311 −0.167906
\(604\) 15.6847i 0.638200i
\(605\) 0 0
\(606\) 16.5616i 0.672767i
\(607\) 9.94602i 0.403697i −0.979417 0.201848i \(-0.935305\pi\)
0.979417 0.201848i \(-0.0646948\pi\)
\(608\) 4.68466i 0.189988i
\(609\) 4.56155i 0.184843i
\(610\) 0 0
\(611\) 0.438447 0.0691303i 0.0177377 0.00279671i
\(612\) 5.68466i 0.229789i
\(613\) −14.6307 −0.590928 −0.295464 0.955354i \(-0.595474\pi\)
−0.295464 + 0.955354i \(0.595474\pi\)
\(614\) 30.9309 1.24827
\(615\) 0 0
\(616\) 11.6847i 0.470788i
\(617\) −15.3153 −0.616572 −0.308286 0.951294i \(-0.599755\pi\)
−0.308286 + 0.951294i \(0.599755\pi\)
\(618\) 1.12311 0.0451779
\(619\) 14.2462i 0.572604i 0.958139 + 0.286302i \(0.0924260\pi\)
−0.958139 + 0.286302i \(0.907574\pi\)
\(620\) 0 0
\(621\) −5.12311 −0.205583
\(622\) 23.1231 0.927152
\(623\) 14.2462i 0.570762i
\(624\) −0.561553 3.56155i −0.0224801 0.142576i
\(625\) 0 0
\(626\) 32.8617i 1.31342i
\(627\) 12.0000i 0.479234i
\(628\) 1.68466i 0.0672252i
\(629\) 2.49242i 0.0993794i
\(630\) 0 0
\(631\) 31.1231i 1.23899i −0.785000 0.619496i \(-0.787337\pi\)
0.785000 0.619496i \(-0.212663\pi\)
\(632\) 14.6847 0.584124
\(633\) 7.36932i 0.292904i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) −49.1771 + 7.75379i −1.94847 + 0.307216i
\(638\) 2.56155i 0.101413i
\(639\) 14.9309i 0.590656i
\(640\) 0 0
\(641\) 42.4233 1.67562 0.837810 0.545962i \(-0.183836\pi\)
0.837810 + 0.545962i \(0.183836\pi\)
\(642\) −12.6847 −0.500624
\(643\) 3.31534 0.130744 0.0653722 0.997861i \(-0.479177\pi\)
0.0653722 + 0.997861i \(0.479177\pi\)
\(644\) 23.3693i 0.920880i
\(645\) 0 0
\(646\) −26.6307 −1.04777
\(647\) 19.1231i 0.751807i 0.926659 + 0.375903i \(0.122668\pi\)
−0.926659 + 0.375903i \(0.877332\pi\)
\(648\) 1.00000 0.0392837
\(649\) 13.9309 0.546834
\(650\) 0 0
\(651\) 6.56155 0.257168
\(652\) −16.4924 −0.645893
\(653\) 7.30019i 0.285678i 0.989746 + 0.142839i \(0.0456232\pi\)
−0.989746 + 0.142839i \(0.954377\pi\)
\(654\) 16.4384 0.642794
\(655\) 0 0
\(656\) 0.438447i 0.0171185i
\(657\) 1.12311 0.0438165
\(658\) −0.561553 −0.0218916
\(659\) 8.43845 0.328715 0.164358 0.986401i \(-0.447445\pi\)
0.164358 + 0.986401i \(0.447445\pi\)
\(660\) 0 0
\(661\) 38.3002i 1.48970i 0.667229 + 0.744852i \(0.267480\pi\)
−0.667229 + 0.744852i \(0.732520\pi\)
\(662\) 8.49242i 0.330067i
\(663\) 20.2462 3.19224i 0.786298 0.123976i
\(664\) −13.9309 −0.540623
\(665\) 0 0
\(666\) −0.438447 −0.0169895
\(667\) 5.12311i 0.198367i
\(668\) 17.5616 0.679477
\(669\) 9.36932i 0.362239i
\(670\) 0 0
\(671\) 29.9309i 1.15547i
\(672\) 4.56155i 0.175966i
\(673\) 13.8769i 0.534915i 0.963570 + 0.267457i \(0.0861835\pi\)
−0.963570 + 0.267457i \(0.913817\pi\)
\(674\) 1.68466i 0.0648906i
\(675\) 0 0
\(676\) 12.3693 4.00000i 0.475743 0.153846i
\(677\) 19.7538i 0.759200i 0.925151 + 0.379600i \(0.123938\pi\)
−0.925151 + 0.379600i \(0.876062\pi\)
\(678\) −12.2462 −0.470313
\(679\) −22.2462 −0.853731
\(680\) 0 0
\(681\) 15.6847i 0.601037i
\(682\) 3.68466 0.141093
\(683\) −42.6695 −1.63270 −0.816352 0.577555i \(-0.804007\pi\)
−0.816352 + 0.577555i \(0.804007\pi\)
\(684\) 4.68466i 0.179122i
\(685\) 0 0
\(686\) 31.0540 1.18565
\(687\) 27.8078 1.06093
\(688\) 2.87689i 0.109681i
\(689\) −2.80776 17.8078i −0.106967 0.678422i
\(690\) 0 0
\(691\) 35.0000i 1.33146i 0.746191 + 0.665731i \(0.231880\pi\)
−0.746191 + 0.665731i \(0.768120\pi\)
\(692\) 8.12311i 0.308794i
\(693\) 11.6847i 0.443863i
\(694\) 14.4384i 0.548076i
\(695\) 0 0
\(696\) 1.00000i 0.0379049i
\(697\) −2.49242 −0.0944072
\(698\) 22.0000i 0.832712i
\(699\) 5.12311 0.193774
\(700\) 0 0
\(701\) 14.3153 0.540683 0.270341 0.962764i \(-0.412863\pi\)
0.270341 + 0.962764i \(0.412863\pi\)
\(702\) 0.561553 + 3.56155i 0.0211944 + 0.134422i
\(703\) 2.05398i 0.0774671i
\(704\) 2.56155i 0.0965422i
\(705\) 0 0
\(706\) −19.5616 −0.736209
\(707\) −75.5464 −2.84121
\(708\) 5.43845 0.204389
\(709\) 26.4924i 0.994944i 0.867480 + 0.497472i \(0.165738\pi\)
−0.867480 + 0.497472i \(0.834262\pi\)
\(710\) 0 0
\(711\) −14.6847 −0.550718
\(712\) 3.12311i 0.117043i
\(713\) −7.36932 −0.275983
\(714\) −25.9309 −0.970438
\(715\) 0 0
\(716\) 10.8769 0.406489
\(717\) 21.9309 0.819023
\(718\) 12.1231i 0.452430i
\(719\) −0.630683 −0.0235205 −0.0117603 0.999931i \(-0.503743\pi\)
−0.0117603 + 0.999931i \(0.503743\pi\)
\(720\) 0 0
\(721\) 5.12311i 0.190794i
\(722\) −2.94602 −0.109640
\(723\) 3.75379 0.139605
\(724\) 20.8078 0.773314
\(725\) 0 0
\(726\) 4.43845i 0.164726i
\(727\) 41.6155i 1.54343i 0.635966 + 0.771717i \(0.280602\pi\)
−0.635966 + 0.771717i \(0.719398\pi\)
\(728\) −16.2462 + 2.56155i −0.602125 + 0.0949375i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −16.3542 −0.604881
\(732\) 11.6847i 0.431877i
\(733\) 27.3153 1.00892 0.504458 0.863436i \(-0.331692\pi\)
0.504458 + 0.863436i \(0.331692\pi\)
\(734\) 7.31534i 0.270014i
\(735\) 0 0
\(736\) 5.12311i 0.188840i
\(737\) 10.5616i 0.389040i
\(738\) 0.438447i 0.0161395i
\(739\) 14.5076i 0.533670i 0.963742 + 0.266835i \(0.0859779\pi\)
−0.963742 + 0.266835i \(0.914022\pi\)
\(740\) 0 0
\(741\) 16.6847 2.63068i 0.612926 0.0966406i
\(742\) 22.8078i 0.837299i
\(743\) 1.63068 0.0598240 0.0299120 0.999553i \(-0.490477\pi\)
0.0299120 + 0.999553i \(0.490477\pi\)
\(744\) 1.43845 0.0527360
\(745\) 0 0
\(746\) 36.5616i 1.33861i
\(747\) 13.9309 0.509704
\(748\) −14.5616 −0.532423
\(749\) 57.8617i 2.11422i
\(750\) 0 0
\(751\) 5.80776 0.211928 0.105964 0.994370i \(-0.466207\pi\)
0.105964 + 0.994370i \(0.466207\pi\)
\(752\) −0.123106 −0.00448920
\(753\) 15.5616i 0.567095i
\(754\) −3.56155 + 0.561553i −0.129704 + 0.0204505i
\(755\) 0 0
\(756\) 4.56155i 0.165902i
\(757\) 25.9309i 0.942473i −0.882007 0.471237i \(-0.843808\pi\)
0.882007 0.471237i \(-0.156192\pi\)
\(758\) 19.6847i 0.714979i
\(759\) 13.1231i 0.476339i
\(760\) 0 0
\(761\) 35.6695i 1.29302i −0.762906 0.646509i \(-0.776228\pi\)
0.762906 0.646509i \(-0.223772\pi\)
\(762\) 15.5616 0.563736
\(763\) 74.9848i 2.71463i
\(764\) 16.4924 0.596675
\(765\) 0 0
\(766\) −14.9309 −0.539474
\(767\) 3.05398 + 19.3693i 0.110273 + 0.699385i
\(768\) 1.00000i 0.0360844i
\(769\) 14.2462i 0.513732i −0.966447 0.256866i \(-0.917310\pi\)
0.966447 0.256866i \(-0.0826898\pi\)
\(770\) 0 0
\(771\) 4.80776 0.173147
\(772\) −13.1231 −0.472311
\(773\) 51.2311 1.84265 0.921327 0.388790i \(-0.127107\pi\)
0.921327 + 0.388790i \(0.127107\pi\)
\(774\) 2.87689i 0.103408i
\(775\) 0 0
\(776\) −4.87689 −0.175070
\(777\) 2.00000i 0.0717496i
\(778\) −13.8078 −0.495032
\(779\) −2.05398 −0.0735913
\(780\) 0 0
\(781\) 38.2462 1.36856
\(782\) 29.1231 1.04144
\(783\) 1.00000i 0.0357371i
\(784\) 13.8078 0.493134
\(785\) 0 0
\(786\) 18.6847i 0.666460i
\(787\) 2.06913 0.0737565 0.0368783 0.999320i \(-0.488259\pi\)
0.0368783 + 0.999320i \(0.488259\pi\)
\(788\) −4.24621 −0.151265
\(789\) 24.7386 0.880719
\(790\) 0 0
\(791\) 55.8617i 1.98621i
\(792\) 2.56155i 0.0910208i
\(793\) −41.6155 + 6.56155i −1.47781 + 0.233008i
\(794\) 2.05398 0.0728929
\(795\) 0 0
\(796\) −22.9309 −0.812763
\(797\) 21.5464i 0.763213i −0.924325 0.381606i \(-0.875371\pi\)
0.924325 0.381606i \(-0.124629\pi\)
\(798\) −21.3693 −0.756466
\(799\) 0.699813i 0.0247576i
\(800\) 0 0
\(801\) 3.12311i 0.110350i
\(802\) 22.0000i 0.776847i
\(803\) 2.87689i 0.101523i
\(804\) 4.12311i 0.145411i
\(805\) 0 0
\(806\) 0.807764 + 5.12311i 0.0284523 + 0.180454i
\(807\) 23.4924i 0.826972i
\(808\) −16.5616 −0.582633
\(809\) −1.12311 −0.0394863 −0.0197431 0.999805i \(-0.506285\pi\)
−0.0197431 + 0.999805i \(0.506285\pi\)
\(810\) 0 0
\(811\) 12.1771i 0.427595i 0.976878 + 0.213798i \(0.0685833\pi\)
−0.976878 + 0.213798i \(0.931417\pi\)
\(812\) 4.56155 0.160079
\(813\) 24.8078 0.870046
\(814\) 1.12311i 0.0393648i
\(815\) 0 0
\(816\) −5.68466 −0.199003
\(817\) −13.4773 −0.471510
\(818\) 13.6155i 0.476056i
\(819\) 16.2462 2.56155i 0.567689 0.0895079i
\(820\) 0 0
\(821\) 29.6155i 1.03359i −0.856109 0.516795i \(-0.827125\pi\)
0.856109 0.516795i \(-0.172875\pi\)
\(822\) 2.43845i 0.0850506i
\(823\) 8.19224i 0.285563i −0.989754 0.142782i \(-0.954395\pi\)
0.989754 0.142782i \(-0.0456047\pi\)
\(824\) 1.12311i 0.0391252i
\(825\) 0 0
\(826\) 24.8078i 0.863173i
\(827\) 53.6847 1.86680 0.933399 0.358840i \(-0.116827\pi\)
0.933399 + 0.358840i \(0.116827\pi\)
\(828\) 5.12311i 0.178040i
\(829\) 1.19224 0.0414081 0.0207040 0.999786i \(-0.493409\pi\)
0.0207040 + 0.999786i \(0.493409\pi\)
\(830\) 0 0
\(831\) 25.6155 0.888593
\(832\) −3.56155 + 0.561553i −0.123475 + 0.0194683i
\(833\) 78.4924i 2.71960i
\(834\) 16.4924i 0.571086i
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) −1.43845 −0.0497200
\(838\) −6.68466 −0.230918
\(839\) 40.0000i 1.38095i −0.723355 0.690477i \(-0.757401\pi\)
0.723355 0.690477i \(-0.242599\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 29.3693i 1.01213i
\(843\) 22.9309 0.789781
\(844\) 7.36932 0.253662
\(845\) 0 0
\(846\) 0.123106 0.00423246
\(847\) 20.2462 0.695668
\(848\) 5.00000i 0.171701i
\(849\) −17.3693 −0.596113
\(850\) 0 0
\(851\) 2.24621i 0.0769991i
\(852\) 14.9309 0.511523
\(853\) −8.19224 −0.280497 −0.140248 0.990116i \(-0.544790\pi\)
−0.140248 + 0.990116i \(0.544790\pi\)
\(854\) 53.3002 1.82389
\(855\) 0 0
\(856\) 12.6847i 0.433553i
\(857\) 9.61553i 0.328460i −0.986422 0.164230i \(-0.947486\pi\)
0.986422 0.164230i \(-0.0525140\pi\)
\(858\) 9.12311 1.43845i 0.311458 0.0491078i
\(859\) 36.7386 1.25351 0.626753 0.779218i \(-0.284384\pi\)
0.626753 + 0.779218i \(0.284384\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 37.1771i 1.26626i
\(863\) 12.8617 0.437819 0.218909 0.975745i \(-0.429750\pi\)
0.218909 + 0.975745i \(0.429750\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 0 0
\(866\) 32.0540i 1.08924i
\(867\) 15.3153i 0.520136i
\(868\) 6.56155i 0.222714i
\(869\) 37.6155i 1.27602i
\(870\) 0 0
\(871\) −14.6847 + 2.31534i −0.497571 + 0.0784524i
\(872\) 16.4384i 0.556676i
\(873\) 4.87689 0.165058
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 1.12311i 0.0379462i
\(877\) −13.1771 −0.444958 −0.222479 0.974937i \(-0.571415\pi\)
−0.222479 + 0.974937i \(0.571415\pi\)
\(878\) −26.3002 −0.887588
\(879\) 25.3693i 0.855686i
\(880\) 0 0
\(881\) 52.0388 1.75323 0.876616 0.481190i \(-0.159795\pi\)
0.876616 + 0.481190i \(0.159795\pi\)
\(882\) −13.8078 −0.464932
\(883\) 23.6155i 0.794726i 0.917662 + 0.397363i \(0.130075\pi\)
−0.917662 + 0.397363i \(0.869925\pi\)
\(884\) −3.19224 20.2462i −0.107367 0.680954i
\(885\) 0 0
\(886\) 24.4384i 0.821026i
\(887\) 1.61553i 0.0542441i 0.999632 + 0.0271221i \(0.00863428\pi\)
−0.999632 + 0.0271221i \(0.991366\pi\)
\(888\) 0.438447i 0.0147133i
\(889\) 70.9848i 2.38076i
\(890\) 0 0
\(891\) 2.56155i 0.0858152i
\(892\) −9.36932 −0.313708
\(893\) 0.576708i 0.0192988i
\(894\) 5.36932 0.179577
\(895\) 0 0
\(896\) 4.56155 0.152391
\(897\) −18.2462 + 2.87689i −0.609223 + 0.0960567i
\(898\) 9.56155i 0.319073i
\(899\) 1.43845i 0.0479749i
\(900\) 0 0
\(901\) −28.4233 −0.946917
\(902\) −1.12311 −0.0373953
\(903\) −13.1231 −0.436710
\(904\) 12.2462i 0.407303i
\(905\) 0 0
\(906\) −15.6847 −0.521088
\(907\) 55.6155i 1.84668i 0.383979 + 0.923342i \(0.374553\pi\)
−0.383979 + 0.923342i \(0.625447\pi\)
\(908\) −15.6847 −0.520514
\(909\) 16.5616 0.549312
\(910\) 0 0
\(911\) −47.3693 −1.56942 −0.784708 0.619866i \(-0.787187\pi\)
−0.784708 + 0.619866i \(0.787187\pi\)
\(912\) −4.68466 −0.155125
\(913\) 35.6847i 1.18099i
\(914\) 20.7386 0.685973
\(915\) 0 0
\(916\) 27.8078i 0.918794i
\(917\) −85.2311 −2.81458
\(918\) 5.68466 0.187622
\(919\) 14.4384 0.476280 0.238140 0.971231i \(-0.423462\pi\)
0.238140 + 0.971231i \(0.423462\pi\)
\(920\) 0 0
\(921\) 30.9309i 1.01921i
\(922\) 5.75379i 0.189491i
\(923\) 8.38447 + 53.1771i 0.275978 + 1.75034i
\(924\) −11.6847 −0.384397
\(925\) 0 0
\(926\) −7.43845 −0.244443
\(927\) 1.12311i 0.0368876i
\(928\) 1.00000 0.0328266
\(929\) 11.4233i 0.374786i 0.982285 + 0.187393i \(0.0600038\pi\)
−0.982285 + 0.187393i \(0.939996\pi\)
\(930\) 0 0
\(931\) 64.6847i 2.11995i
\(932\) 5.12311i 0.167813i
\(933\) 23.1231i 0.757016i
\(934\) 20.1922i 0.660710i
\(935\) 0 0
\(936\) 3.56155 0.561553i 0.116413 0.0183549i
\(937\) 39.9309i 1.30448i 0.758011 + 0.652242i \(0.226172\pi\)
−0.758011 + 0.652242i \(0.773828\pi\)
\(938\) 18.8078 0.614095
\(939\) 32.8617 1.07240
\(940\) 0 0
\(941\) 8.73863i 0.284871i 0.989804 + 0.142436i \(0.0454934\pi\)
−0.989804 + 0.142436i \(0.954507\pi\)
\(942\) 1.68466 0.0548891
\(943\) 2.24621 0.0731467
\(944\) 5.43845i 0.177006i
\(945\) 0 0
\(946\) −7.36932 −0.239597
\(947\) 11.9309 0.387701 0.193851 0.981031i \(-0.437902\pi\)
0.193851 + 0.981031i \(0.437902\pi\)
\(948\) 14.6847i 0.476936i
\(949\) 4.00000 0.630683i 0.129845 0.0204728i
\(950\) 0 0
\(951\) 30.0000i 0.972817i
\(952\) 25.9309i 0.840424i
\(953\) 18.8078i 0.609243i −0.952473 0.304622i \(-0.901470\pi\)
0.952473 0.304622i \(-0.0985300\pi\)
\(954\) 5.00000i 0.161881i
\(955\) 0 0
\(956\) 21.9309i 0.709295i
\(957\) −2.56155 −0.0828032
\(958\) 23.8769i 0.771427i
\(959\) 11.1231 0.359184
\(960\) 0 0
\(961\) 28.9309 0.933254
\(962\) −1.56155 + 0.246211i −0.0503465 + 0.00793817i
\(963\) 12.6847i 0.408757i
\(964\) 3.75379i 0.120901i
\(965\) 0 0
\(966\) 23.3693 0.751895
\(967\) 34.6695 1.11490 0.557448 0.830212i \(-0.311781\pi\)
0.557448 + 0.830212i \(0.311781\pi\)
\(968\) 4.43845 0.142657
\(969\) 26.6307i 0.855501i
\(970\) 0 0
\(971\) −26.9309 −0.864253 −0.432126 0.901813i \(-0.642237\pi\)
−0.432126 + 0.901813i \(0.642237\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 75.2311 2.41180
\(974\) 5.68466 0.182148
\(975\) 0 0
\(976\) 11.6847 0.374017
\(977\) 6.63068 0.212134 0.106067 0.994359i \(-0.466174\pi\)
0.106067 + 0.994359i \(0.466174\pi\)
\(978\) 16.4924i 0.527370i
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 16.4384i 0.524839i
\(982\) −10.7386 −0.342684
\(983\) 37.9309 1.20981 0.604903 0.796299i \(-0.293212\pi\)
0.604903 + 0.796299i \(0.293212\pi\)
\(984\) −0.438447 −0.0139772
\(985\) 0 0
\(986\) 5.68466i 0.181036i
\(987\) 0.561553i 0.0178744i
\(988\) −2.63068 16.6847i −0.0836932 0.530810i
\(989\) 14.7386 0.468661
\(990\) 0 0
\(991\) −13.3153 −0.422976 −0.211488 0.977381i \(-0.567831\pi\)
−0.211488 + 0.977381i \(0.567831\pi\)
\(992\) 1.43845i 0.0456707i
\(993\) −8.49242 −0.269499
\(994\) 68.1080i 2.16025i
\(995\) 0 0
\(996\) 13.9309i 0.441416i
\(997\) 24.6695i 0.781291i −0.920541 0.390646i \(-0.872252\pi\)
0.920541 0.390646i \(-0.127748\pi\)
\(998\) 19.8769i 0.629192i
\(999\) 0.438447i 0.0138719i
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 1950.2.f.p.649.4 4
5.2 odd 4 1950.2.b.j.1351.4 yes 4
5.3 odd 4 1950.2.b.i.1351.1 4
5.4 even 2 1950.2.f.k.649.1 4
13.12 even 2 1950.2.f.k.649.3 4
65.12 odd 4 1950.2.b.j.1351.1 yes 4
65.38 odd 4 1950.2.b.i.1351.4 yes 4
65.64 even 2 inner 1950.2.f.p.649.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.b.i.1351.1 4 5.3 odd 4
1950.2.b.i.1351.4 yes 4 65.38 odd 4
1950.2.b.j.1351.1 yes 4 65.12 odd 4
1950.2.b.j.1351.4 yes 4 5.2 odd 4
1950.2.f.k.649.1 4 5.4 even 2
1950.2.f.k.649.3 4 13.12 even 2
1950.2.f.p.649.2 4 65.64 even 2 inner
1950.2.f.p.649.4 4 1.1 even 1 trivial